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sjplimp 2014-10-30 15:22:01 +00:00
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commit a79274aac4
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9
lib/linalg/Makefile.gfortran
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@ -1,4 +1,4 @@
# *
# -*- makefile -*-
# *_________________________________________________________________________*
# * Minimal BLAS/LAPACK Library for use by other LAMMPS packages
@ -6,12 +6,7 @@ SHELL = /bin/sh
# ------ FILES ------
SRC = dasum.f daxpy.f dcopy.f ddot.f dgecon.f dgemm.f dgemv.f dger.f \
dgetf2.f dgetrf.f dgetri.f disnan.f dlabad.f dlaisnan.f dlamch.f\
dlacn2.f dlange.f dlassq.f dlaswp.f dlatrs.f drscl.f dscal.f \
dswap.f dtrmm.f dtrmv.f dtrsm.f dtrsv.f dtrti2.f dtrtri.f \
idamax.f ieeeck.f ilaenv.f iparmq.f lsame.f xerbla.f zdotc.f \
zdscal.f zhpr.f zpptrf.f zpptri.f zscal.f ztpmv.f ztpsv.f ztptri.f
SRC = $(wildcard *.f)
FILES = $(SRC) Makefile.* README

24
lib/linalg/Makefile.mingw32-cross
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@ -1,30 +1,12 @@
# *
# -*- makefile -*-
# *_________________________________________________________________________*
# * Minimal BLAS/LAPACK Library for ATC and AWPMD
# To compile and link LAMMPS to the linalg library generated by this Makefile,
# adjust the corresponding definitions in the library's Makefile.lammps file.
#
# for lib/atc/Makefile.lammps use:
# user-atc_SYSINC =
# user-atc_SYSLIB = ../../lib/linalg/$(LIBOBJDIR)liblinalg.a -lgfortran
# user-atc_SYSPATH =
#
# for lib/awpmd/Makefile.lammps use:
# user-awpmd_SYSINC =
# user-awpmd_SYSLIB = ../../lib/linalg/$(LIBOBJDIR)liblinalg.a -lgfortran
# user-awpmd_SYSPATH =
# * Minimal BLAS/LAPACK Library for use by other LAMMPS packages
SHELL = /bin/sh
# ------ FILES ------
SRC = dasum.f daxpy.f dcopy.f ddot.f dgecon.f dgemm.f dgemv.f dger.f \
dgetf2.f dgetrf.f dgetri.f disnan.f dlabad.f dlaisnan.f dlamch.f\
dlacn2.f dlange.f dlassq.f dlaswp.f dlatrs.f drscl.f dscal.f \
dswap.f dtrmm.f dtrmv.f dtrsm.f dtrsv.f dtrti2.f dtrtri.f \
idamax.f ieeeck.f ilaenv.f iparmq.f lsame.f xerbla.f zdotc.f \
zdscal.f zhpr.f zpptrf.f zpptri.f zscal.f ztpmv.f ztpsv.f ztptri.f
SRC = $(wildcard *.f)
FILES = $(SRC) Makefile.* README

26
lib/linalg/Makefile.mingw64-cross
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@ -1,30 +1,12 @@
# *
# -*- makefile -*-
# *_________________________________________________________________________*
# * Minimal BLAS/LAPACK Library for ATC and AWPMD
# To compile and link LAMMPS to the linalg library generated by this Makefile,
# adjust the corresponding definitions in the library's Makefile.lammps file.
#
# for lib/atc/Makefile.lammps use:
# user-atc_SYSINC =
# user-atc_SYSLIB = ../../lib/linalg/$(LIBOBJDIR)liblinalg.a -lgfortran
# user-atc_SYSPATH =
#
# for lib/awpmd/Makefile.lammps use:
# user-awpmd_SYSINC =
# user-awpmd_SYSLIB = ../../lib/linalg/$(LIBOBJDIR)liblinalg.a -lgfortran
# user-awpmd_SYSPATH =
# * Minimal BLAS/LAPACK Library for use by other LAMMPS packages
SHELL = /bin/sh
# ------ FILES ------
SRC = dasum.f daxpy.f dcopy.f ddot.f dgecon.f dgemm.f dgemv.f dger.f \
dgetf2.f dgetrf.f dgetri.f disnan.f dlabad.f dlaisnan.f dlamch.f\
dlacn2.f dlange.f dlassq.f dlaswp.f dlatrs.f drscl.f dscal.f \
dswap.f dtrmm.f dtrmv.f dtrsm.f dtrsv.f dtrti2.f dtrtri.f \
idamax.f ieeeck.f ilaenv.f iparmq.f lsame.f xerbla.f zdotc.f \
zdscal.f zhpr.f zpptrf.f zpptri.f zscal.f ztpmv.f ztpsv.f ztptri.f
SRC = $(wildcard *.f)
FILES = $(SRC) Makefile.* README
@ -77,7 +59,7 @@ $(DIR)dlamch.o: dlamch.f
# ------ CLEAN ------
clean:
-rm $(DIR)*.o $(DOR)*.mod *~ $(LIB)
-rm $(DIR)*.o $(DIR)*.mod *~ $(LIB)
-rmdir $(DIR)
tar:

850
lib/linalg/dbdsqr.f Normal file
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@ -0,0 +1,850 @@
*> \brief \b DBDSQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBDSQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbdsqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbdsqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbdsqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
* LDU, C, LDC, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DBDSQR computes the singular values and, optionally, the right and/or
*> left singular vectors from the singular value decomposition (SVD) of
*> a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
*> zero-shift QR algorithm. The SVD of B has the form
*>
*> B = Q * S * P**T
*>
*> where S is the diagonal matrix of singular values, Q is an orthogonal
*> matrix of left singular vectors, and P is an orthogonal matrix of
*> right singular vectors. If left singular vectors are requested, this
*> subroutine actually returns U*Q instead of Q, and, if right singular
*> vectors are requested, this subroutine returns P**T*VT instead of
*> P**T, for given real input matrices U and VT. When U and VT are the
*> orthogonal matrices that reduce a general matrix A to bidiagonal
*> form: A = U*B*VT, as computed by DGEBRD, then
*>
*> A = (U*Q) * S * (P**T*VT)
*>
*> is the SVD of A. Optionally, the subroutine may also compute Q**T*C
*> for a given real input matrix C.
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices With
*> Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
*> LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
*> no. 5, pp. 873-912, Sept 1990) and
*> "Accurate singular values and differential qd algorithms," by
*> B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
*> Department, University of California at Berkeley, July 1992
*> for a detailed description of the algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': B is upper bidiagonal;
*> = 'L': B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix B. N >= 0.
*> \endverbatim
*>
*> \param[in] NCVT
*> \verbatim
*> NCVT is INTEGER
*> The number of columns of the matrix VT. NCVT >= 0.
*> \endverbatim
*>
*> \param[in] NRU
*> \verbatim
*> NRU is INTEGER
*> The number of rows of the matrix U. NRU >= 0.
*> \endverbatim
*>
*> \param[in] NCC
*> \verbatim
*> NCC is INTEGER
*> The number of columns of the matrix C. NCC >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the bidiagonal matrix B.
*> On exit, if INFO=0, the singular values of B in decreasing
*> order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the N-1 offdiagonal elements of the bidiagonal
*> matrix B.
*> On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
*> will contain the diagonal and superdiagonal elements of a
*> bidiagonal matrix orthogonally equivalent to the one given
*> as input.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
*> On entry, an N-by-NCVT matrix VT.
*> On exit, VT is overwritten by P**T * VT.
*> Not referenced if NCVT = 0.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT.
*> LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU, N)
*> On entry, an NRU-by-N matrix U.
*> On exit, U is overwritten by U * Q.
*> Not referenced if NRU = 0.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,NRU).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC, NCC)
*> On entry, an N-by-NCC matrix C.
*> On exit, C is overwritten by Q**T * C.
*> Not referenced if NCC = 0.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C.
*> LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: If INFO = -i, the i-th argument had an illegal value
*> > 0:
*> if NCVT = NRU = NCC = 0,
*> = 1, a split was marked by a positive value in E
*> = 2, current block of Z not diagonalized after 30*N
*> iterations (in inner while loop)
*> = 3, termination criterion of outer while loop not met
*> (program created more than N unreduced blocks)
*> else NCVT = NRU = NCC = 0,
*> the algorithm did not converge; D and E contain the
*> elements of a bidiagonal matrix which is orthogonally
*> similar to the input matrix B; if INFO = i, i
*> elements of E have not converged to zero.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> TOLMUL DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
*> TOLMUL controls the convergence criterion of the QR loop.
*> If it is positive, TOLMUL*EPS is the desired relative
*> precision in the computed singular values.
*> If it is negative, abs(TOLMUL*EPS*sigma_max) is the
*> desired absolute accuracy in the computed singular
*> values (corresponds to relative accuracy
*> abs(TOLMUL*EPS) in the largest singular value.
*> abs(TOLMUL) should be between 1 and 1/EPS, and preferably
*> between 10 (for fast convergence) and .1/EPS
*> (for there to be some accuracy in the results).
*> Default is to lose at either one eighth or 2 of the
*> available decimal digits in each computed singular value
*> (whichever is smaller).
*>
*> MAXITR INTEGER, default = 6
*> MAXITR controls the maximum number of passes of the
*> algorithm through its inner loop. The algorithms stops
*> (and so fails to converge) if the number of passes
*> through the inner loop exceeds MAXITR*N**2.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
$ LDU, C, LDC, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION NEGONE
PARAMETER ( NEGONE = -1.0D0 )
DOUBLE PRECISION HNDRTH
PARAMETER ( HNDRTH = 0.01D0 )
DOUBLE PRECISION TEN
PARAMETER ( TEN = 10.0D0 )
DOUBLE PRECISION HNDRD
PARAMETER ( HNDRD = 100.0D0 )
DOUBLE PRECISION MEIGTH
PARAMETER ( MEIGTH = -0.125D0 )
INTEGER MAXITR
PARAMETER ( MAXITR = 6 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, ROTATE
INTEGER I, IDIR, ISUB, ITER, J, LL, LLL, M, MAXIT, NM1,
$ NM12, NM13, OLDLL, OLDM
DOUBLE PRECISION ABSE, ABSS, COSL, COSR, CS, EPS, F, G, H, MU,
$ OLDCS, OLDSN, R, SHIFT, SIGMN, SIGMX, SINL,
$ SINR, SLL, SMAX, SMIN, SMINL, SMINOA,
$ SN, THRESH, TOL, TOLMUL, UNFL
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLARTG, DLAS2, DLASQ1, DLASR, DLASV2, DROT,
$ DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LOWER ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NCVT.LT.0 ) THEN
INFO = -3
ELSE IF( NRU.LT.0 ) THEN
INFO = -4
ELSE IF( NCC.LT.0 ) THEN
INFO = -5
ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR.
$ ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN
INFO = -9
ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN
INFO = -11
ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR.
$ ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DBDSQR', -INFO )
RETURN
END IF
IF( N.EQ.0 )
$ RETURN
IF( N.EQ.1 )
$ GO TO 160
*
* ROTATE is true if any singular vectors desired, false otherwise
*
ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 )
*
* If no singular vectors desired, use qd algorithm
*
IF( .NOT.ROTATE ) THEN
CALL DLASQ1( N, D, E, WORK, INFO )
*
* If INFO equals 2, dqds didn't finish, try to finish
*
IF( INFO .NE. 2 ) RETURN
INFO = 0
END IF
*
NM1 = N - 1
NM12 = NM1 + NM1
NM13 = NM12 + NM1
IDIR = 0
*
* Get machine constants
*
EPS = DLAMCH( 'Epsilon' )
UNFL = DLAMCH( 'Safe minimum' )
*
* If matrix lower bidiagonal, rotate to be upper bidiagonal
* by applying Givens rotations on the left
*
IF( LOWER ) THEN
DO 10 I = 1, N - 1
CALL DLARTG( D( I ), E( I ), CS, SN, R )
D( I ) = R
E( I ) = SN*D( I+1 )
D( I+1 ) = CS*D( I+1 )
WORK( I ) = CS
WORK( NM1+I ) = SN
10 CONTINUE
*
* Update singular vectors if desired
*
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ), WORK( N ), U,
$ LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ), WORK( N ), C,
$ LDC )
END IF
*
* Compute singular values to relative accuracy TOL
* (By setting TOL to be negative, algorithm will compute
* singular values to absolute accuracy ABS(TOL)*norm(input matrix))
*
TOLMUL = MAX( TEN, MIN( HNDRD, EPS**MEIGTH ) )
TOL = TOLMUL*EPS
*
* Compute approximate maximum, minimum singular values
*
SMAX = ZERO
DO 20 I = 1, N
SMAX = MAX( SMAX, ABS( D( I ) ) )
20 CONTINUE
DO 30 I = 1, N - 1
SMAX = MAX( SMAX, ABS( E( I ) ) )
30 CONTINUE
SMINL = ZERO
IF( TOL.GE.ZERO ) THEN
*
* Relative accuracy desired
*
SMINOA = ABS( D( 1 ) )
IF( SMINOA.EQ.ZERO )
$ GO TO 50
MU = SMINOA
DO 40 I = 2, N
MU = ABS( D( I ) )*( MU / ( MU+ABS( E( I-1 ) ) ) )
SMINOA = MIN( SMINOA, MU )
IF( SMINOA.EQ.ZERO )
$ GO TO 50
40 CONTINUE
50 CONTINUE
SMINOA = SMINOA / SQRT( DBLE( N ) )
THRESH = MAX( TOL*SMINOA, MAXITR*N*N*UNFL )
ELSE
*
* Absolute accuracy desired
*
THRESH = MAX( ABS( TOL )*SMAX, MAXITR*N*N*UNFL )
END IF
*
* Prepare for main iteration loop for the singular values
* (MAXIT is the maximum number of passes through the inner
* loop permitted before nonconvergence signalled.)
*
MAXIT = MAXITR*N*N
ITER = 0
OLDLL = -1
OLDM = -1
*
* M points to last element of unconverged part of matrix
*
M = N
*
* Begin main iteration loop
*
60 CONTINUE
*
* Check for convergence or exceeding iteration count
*
IF( M.LE.1 )
$ GO TO 160
IF( ITER.GT.MAXIT )
$ GO TO 200
*
* Find diagonal block of matrix to work on
*
IF( TOL.LT.ZERO .AND. ABS( D( M ) ).LE.THRESH )
$ D( M ) = ZERO
SMAX = ABS( D( M ) )
SMIN = SMAX
DO 70 LLL = 1, M - 1
LL = M - LLL
ABSS = ABS( D( LL ) )
ABSE = ABS( E( LL ) )
IF( TOL.LT.ZERO .AND. ABSS.LE.THRESH )
$ D( LL ) = ZERO
IF( ABSE.LE.THRESH )
$ GO TO 80
SMIN = MIN( SMIN, ABSS )
SMAX = MAX( SMAX, ABSS, ABSE )
70 CONTINUE
LL = 0
GO TO 90
80 CONTINUE
E( LL ) = ZERO
*
* Matrix splits since E(LL) = 0
*
IF( LL.EQ.M-1 ) THEN
*
* Convergence of bottom singular value, return to top of loop
*
M = M - 1
GO TO 60
END IF
90 CONTINUE
LL = LL + 1
*
* E(LL) through E(M-1) are nonzero, E(LL-1) is zero
*
IF( LL.EQ.M-1 ) THEN
*
* 2 by 2 block, handle separately
*
CALL DLASV2( D( M-1 ), E( M-1 ), D( M ), SIGMN, SIGMX, SINR,
$ COSR, SINL, COSL )
D( M-1 ) = SIGMX
E( M-1 ) = ZERO
D( M ) = SIGMN
*
* Compute singular vectors, if desired
*
IF( NCVT.GT.0 )
$ CALL DROT( NCVT, VT( M-1, 1 ), LDVT, VT( M, 1 ), LDVT, COSR,
$ SINR )
IF( NRU.GT.0 )
$ CALL DROT( NRU, U( 1, M-1 ), 1, U( 1, M ), 1, COSL, SINL )
IF( NCC.GT.0 )
$ CALL DROT( NCC, C( M-1, 1 ), LDC, C( M, 1 ), LDC, COSL,
$ SINL )
M = M - 2
GO TO 60
END IF
*
* If working on new submatrix, choose shift direction
* (from larger end diagonal element towards smaller)
*
IF( LL.GT.OLDM .OR. M.LT.OLDLL ) THEN
IF( ABS( D( LL ) ).GE.ABS( D( M ) ) ) THEN
*
* Chase bulge from top (big end) to bottom (small end)
*
IDIR = 1
ELSE
*
* Chase bulge from bottom (big end) to top (small end)
*
IDIR = 2
END IF
END IF
*
* Apply convergence tests
*
IF( IDIR.EQ.1 ) THEN
*
* Run convergence test in forward direction
* First apply standard test to bottom of matrix
*
IF( ABS( E( M-1 ) ).LE.ABS( TOL )*ABS( D( M ) ) .OR.
$ ( TOL.LT.ZERO .AND. ABS( E( M-1 ) ).LE.THRESH ) ) THEN
E( M-1 ) = ZERO
GO TO 60
END IF
*
IF( TOL.GE.ZERO ) THEN
*
* If relative accuracy desired,
* apply convergence criterion forward
*
MU = ABS( D( LL ) )
SMINL = MU
DO 100 LLL = LL, M - 1
IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
E( LLL ) = ZERO
GO TO 60
END IF
MU = ABS( D( LLL+1 ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
SMINL = MIN( SMINL, MU )
100 CONTINUE
END IF
*
ELSE
*
* Run convergence test in backward direction
* First apply standard test to top of matrix
*
IF( ABS( E( LL ) ).LE.ABS( TOL )*ABS( D( LL ) ) .OR.
$ ( TOL.LT.ZERO .AND. ABS( E( LL ) ).LE.THRESH ) ) THEN
E( LL ) = ZERO
GO TO 60
END IF
*
IF( TOL.GE.ZERO ) THEN
*
* If relative accuracy desired,
* apply convergence criterion backward
*
MU = ABS( D( M ) )
SMINL = MU
DO 110 LLL = M - 1, LL, -1
IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
E( LLL ) = ZERO
GO TO 60
END IF
MU = ABS( D( LLL ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
SMINL = MIN( SMINL, MU )
110 CONTINUE
END IF
END IF
OLDLL = LL
OLDM = M
*
* Compute shift. First, test if shifting would ruin relative
* accuracy, and if so set the shift to zero.
*
IF( TOL.GE.ZERO .AND. N*TOL*( SMINL / SMAX ).LE.
$ MAX( EPS, HNDRTH*TOL ) ) THEN
*
* Use a zero shift to avoid loss of relative accuracy
*
SHIFT = ZERO
ELSE
*
* Compute the shift from 2-by-2 block at end of matrix
*
IF( IDIR.EQ.1 ) THEN
SLL = ABS( D( LL ) )
CALL DLAS2( D( M-1 ), E( M-1 ), D( M ), SHIFT, R )
ELSE
SLL = ABS( D( M ) )
CALL DLAS2( D( LL ), E( LL ), D( LL+1 ), SHIFT, R )
END IF
*
* Test if shift negligible, and if so set to zero
*
IF( SLL.GT.ZERO ) THEN
IF( ( SHIFT / SLL )**2.LT.EPS )
$ SHIFT = ZERO
END IF
END IF
*
* Increment iteration count
*
ITER = ITER + M - LL
*
* If SHIFT = 0, do simplified QR iteration
*
IF( SHIFT.EQ.ZERO ) THEN
IF( IDIR.EQ.1 ) THEN
*
* Chase bulge from top to bottom
* Save cosines and sines for later singular vector updates
*
CS = ONE
OLDCS = ONE
DO 120 I = LL, M - 1
CALL DLARTG( D( I )*CS, E( I ), CS, SN, R )
IF( I.GT.LL )
$ E( I-1 ) = OLDSN*R
CALL DLARTG( OLDCS*R, D( I+1 )*SN, OLDCS, OLDSN, D( I ) )
WORK( I-LL+1 ) = CS
WORK( I-LL+1+NM1 ) = SN
WORK( I-LL+1+NM12 ) = OLDCS
WORK( I-LL+1+NM13 ) = OLDSN
120 CONTINUE
H = D( M )*CS
D( M ) = H*OLDCS
E( M-1 ) = H*OLDSN
*
* Update singular vectors
*
IF( NCVT.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ),
$ WORK( N ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1, WORK( NM12+1 ),
$ WORK( NM13+1 ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC, WORK( NM12+1 ),
$ WORK( NM13+1 ), C( LL, 1 ), LDC )
*
* Test convergence
*
IF( ABS( E( M-1 ) ).LE.THRESH )
$ E( M-1 ) = ZERO
*
ELSE
*
* Chase bulge from bottom to top
* Save cosines and sines for later singular vector updates
*
CS = ONE
OLDCS = ONE
DO 130 I = M, LL + 1, -1
CALL DLARTG( D( I )*CS, E( I-1 ), CS, SN, R )
IF( I.LT.M )
$ E( I ) = OLDSN*R
CALL DLARTG( OLDCS*R, D( I-1 )*SN, OLDCS, OLDSN, D( I ) )
WORK( I-LL ) = CS
WORK( I-LL+NM1 ) = -SN
WORK( I-LL+NM12 ) = OLDCS
WORK( I-LL+NM13 ) = -OLDSN
130 CONTINUE
H = D( LL )*CS
D( LL ) = H*OLDCS
E( LL ) = H*OLDSN
*
* Update singular vectors
*
IF( NCVT.GT.0 )
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT, WORK( NM12+1 ),
$ WORK( NM13+1 ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ),
$ WORK( N ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ),
$ WORK( N ), C( LL, 1 ), LDC )
*
* Test convergence
*
IF( ABS( E( LL ) ).LE.THRESH )
$ E( LL ) = ZERO
END IF
ELSE
*
* Use nonzero shift
*
IF( IDIR.EQ.1 ) THEN
*
* Chase bulge from top to bottom
* Save cosines and sines for later singular vector updates
*
F = ( ABS( D( LL ) )-SHIFT )*
$ ( SIGN( ONE, D( LL ) )+SHIFT / D( LL ) )
G = E( LL )
DO 140 I = LL, M - 1
CALL DLARTG( F, G, COSR, SINR, R )
IF( I.GT.LL )
$ E( I-1 ) = R
F = COSR*D( I ) + SINR*E( I )
E( I ) = COSR*E( I ) - SINR*D( I )
G = SINR*D( I+1 )
D( I+1 ) = COSR*D( I+1 )
CALL DLARTG( F, G, COSL, SINL, R )
D( I ) = R
F = COSL*E( I ) + SINL*D( I+1 )
D( I+1 ) = COSL*D( I+1 ) - SINL*E( I )
IF( I.LT.M-1 ) THEN
G = SINL*E( I+1 )
E( I+1 ) = COSL*E( I+1 )
END IF
WORK( I-LL+1 ) = COSR
WORK( I-LL+1+NM1 ) = SINR
WORK( I-LL+1+NM12 ) = COSL
WORK( I-LL+1+NM13 ) = SINL
140 CONTINUE
E( M-1 ) = F
*
* Update singular vectors
*
IF( NCVT.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ),
$ WORK( N ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1, WORK( NM12+1 ),
$ WORK( NM13+1 ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC, WORK( NM12+1 ),
$ WORK( NM13+1 ), C( LL, 1 ), LDC )
*
* Test convergence
*
IF( ABS( E( M-1 ) ).LE.THRESH )
$ E( M-1 ) = ZERO
*
ELSE
*
* Chase bulge from bottom to top
* Save cosines and sines for later singular vector updates
*
F = ( ABS( D( M ) )-SHIFT )*( SIGN( ONE, D( M ) )+SHIFT /
$ D( M ) )
G = E( M-1 )
DO 150 I = M, LL + 1, -1
CALL DLARTG( F, G, COSR, SINR, R )
IF( I.LT.M )
$ E( I ) = R
F = COSR*D( I ) + SINR*E( I-1 )
E( I-1 ) = COSR*E( I-1 ) - SINR*D( I )
G = SINR*D( I-1 )
D( I-1 ) = COSR*D( I-1 )
CALL DLARTG( F, G, COSL, SINL, R )
D( I ) = R
F = COSL*E( I-1 ) + SINL*D( I-1 )
D( I-1 ) = COSL*D( I-1 ) - SINL*E( I-1 )
IF( I.GT.LL+1 ) THEN
G = SINL*E( I-2 )
E( I-2 ) = COSL*E( I-2 )
END IF
WORK( I-LL ) = COSR
WORK( I-LL+NM1 ) = -SINR
WORK( I-LL+NM12 ) = COSL
WORK( I-LL+NM13 ) = -SINL
150 CONTINUE
E( LL ) = F
*
* Test convergence
*
IF( ABS( E( LL ) ).LE.THRESH )
$ E( LL ) = ZERO
*
* Update singular vectors if desired
*
IF( NCVT.GT.0 )
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT, WORK( NM12+1 ),
$ WORK( NM13+1 ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ),
$ WORK( N ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ),
$ WORK( N ), C( LL, 1 ), LDC )
END IF
END IF
*
* QR iteration finished, go back and check convergence
*
GO TO 60
*
* All singular values converged, so make them positive
*
160 CONTINUE
DO 170 I = 1, N
IF( D( I ).LT.ZERO ) THEN
D( I ) = -D( I )
*
* Change sign of singular vectors, if desired
*
IF( NCVT.GT.0 )
$ CALL DSCAL( NCVT, NEGONE, VT( I, 1 ), LDVT )
END IF
170 CONTINUE
*
* Sort the singular values into decreasing order (insertion sort on
* singular values, but only one transposition per singular vector)
*
DO 190 I = 1, N - 1
*
* Scan for smallest D(I)
*
ISUB = 1
SMIN = D( 1 )
DO 180 J = 2, N + 1 - I
IF( D( J ).LE.SMIN ) THEN
ISUB = J
SMIN = D( J )
END IF
180 CONTINUE
IF( ISUB.NE.N+1-I ) THEN
*
* Swap singular values and vectors
*
D( ISUB ) = D( N+1-I )
D( N+1-I ) = SMIN
IF( NCVT.GT.0 )
$ CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( N+1-I, 1 ),
$ LDVT )
IF( NRU.GT.0 )
$ CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, N+1-I ), 1 )
IF( NCC.GT.0 )
$ CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( N+1-I, 1 ), LDC )
END IF
190 CONTINUE
GO TO 220
*
* Maximum number of iterations exceeded, failure to converge
*
200 CONTINUE
INFO = 0
DO 210 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
210 CONTINUE
220 CONTINUE
RETURN
*
* End of DBDSQR
*
END

320
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*> \brief \b DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
* $ TAUQ( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBD2 reduces a real general m by n matrix A to upper or lower
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows in the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns in the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n general matrix to be reduced.
*> On exit,
*> if m >= n, the diagonal and the first superdiagonal are
*> overwritten with the upper bidiagonal matrix B; the
*> elements below the diagonal, with the array TAUQ, represent
*> the orthogonal matrix Q as a product of elementary
*> reflectors, and the elements above the first superdiagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors;
*> if m < n, the diagonal and the first subdiagonal are
*> overwritten with the lower bidiagonal matrix B; the
*> elements below the first subdiagonal, with the array TAUQ,
*> represent the orthogonal matrix Q as a product of
*> elementary reflectors, and the elements above the diagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (min(M,N))
*> The diagonal elements of the bidiagonal matrix B:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
*> The off-diagonal elements of the bidiagonal matrix B:
*> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
*> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*> TAUQ is DOUBLE PRECISION array dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*> TAUP is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (max(M,N))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrices Q and P are represented as products of elementary
*> reflectors:
*>
*> If m >= n,
*>
*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
*> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> If m < n,
*>
*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
*> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> The contents of A on exit are illustrated by the following examples:
*>
*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*>
*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
*> ( v1 v2 v3 v4 v5 )
*>
*> where d and e denote diagonal and off-diagonal elements of B, vi
*> denotes an element of the vector defining H(i), and ui an element of
*> the vector defining G(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
$ TAUQ( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.LT.0 ) THEN
CALL XERBLA( 'DGEBD2', -INFO )
RETURN
END IF
*
IF( M.GE.N ) THEN
*
* Reduce to upper bidiagonal form
*
DO 10 I = 1, N
*
* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ TAUQ( I ) )
D( I ) = A( I, I )
A( I, I ) = ONE
*
* Apply H(i) to A(i:m,i+1:n) from the left
*
IF( I.LT.N )
$ CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ),
$ A( I, I+1 ), LDA, WORK )
A( I, I ) = D( I )
*
IF( I.LT.N ) THEN
*
* Generate elementary reflector G(i) to annihilate
* A(i,i+2:n)
*
CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
$ LDA, TAUP( I ) )
E( I ) = A( I, I+1 )
A( I, I+1 ) = ONE
*
* Apply G(i) to A(i+1:m,i+1:n) from the right
*
CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
$ TAUP( I ), A( I+1, I+1 ), LDA, WORK )
A( I, I+1 ) = E( I )
ELSE
TAUP( I ) = ZERO
END IF
10 CONTINUE
ELSE
*
* Reduce to lower bidiagonal form
*
DO 20 I = 1, M
*
* Generate elementary reflector G(i) to annihilate A(i,i+1:n)
*
CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
$ TAUP( I ) )
D( I ) = A( I, I )
A( I, I ) = ONE
*
* Apply G(i) to A(i+1:m,i:n) from the right
*
IF( I.LT.M )
$ CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
$ TAUP( I ), A( I+1, I ), LDA, WORK )
A( I, I ) = D( I )
*
IF( I.LT.M ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(i+2:m,i)
*
CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
$ TAUQ( I ) )
E( I ) = A( I+1, I )
A( I+1, I ) = ONE
*
* Apply H(i) to A(i+1:m,i+1:n) from the left
*
CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ),
$ A( I+1, I+1 ), LDA, WORK )
A( I+1, I ) = E( I )
ELSE
TAUQ( I ) = ZERO
END IF
20 CONTINUE
END IF
RETURN
*
* End of DGEBD2
*
END

353
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*> \brief \b DGEBRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEBRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
* $ TAUQ( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBRD reduces a general real M-by-N matrix A to upper or lower
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows in the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns in the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N general matrix to be reduced.
*> On exit,
*> if m >= n, the diagonal and the first superdiagonal are
*> overwritten with the upper bidiagonal matrix B; the
*> elements below the diagonal, with the array TAUQ, represent
*> the orthogonal matrix Q as a product of elementary
*> reflectors, and the elements above the first superdiagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors;
*> if m < n, the diagonal and the first subdiagonal are
*> overwritten with the lower bidiagonal matrix B; the
*> elements below the first subdiagonal, with the array TAUQ,
*> represent the orthogonal matrix Q as a product of
*> elementary reflectors, and the elements above the diagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (min(M,N))
*> The diagonal elements of the bidiagonal matrix B:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
*> The off-diagonal elements of the bidiagonal matrix B:
*> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
*> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*> TAUQ is DOUBLE PRECISION array dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*> TAUP is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK. LWORK >= max(1,M,N).
*> For optimum performance LWORK >= (M+N)*NB, where NB
*> is the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrices Q and P are represented as products of elementary
*> reflectors:
*>
*> If m >= n,
*>
*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
*> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> If m < n,
*>
*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
*> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> The contents of A on exit are illustrated by the following examples:
*>
*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*>
*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
*> ( v1 v2 v3 v4 v5 )
*>
*> where d and e denote diagonal and off-diagonal elements of B, vi
*> denotes an element of the vector defining H(i), and ui an element of
*> the vector defining G(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
$ INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
$ TAUQ( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
$ NBMIN, NX
DOUBLE PRECISION WS
* ..
* .. External Subroutines ..
EXTERNAL DGEBD2, DGEMM, DLABRD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
LWKOPT = ( M+N )*NB
WORK( 1 ) = DBLE( LWKOPT )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
IF( INFO.LT.0 ) THEN
CALL XERBLA( 'DGEBRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
MINMN = MIN( M, N )
IF( MINMN.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
WS = MAX( M, N )
LDWRKX = M
LDWRKY = N
*
IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN
*
* Set the crossover point NX.
*
NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )
*
* Determine when to switch from blocked to unblocked code.
*
IF( NX.LT.MINMN ) THEN
WS = ( M+N )*NB
IF( LWORK.LT.WS ) THEN
*
* Not enough work space for the optimal NB, consider using
* a smaller block size.
*
NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )
IF( LWORK.GE.( M+N )*NBMIN ) THEN
NB = LWORK / ( M+N )
ELSE
NB = 1
NX = MINMN
END IF
END IF
END IF
ELSE
NX = MINMN
END IF
*
DO 30 I = 1, MINMN - NX, NB
*
* Reduce rows and columns i:i+nb-1 to bidiagonal form and return
* the matrices X and Y which are needed to update the unreduced
* part of the matrix
*
CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ),
$ TAUQ( I ), TAUP( I ), WORK, LDWRKX,
$ WORK( LDWRKX*NB+1 ), LDWRKY )
*
* Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
* of the form A := A - V*Y**T - X*U**T
*
CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,
$ NB, -ONE, A( I+NB, I ), LDA,
$ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,
$ A( I+NB, I+NB ), LDA )
CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1,
$ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,
$ ONE, A( I+NB, I+NB ), LDA )
*
* Copy diagonal and off-diagonal elements of B back into A
*
IF( M.GE.N ) THEN
DO 10 J = I, I + NB - 1
A( J, J ) = D( J )
A( J, J+1 ) = E( J )
10 CONTINUE
ELSE
DO 20 J = I, I + NB - 1
A( J, J ) = D( J )
A( J+1, J ) = E( J )
20 CONTINUE
END IF
30 CONTINUE
*
* Use unblocked code to reduce the remainder of the matrix
*
CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),
$ TAUQ( I ), TAUP( I ), WORK, IINFO )
WORK( 1 ) = WS
RETURN
*
* End of DGEBRD
*
END

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*> \brief \b DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELQ2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelq2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelq2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelq2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELQ2( M, N, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGELQ2 computes an LQ factorization of a real m by n matrix A:
*> A = L * Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n matrix A.
*> On exit, the elements on and below the diagonal of the array
*> contain the m by min(m,n) lower trapezoidal matrix L (L is
*> lower triangular if m <= n); the elements above the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
*> and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGELQ2( M, N, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K
DOUBLE PRECISION AII
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELQ2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO 10 I = 1, K
*
* Generate elementary reflector H(i) to annihilate A(i,i+1:n)
*
CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
$ TAU( I ) )
IF( I.LT.M ) THEN
*
* Apply H(i) to A(i+1:m,i:n) from the right
*
AII = A( I, I )
A( I, I ) = ONE
CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, TAU( I ),
$ A( I+1, I ), LDA, WORK )
A( I, I ) = AII
END IF
10 CONTINUE
RETURN
*
* End of DGELQ2
*
END

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*> \brief \b DGELQF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGELQF computes an LQ factorization of a real M-by-N matrix A:
*> A = L * Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the elements on and below the diagonal of the array
*> contain the m-by-min(m,n) lower trapezoidal matrix L (L is
*> lower triangular if m <= n); the elements above the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,M).
*> For optimum performance LWORK >= M*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
*> and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DGELQ2, DLARFB, DLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
LWKOPT = M*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELQF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
K = MIN( M, N )
IF( K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = M
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DGELQF', ' ', M, N, -1, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = M
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGELQF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code initially
*
DO 10 I = 1, K - NX, NB
IB = MIN( K-I+1, NB )
*
* Compute the LQ factorization of the current block
* A(i:i+ib-1,i:n)
*
CALL DGELQ2( IB, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
IF( I+IB.LE.M ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I, I ),
$ LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(i+ib:m,i:n) from the right
*
CALL DLARFB( 'Right', 'No transpose', 'Forward',
$ 'Rowwise', M-I-IB+1, N-I+1, IB, A( I, I ),
$ LDA, WORK, LDWORK, A( I+IB, I ), LDA,
$ WORK( IB+1 ), LDWORK )
END IF
10 CONTINUE
ELSE
I = 1
END IF
*
* Use unblocked code to factor the last or only block.
*
IF( I.LE.K )
$ CALL DGELQ2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
*
WORK( 1 ) = IWS
RETURN
*
* End of DGELQF
*
END

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*> \brief \b DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQR2( M, N, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQR2 computes a QR factorization of a real m by n matrix A:
*> A = Q * R.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(m,n) by n upper trapezoidal matrix R (R is
*> upper triangular if m >= n); the elements below the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*> and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQR2( M, N, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K
DOUBLE PRECISION AII
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQR2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO 10 I = 1, K
*
* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ TAU( I ) )
IF( I.LT.N ) THEN
*
* Apply H(i) to A(i:m,i+1:n) from the left
*
AII = A( I, I )
A( I, I ) = ONE
CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
$ A( I, I+1 ), LDA, WORK )
A( I, I ) = AII
END IF
10 CONTINUE
RETURN
*
* End of DGEQR2
*
END

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*> \brief \b DGEQRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEQRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQRF computes a QR factorization of a real M-by-N matrix A:
*> A = Q * R.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(M,N)-by-N upper trapezoidal matrix R (R is
*> upper triangular if m >= n); the elements below the diagonal,
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of min(m,n) elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*> For optimum performance LWORK >= N*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*> and tau in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DGEQR2, DLARFB, DLARFT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
LWKOPT = N*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQRF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
K = MIN( M, N )
IF( K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DGEQRF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code initially
*
DO 10 I = 1, K - NX, NB
IB = MIN( K-I+1, NB )
*
* Compute the QR factorization of the current block
* A(i:m,i:i+ib-1)
*
CALL DGEQR2( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
IF( I+IB.LE.N ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
$ A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H**T to A(i:m,i+ib:n) from the left
*
CALL DLARFB( 'Left', 'Transpose', 'Forward',
$ 'Columnwise', M-I+1, N-I-IB+1, IB,
$ A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
$ LDA, WORK( IB+1 ), LDWORK )
END IF
10 CONTINUE
ELSE
I = 1
END IF
*
* Use unblocked code to factor the last or only block.
*
IF( I.LE.K )
$ CALL DGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
*
WORK( 1 ) = IWS
RETURN
*
* End of DGEQRF
*
END

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*> \brief \b DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLABRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlabrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlabrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlabrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
* LDY )
*
* .. Scalar Arguments ..
* INTEGER LDA, LDX, LDY, M, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
* $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLABRD reduces the first NB rows and columns of a real general
*> m by n matrix A to upper or lower bidiagonal form by an orthogonal
*> transformation Q**T * A * P, and returns the matrices X and Y which
*> are needed to apply the transformation to the unreduced part of A.
*>
*> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
*> bidiagonal form.
*>
*> This is an auxiliary routine called by DGEBRD
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows in the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns in the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of leading rows and columns of A to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the m by n general matrix to be reduced.
*> On exit, the first NB rows and columns of the matrix are
*> overwritten; the rest of the array is unchanged.
*> If m >= n, elements on and below the diagonal in the first NB
*> columns, with the array TAUQ, represent the orthogonal
*> matrix Q as a product of elementary reflectors; and
*> elements above the diagonal in the first NB rows, with the
*> array TAUP, represent the orthogonal matrix P as a product
*> of elementary reflectors.
*> If m < n, elements below the diagonal in the first NB
*> columns, with the array TAUQ, represent the orthogonal
*> matrix Q as a product of elementary reflectors, and
*> elements on and above the diagonal in the first NB rows,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (NB)
*> The diagonal elements of the first NB rows and columns of
*> the reduced matrix. D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (NB)
*> The off-diagonal elements of the first NB rows and columns of
*> the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*> TAUQ is DOUBLE PRECISION array dimension (NB)
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*> TAUP is DOUBLE PRECISION array, dimension (NB)
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NB)
*> The m-by-nb matrix X required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,M).
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array, dimension (LDY,NB)
*> The n-by-nb matrix Y required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*> LDY is INTEGER
*> The leading dimension of the array Y. LDY >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrices Q and P are represented as products of elementary
*> reflectors:
*>
*> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors.
*>
*> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
*> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
*> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
*> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
*> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> The elements of the vectors v and u together form the m-by-nb matrix
*> V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
*> the transformation to the unreduced part of the matrix, using a block
*> update of the form: A := A - V*Y**T - X*U**T.
*>
*> The contents of A on exit are illustrated by the following examples
*> with nb = 2:
*>
*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*>
*> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
*> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
*> ( v1 v2 a a a ) ( v1 1 a a a a )
*> ( v1 v2 a a a ) ( v1 v2 a a a a )
*> ( v1 v2 a a a ) ( v1 v2 a a a a )
*> ( v1 v2 a a a )
*>
*> where a denotes an element of the original matrix which is unchanged,
*> vi denotes an element of the vector defining H(i), and ui an element
*> of the vector defining G(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
$ LDY )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER LDA, LDX, LDY, M, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
$ TAUQ( * ), X( LDX, * ), Y( LDY, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DLARFG, DSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
IF( M.GE.N ) THEN
*
* Reduce to upper bidiagonal form
*
DO 10 I = 1, NB
*
* Update A(i:m,i)
*
CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
$ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
$ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
*
* Generate reflection Q(i) to annihilate A(i+1:m,i)
*
CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ TAUQ( I ) )
D( I ) = A( I, I )
IF( I.LT.N ) THEN
A( I, I ) = ONE
*
* Compute Y(i+1:n,i)
*
CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),
$ LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA,
$ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
$ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX,
$ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
$ LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
*
* Update A(i,i+1:n)
*
CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
$ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
$ LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
*
* Generate reflection P(i) to annihilate A(i,i+2:n)
*
CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
$ LDA, TAUP( I ) )
E( I ) = A( I, I+1 )
A( I, I+1 ) = ONE
*
* Compute X(i+1:m,i)
*
CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
$ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY,
$ A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
$ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
$ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
$ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
END IF
10 CONTINUE
ELSE
*
* Reduce to lower bidiagonal form
*
DO 20 I = 1, NB
*
* Update A(i,i:n)
*
CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
$ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA,
$ X( I, 1 ), LDX, ONE, A( I, I ), LDA )
*
* Generate reflection P(i) to annihilate A(i,i+1:n)
*
CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
$ TAUP( I ) )
D( I ) = A( I, I )
IF( I.LT.M ) THEN
A( I, I ) = ONE
*
* Compute X(i+1:m,i)
*
CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
$ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY,
$ A( I, I ), LDA, ZERO, X( 1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
$ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
$ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
$ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
*
* Update A(i+1:m,i)
*
CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
$ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
$ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
*
* Generate reflection Q(i) to annihilate A(i+2:m,i)
*
CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
$ TAUQ( I ) )
E( I ) = A( I+1, I )
A( I+1, I ) = ONE
*
* Compute Y(i+1:n,i)
*
CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),
$ LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA,
$ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
$ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX,
$ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA,
$ Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
END IF
20 CONTINUE
END IF
RETURN
*
* End of DLABRD
*
END

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*> \brief \b DLACPY copies all or part of one two-dimensional array to another.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLACPY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlacpy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlacpy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlacpy.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDB, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLACPY copies all or part of a two-dimensional matrix A to another
*> matrix B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies the part of the matrix A to be copied to B.
*> = 'U': Upper triangular part
*> = 'L': Lower triangular part
*> Otherwise: All of the matrix A
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A. If UPLO = 'U', only the upper triangle
*> or trapezoid is accessed; if UPLO = 'L', only the lower
*> triangle or trapezoid is accessed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On exit, B = A in the locations specified by UPLO.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = 1, MIN( J, M )
B( I, J ) = A( I, J )
10 CONTINUE
20 CONTINUE
ELSE IF( LSAME( UPLO, 'L' ) ) THEN
DO 40 J = 1, N
DO 30 I = J, M
B( I, J ) = A( I, J )
30 CONTINUE
40 CONTINUE
ELSE
DO 60 J = 1, N
DO 50 I = 1, M
B( I, J ) = A( I, J )
50 CONTINUE
60 CONTINUE
END IF
RETURN
*
* End of DLACPY
*
END

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*> \brief \b DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAE2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlae2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlae2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlae2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, RT1, RT2
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix
*> [ A B ]
*> [ B C ].
*> On return, RT1 is the eigenvalue of larger absolute value, and RT2
*> is the eigenvalue of smaller absolute value.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION
*> The (1,2) and (2,1) elements of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] RT1
*> \verbatim
*> RT1 is DOUBLE PRECISION
*> The eigenvalue of larger absolute value.
*> \endverbatim
*>
*> \param[out] RT2
*> \verbatim
*> RT2 is DOUBLE PRECISION
*> The eigenvalue of smaller absolute value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> RT1 is accurate to a few ulps barring over/underflow.
*>
*> RT2 may be inaccurate if there is massive cancellation in the
*> determinant A*C-B*B; higher precision or correctly rounded or
*> correctly truncated arithmetic would be needed to compute RT2
*> accurately in all cases.
*>
*> Overflow is possible only if RT1 is within a factor of 5 of overflow.
*> Underflow is harmless if the input data is 0 or exceeds
*> underflow_threshold / macheps.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, RT1, RT2
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AB, ACMN, ACMX, ADF, DF, RT, SM, TB
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
* Compute the eigenvalues
*
SM = A + C
DF = A - C
ADF = ABS( DF )
TB = B + B
AB = ABS( TB )
IF( ABS( A ).GT.ABS( C ) ) THEN
ACMX = A
ACMN = C
ELSE
ACMX = C
ACMN = A
END IF
IF( ADF.GT.AB ) THEN
RT = ADF*SQRT( ONE+( AB / ADF )**2 )
ELSE IF( ADF.LT.AB ) THEN
RT = AB*SQRT( ONE+( ADF / AB )**2 )
ELSE
*
* Includes case AB=ADF=0
*
RT = AB*SQRT( TWO )
END IF
IF( SM.LT.ZERO ) THEN
RT1 = HALF*( SM-RT )
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE IF( SM.GT.ZERO ) THEN
RT1 = HALF*( SM+RT )
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE
*
* Includes case RT1 = RT2 = 0
*
RT1 = HALF*RT
RT2 = -HALF*RT
END IF
RETURN
*
* End of DLAE2
*
END

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*> \brief \b DLAED0 used by sstedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED0( ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER ICOMPQ, INFO, LDQ, LDQS, N, QSIZ
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), Q( LDQ, * ), QSTORE( LDQS, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED0 computes all eigenvalues and corresponding eigenvectors of a
*> symmetric tridiagonal matrix using the divide and conquer method.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> = 0: Compute eigenvalues only.
*> = 1: Compute eigenvectors of original dense symmetric matrix
*> also. On entry, Q contains the orthogonal matrix used
*> to reduce the original matrix to tridiagonal form.
*> = 2: Compute eigenvalues and eigenvectors of tridiagonal
*> matrix.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*> QSIZ is INTEGER
*> The dimension of the orthogonal matrix used to reduce
*> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the main diagonal of the tridiagonal matrix.
*> On exit, its eigenvalues.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, Q must contain an N-by-N orthogonal matrix.
*> If ICOMPQ = 0 Q is not referenced.
*> If ICOMPQ = 1 On entry, Q is a subset of the columns of the
*> orthogonal matrix used to reduce the full
*> matrix to tridiagonal form corresponding to
*> the subset of the full matrix which is being
*> decomposed at this time.
*> If ICOMPQ = 2 On entry, Q will be the identity matrix.
*> On exit, Q contains the eigenvectors of the
*> tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. If eigenvectors are
*> desired, then LDQ >= max(1,N). In any case, LDQ >= 1.
*> \endverbatim
*>
*> \param[out] QSTORE
*> \verbatim
*> QSTORE is DOUBLE PRECISION array, dimension (LDQS, N)
*> Referenced only when ICOMPQ = 1. Used to store parts of
*> the eigenvector matrix when the updating matrix multiplies
*> take place.
*> \endverbatim
*>
*> \param[in] LDQS
*> \verbatim
*> LDQS is INTEGER
*> The leading dimension of the array QSTORE. If ICOMPQ = 1,
*> then LDQS >= max(1,N). In any case, LDQS >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> If ICOMPQ = 0 or 1, the dimension of WORK must be at least
*> 1 + 3*N + 2*N*lg N + 3*N**2
*> ( lg( N ) = smallest integer k
*> such that 2^k >= N )
*> If ICOMPQ = 2, the dimension of WORK must be at least
*> 4*N + N**2.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array,
*> If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
*> 6 + 6*N + 5*N*lg N.
*> ( lg( N ) = smallest integer k
*> such that 2^k >= N )
*> If ICOMPQ = 2, the dimension of IWORK must be at least
*> 3 + 5*N.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: The algorithm failed to compute an eigenvalue while
*> working on the submatrix lying in rows and columns
*> INFO/(N+1) through mod(INFO,N+1).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED0( ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS,
$ WORK, IWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER ICOMPQ, INFO, LDQ, LDQS, N, QSIZ
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), Q( LDQ, * ), QSTORE( LDQS, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.D0, ONE = 1.D0, TWO = 2.D0 )
* ..
* .. Local Scalars ..
INTEGER CURLVL, CURPRB, CURR, I, IGIVCL, IGIVNM,
$ IGIVPT, INDXQ, IPERM, IPRMPT, IQ, IQPTR, IWREM,
$ J, K, LGN, MATSIZ, MSD2, SMLSIZ, SMM1, SPM1,
$ SPM2, SUBMAT, SUBPBS, TLVLS
DOUBLE PRECISION TEMP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLACPY, DLAED1, DLAED7, DSTEQR,
$ XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.2 ) THEN
INFO = -1
ELSE IF( ( ICOMPQ.EQ.1 ) .AND. ( QSIZ.LT.MAX( 0, N ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDQS.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED0', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
SMLSIZ = ILAENV( 9, 'DLAED0', ' ', 0, 0, 0, 0 )
*
* Determine the size and placement of the submatrices, and save in
* the leading elements of IWORK.
*
IWORK( 1 ) = N
SUBPBS = 1
TLVLS = 0
10 CONTINUE
IF( IWORK( SUBPBS ).GT.SMLSIZ ) THEN
DO 20 J = SUBPBS, 1, -1
IWORK( 2*J ) = ( IWORK( J )+1 ) / 2
IWORK( 2*J-1 ) = IWORK( J ) / 2
20 CONTINUE
TLVLS = TLVLS + 1
SUBPBS = 2*SUBPBS
GO TO 10
END IF
DO 30 J = 2, SUBPBS
IWORK( J ) = IWORK( J ) + IWORK( J-1 )
30 CONTINUE
*
* Divide the matrix into SUBPBS submatrices of size at most SMLSIZ+1
* using rank-1 modifications (cuts).
*
SPM1 = SUBPBS - 1
DO 40 I = 1, SPM1
SUBMAT = IWORK( I ) + 1
SMM1 = SUBMAT - 1
D( SMM1 ) = D( SMM1 ) - ABS( E( SMM1 ) )
D( SUBMAT ) = D( SUBMAT ) - ABS( E( SMM1 ) )
40 CONTINUE
*
INDXQ = 4*N + 3
IF( ICOMPQ.NE.2 ) THEN
*
* Set up workspaces for eigenvalues only/accumulate new vectors
* routine
*
TEMP = LOG( DBLE( N ) ) / LOG( TWO )
LGN = INT( TEMP )
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IPRMPT = INDXQ + N + 1
IPERM = IPRMPT + N*LGN
IQPTR = IPERM + N*LGN
IGIVPT = IQPTR + N + 2
IGIVCL = IGIVPT + N*LGN
*
IGIVNM = 1
IQ = IGIVNM + 2*N*LGN
IWREM = IQ + N**2 + 1
*
* Initialize pointers
*
DO 50 I = 0, SUBPBS
IWORK( IPRMPT+I ) = 1
IWORK( IGIVPT+I ) = 1
50 CONTINUE
IWORK( IQPTR ) = 1
END IF
*
* Solve each submatrix eigenproblem at the bottom of the divide and
* conquer tree.
*
CURR = 0
DO 70 I = 0, SPM1
IF( I.EQ.0 ) THEN
SUBMAT = 1
MATSIZ = IWORK( 1 )
ELSE
SUBMAT = IWORK( I ) + 1
MATSIZ = IWORK( I+1 ) - IWORK( I )
END IF
IF( ICOMPQ.EQ.2 ) THEN
CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ),
$ Q( SUBMAT, SUBMAT ), LDQ, WORK, INFO )
IF( INFO.NE.0 )
$ GO TO 130
ELSE
CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ),
$ WORK( IQ-1+IWORK( IQPTR+CURR ) ), MATSIZ, WORK,
$ INFO )
IF( INFO.NE.0 )
$ GO TO 130
IF( ICOMPQ.EQ.1 ) THEN
CALL DGEMM( 'N', 'N', QSIZ, MATSIZ, MATSIZ, ONE,
$ Q( 1, SUBMAT ), LDQ, WORK( IQ-1+IWORK( IQPTR+
$ CURR ) ), MATSIZ, ZERO, QSTORE( 1, SUBMAT ),
$ LDQS )
END IF
IWORK( IQPTR+CURR+1 ) = IWORK( IQPTR+CURR ) + MATSIZ**2
CURR = CURR + 1
END IF
K = 1
DO 60 J = SUBMAT, IWORK( I+1 )
IWORK( INDXQ+J ) = K
K = K + 1
60 CONTINUE
70 CONTINUE
*
* Successively merge eigensystems of adjacent submatrices
* into eigensystem for the corresponding larger matrix.
*
* while ( SUBPBS > 1 )
*
CURLVL = 1
80 CONTINUE
IF( SUBPBS.GT.1 ) THEN
SPM2 = SUBPBS - 2
DO 90 I = 0, SPM2, 2
IF( I.EQ.0 ) THEN
SUBMAT = 1
MATSIZ = IWORK( 2 )
MSD2 = IWORK( 1 )
CURPRB = 0
ELSE
SUBMAT = IWORK( I ) + 1
MATSIZ = IWORK( I+2 ) - IWORK( I )
MSD2 = MATSIZ / 2
CURPRB = CURPRB + 1
END IF
*
* Merge lower order eigensystems (of size MSD2 and MATSIZ - MSD2)
* into an eigensystem of size MATSIZ.
* DLAED1 is used only for the full eigensystem of a tridiagonal
* matrix.
* DLAED7 handles the cases in which eigenvalues only or eigenvalues
* and eigenvectors of a full symmetric matrix (which was reduced to
* tridiagonal form) are desired.
*
IF( ICOMPQ.EQ.2 ) THEN
CALL DLAED1( MATSIZ, D( SUBMAT ), Q( SUBMAT, SUBMAT ),
$ LDQ, IWORK( INDXQ+SUBMAT ),
$ E( SUBMAT+MSD2-1 ), MSD2, WORK,
$ IWORK( SUBPBS+1 ), INFO )
ELSE
CALL DLAED7( ICOMPQ, MATSIZ, QSIZ, TLVLS, CURLVL, CURPRB,
$ D( SUBMAT ), QSTORE( 1, SUBMAT ), LDQS,
$ IWORK( INDXQ+SUBMAT ), E( SUBMAT+MSD2-1 ),
$ MSD2, WORK( IQ ), IWORK( IQPTR ),
$ IWORK( IPRMPT ), IWORK( IPERM ),
$ IWORK( IGIVPT ), IWORK( IGIVCL ),
$ WORK( IGIVNM ), WORK( IWREM ),
$ IWORK( SUBPBS+1 ), INFO )
END IF
IF( INFO.NE.0 )
$ GO TO 130
IWORK( I / 2+1 ) = IWORK( I+2 )
90 CONTINUE
SUBPBS = SUBPBS / 2
CURLVL = CURLVL + 1
GO TO 80
END IF
*
* end while
*
* Re-merge the eigenvalues/vectors which were deflated at the final
* merge step.
*
IF( ICOMPQ.EQ.1 ) THEN
DO 100 I = 1, N
J = IWORK( INDXQ+I )
WORK( I ) = D( J )
CALL DCOPY( QSIZ, QSTORE( 1, J ), 1, Q( 1, I ), 1 )
100 CONTINUE
CALL DCOPY( N, WORK, 1, D, 1 )
ELSE IF( ICOMPQ.EQ.2 ) THEN
DO 110 I = 1, N
J = IWORK( INDXQ+I )
WORK( I ) = D( J )
CALL DCOPY( N, Q( 1, J ), 1, WORK( N*I+1 ), 1 )
110 CONTINUE
CALL DCOPY( N, WORK, 1, D, 1 )
CALL DLACPY( 'A', N, N, WORK( N+1 ), N, Q, LDQ )
ELSE
DO 120 I = 1, N
J = IWORK( INDXQ+I )
WORK( I ) = D( J )
120 CONTINUE
CALL DCOPY( N, WORK, 1, D, 1 )
END IF
GO TO 140
*
130 CONTINUE
INFO = SUBMAT*( N+1 ) + SUBMAT + MATSIZ - 1
*
140 CONTINUE
RETURN
*
* End of DLAED0
*
END

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*> \brief \b DLAED1 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER CUTPNT, INFO, LDQ, N
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER INDXQ( * ), IWORK( * )
* DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED1 computes the updated eigensystem of a diagonal
*> matrix after modification by a rank-one symmetric matrix. This
*> routine is used only for the eigenproblem which requires all
*> eigenvalues and eigenvectors of a tridiagonal matrix. DLAED7 handles
*> the case in which eigenvalues only or eigenvalues and eigenvectors
*> of a full symmetric matrix (which was reduced to tridiagonal form)
*> are desired.
*>
*> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
*>
*> where Z = Q**T*u, u is a vector of length N with ones in the
*> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
*>
*> The eigenvectors of the original matrix are stored in Q, and the
*> eigenvalues are in D. The algorithm consists of three stages:
*>
*> The first stage consists of deflating the size of the problem
*> when there are multiple eigenvalues or if there is a zero in
*> the Z vector. For each such occurence the dimension of the
*> secular equation problem is reduced by one. This stage is
*> performed by the routine DLAED2.
*>
*> The second stage consists of calculating the updated
*> eigenvalues. This is done by finding the roots of the secular
*> equation via the routine DLAED4 (as called by DLAED3).
*> This routine also calculates the eigenvectors of the current
*> problem.
*>
*> The final stage consists of computing the updated eigenvectors
*> directly using the updated eigenvalues. The eigenvectors for
*> the current problem are multiplied with the eigenvectors from
*> the overall problem.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the eigenvalues of the rank-1-perturbed matrix.
*> On exit, the eigenvalues of the repaired matrix.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, the eigenvectors of the rank-1-perturbed matrix.
*> On exit, the eigenvectors of the repaired tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> On entry, the permutation which separately sorts the two
*> subproblems in D into ascending order.
*> On exit, the permutation which will reintegrate the
*> subproblems back into sorted order,
*> i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The subdiagonal entry used to create the rank-1 modification.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*> CUTPNT is INTEGER
*> The location of the last eigenvalue in the leading sub-matrix.
*> min(1,N) <= CUTPNT <= N/2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N + N**2)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER CUTPNT, INFO, LDQ, N
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER INDXQ( * ), IWORK( * )
DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER COLTYP, I, IDLMDA, INDX, INDXC, INDXP, IQ2, IS,
$ IW, IZ, K, N1, N2, ZPP1
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAED2, DLAED3, DLAMRG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED1', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* The following values are integer pointers which indicate
* the portion of the workspace
* used by a particular array in DLAED2 and DLAED3.
*
IZ = 1
IDLMDA = IZ + N
IW = IDLMDA + N
IQ2 = IW + N
*
INDX = 1
INDXC = INDX + N
COLTYP = INDXC + N
INDXP = COLTYP + N
*
*
* Form the z-vector which consists of the last row of Q_1 and the
* first row of Q_2.
*
CALL DCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
ZPP1 = CUTPNT + 1
CALL DCOPY( N-CUTPNT, Q( ZPP1, ZPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 )
*
* Deflate eigenvalues.
*
CALL DLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
$ WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
$ IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
$ IWORK( COLTYP ), INFO )
*
IF( INFO.NE.0 )
$ GO TO 20
*
* Solve Secular Equation.
*
IF( K.NE.0 ) THEN
IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
$ ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
CALL DLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
$ WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
$ WORK( IW ), WORK( IS ), INFO )
IF( INFO.NE.0 )
$ GO TO 20
*
* Prepare the INDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
ELSE
DO 10 I = 1, N
INDXQ( I ) = I
10 CONTINUE
END IF
*
20 CONTINUE
RETURN
*
* End of DLAED1
*
END

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*> \brief \b DLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
* Q2, INDX, INDXC, INDXP, COLTYP, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDQ, N, N1
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
* $ INDXQ( * )
* DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
* $ W( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED2 merges the two sets of eigenvalues together into a single
*> sorted set. Then it tries to deflate the size of the problem.
*> There are two ways in which deflation can occur: when two or more
*> eigenvalues are close together or if there is a tiny entry in the
*> Z vector. For each such occurrence the order of the related secular
*> equation problem is reduced by one.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> The number of non-deflated eigenvalues, and the order of the
*> related secular equation. 0 <= K <=N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> The location of the last eigenvalue in the leading sub-matrix.
*> min(1,N) <= N1 <= N/2.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, D contains the eigenvalues of the two submatrices to
*> be combined.
*> On exit, D contains the trailing (N-K) updated eigenvalues
*> (those which were deflated) sorted into increasing order.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, Q contains the eigenvectors of two submatrices in
*> the two square blocks with corners at (1,1), (N1,N1)
*> and (N1+1, N1+1), (N,N).
*> On exit, Q contains the trailing (N-K) updated eigenvectors
*> (those which were deflated) in its last N-K columns.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> The permutation which separately sorts the two sub-problems
*> in D into ascending order. Note that elements in the second
*> half of this permutation must first have N1 added to their
*> values. Destroyed on exit.
*> \endverbatim
*>
*> \param[in,out] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> On entry, the off-diagonal element associated with the rank-1
*> cut which originally split the two submatrices which are now
*> being recombined.
*> On exit, RHO has been modified to the value required by
*> DLAED3.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (N)
*> On entry, Z contains the updating vector (the last
*> row of the first sub-eigenvector matrix and the first row of
*> the second sub-eigenvector matrix).
*> On exit, the contents of Z have been destroyed by the updating
*> process.
*> \endverbatim
*>
*> \param[out] DLAMDA
*> \verbatim
*> DLAMDA is DOUBLE PRECISION array, dimension (N)
*> A copy of the first K eigenvalues which will be used by
*> DLAED3 to form the secular equation.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first k values of the final deflation-altered z-vector
*> which will be passed to DLAED3.
*> \endverbatim
*>
*> \param[out] Q2
*> \verbatim
*> Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
*> A copy of the first K eigenvectors which will be used by
*> DLAED3 in a matrix multiply (DGEMM) to solve for the new
*> eigenvectors.
*> \endverbatim
*>
*> \param[out] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> The permutation used to sort the contents of DLAMDA into
*> ascending order.
*> \endverbatim
*>
*> \param[out] INDXC
*> \verbatim
*> INDXC is INTEGER array, dimension (N)
*> The permutation used to arrange the columns of the deflated
*> Q matrix into three groups: the first group contains non-zero
*> elements only at and above N1, the second contains
*> non-zero elements only below N1, and the third is dense.
*> \endverbatim
*>
*> \param[out] INDXP
*> \verbatim
*> INDXP is INTEGER array, dimension (N)
*> The permutation used to place deflated values of D at the end
*> of the array. INDXP(1:K) points to the nondeflated D-values
*> and INDXP(K+1:N) points to the deflated eigenvalues.
*> \endverbatim
*>
*> \param[out] COLTYP
*> \verbatim
*> COLTYP is INTEGER array, dimension (N)
*> During execution, a label which will indicate which of the
*> following types a column in the Q2 matrix is:
*> 1 : non-zero in the upper half only;
*> 2 : dense;
*> 3 : non-zero in the lower half only;
*> 4 : deflated.
*> On exit, COLTYP(i) is the number of columns of type i,
*> for i=1 to 4 only.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
$ Q2, INDX, INDXC, INDXP, COLTYP, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDQ, N, N1
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
$ INDXQ( * )
DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
$ W( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, EIGHT = 8.0D0 )
* ..
* .. Local Arrays ..
INTEGER CTOT( 4 ), PSM( 4 )
* ..
* .. Local Scalars ..
INTEGER CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
$ N2, NJ, PJ
DOUBLE PRECISION C, EPS, S, T, TAU, TOL
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL IDAMAX, DLAMCH, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DLAMRG, DROT, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( MIN( 1, ( N / 2 ) ).GT.N1 .OR. ( N / 2 ).LT.N1 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
N2 = N - N1
N1P1 = N1 + 1
*
IF( RHO.LT.ZERO ) THEN
CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
END IF
*
* Normalize z so that norm(z) = 1. Since z is the concatenation of
* two normalized vectors, norm2(z) = sqrt(2).
*
T = ONE / SQRT( TWO )
CALL DSCAL( N, T, Z, 1 )
*
* RHO = ABS( norm(z)**2 * RHO )
*
RHO = ABS( TWO*RHO )
*
* Sort the eigenvalues into increasing order
*
DO 10 I = N1P1, N
INDXQ( I ) = INDXQ( I ) + N1
10 CONTINUE
*
* re-integrate the deflated parts from the last pass
*
DO 20 I = 1, N
DLAMDA( I ) = D( INDXQ( I ) )
20 CONTINUE
CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDXC )
DO 30 I = 1, N
INDX( I ) = INDXQ( INDXC( I ) )
30 CONTINUE
*
* Calculate the allowable deflation tolerance
*
IMAX = IDAMAX( N, Z, 1 )
JMAX = IDAMAX( N, D, 1 )
EPS = DLAMCH( 'Epsilon' )
TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) )
*
* If the rank-1 modifier is small enough, no more needs to be done
* except to reorganize Q so that its columns correspond with the
* elements in D.
*
IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
K = 0
IQ2 = 1
DO 40 J = 1, N
I = INDX( J )
CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 )
DLAMDA( J ) = D( I )
IQ2 = IQ2 + N
40 CONTINUE
CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ )
CALL DCOPY( N, DLAMDA, 1, D, 1 )
GO TO 190
END IF
*
* If there are multiple eigenvalues then the problem deflates. Here
* the number of equal eigenvalues are found. As each equal
* eigenvalue is found, an elementary reflector is computed to rotate
* the corresponding eigensubspace so that the corresponding
* components of Z are zero in this new basis.
*
DO 50 I = 1, N1
COLTYP( I ) = 1
50 CONTINUE
DO 60 I = N1P1, N
COLTYP( I ) = 3
60 CONTINUE
*
*
K = 0
K2 = N + 1
DO 70 J = 1, N
NJ = INDX( J )
IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
COLTYP( NJ ) = 4
INDXP( K2 ) = NJ
IF( J.EQ.N )
$ GO TO 100
ELSE
PJ = NJ
GO TO 80
END IF
70 CONTINUE
80 CONTINUE
J = J + 1
NJ = INDX( J )
IF( J.GT.N )
$ GO TO 100
IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
COLTYP( NJ ) = 4
INDXP( K2 ) = NJ
ELSE
*
* Check if eigenvalues are close enough to allow deflation.
*
S = Z( PJ )
C = Z( NJ )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = DLAPY2( C, S )
T = D( NJ ) - D( PJ )
C = C / TAU
S = -S / TAU
IF( ABS( T*C*S ).LE.TOL ) THEN
*
* Deflation is possible.
*
Z( NJ ) = TAU
Z( PJ ) = ZERO
IF( COLTYP( NJ ).NE.COLTYP( PJ ) )
$ COLTYP( NJ ) = 2
COLTYP( PJ ) = 4
CALL DROT( N, Q( 1, PJ ), 1, Q( 1, NJ ), 1, C, S )
T = D( PJ )*C**2 + D( NJ )*S**2
D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2
D( PJ ) = T
K2 = K2 - 1
I = 1
90 CONTINUE
IF( K2+I.LE.N ) THEN
IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN
INDXP( K2+I-1 ) = INDXP( K2+I )
INDXP( K2+I ) = PJ
I = I + 1
GO TO 90
ELSE
INDXP( K2+I-1 ) = PJ
END IF
ELSE
INDXP( K2+I-1 ) = PJ
END IF
PJ = NJ
ELSE
K = K + 1
DLAMDA( K ) = D( PJ )
W( K ) = Z( PJ )
INDXP( K ) = PJ
PJ = NJ
END IF
END IF
GO TO 80
100 CONTINUE
*
* Record the last eigenvalue.
*
K = K + 1
DLAMDA( K ) = D( PJ )
W( K ) = Z( PJ )
INDXP( K ) = PJ
*
* Count up the total number of the various types of columns, then
* form a permutation which positions the four column types into
* four uniform groups (although one or more of these groups may be
* empty).
*
DO 110 J = 1, 4
CTOT( J ) = 0
110 CONTINUE
DO 120 J = 1, N
CT = COLTYP( J )
CTOT( CT ) = CTOT( CT ) + 1
120 CONTINUE
*
* PSM(*) = Position in SubMatrix (of types 1 through 4)
*
PSM( 1 ) = 1
PSM( 2 ) = 1 + CTOT( 1 )
PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
K = N - CTOT( 4 )
*
* Fill out the INDXC array so that the permutation which it induces
* will place all type-1 columns first, all type-2 columns next,
* then all type-3's, and finally all type-4's.
*
DO 130 J = 1, N
JS = INDXP( J )
CT = COLTYP( JS )
INDX( PSM( CT ) ) = JS
INDXC( PSM( CT ) ) = J
PSM( CT ) = PSM( CT ) + 1
130 CONTINUE
*
* Sort the eigenvalues and corresponding eigenvectors into DLAMDA
* and Q2 respectively. The eigenvalues/vectors which were not
* deflated go into the first K slots of DLAMDA and Q2 respectively,
* while those which were deflated go into the last N - K slots.
*
I = 1
IQ1 = 1
IQ2 = 1 + ( CTOT( 1 )+CTOT( 2 ) )*N1
DO 140 J = 1, CTOT( 1 )
JS = INDX( I )
CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
Z( I ) = D( JS )
I = I + 1
IQ1 = IQ1 + N1
140 CONTINUE
*
DO 150 J = 1, CTOT( 2 )
JS = INDX( I )
CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
Z( I ) = D( JS )
I = I + 1
IQ1 = IQ1 + N1
IQ2 = IQ2 + N2
150 CONTINUE
*
DO 160 J = 1, CTOT( 3 )
JS = INDX( I )
CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
Z( I ) = D( JS )
I = I + 1
IQ2 = IQ2 + N2
160 CONTINUE
*
IQ1 = IQ2
DO 170 J = 1, CTOT( 4 )
JS = INDX( I )
CALL DCOPY( N, Q( 1, JS ), 1, Q2( IQ2 ), 1 )
IQ2 = IQ2 + N
Z( I ) = D( JS )
I = I + 1
170 CONTINUE
*
* The deflated eigenvalues and their corresponding vectors go back
* into the last N - K slots of D and Q respectively.
*
IF( K.LT.N ) THEN
CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N,
$ Q( 1, K+1 ), LDQ )
CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )
END IF
*
* Copy CTOT into COLTYP for referencing in DLAED3.
*
DO 180 J = 1, 4
COLTYP( J ) = CTOT( J )
180 CONTINUE
*
190 CONTINUE
RETURN
*
* End of DLAED2
*
END

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*> \brief \b DLAED3 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
* CTOT, W, S, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDQ, N, N1
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER CTOT( * ), INDX( * )
* DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
* $ S( * ), W( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED3 finds the roots of the secular equation, as defined by the
*> values in D, W, and RHO, between 1 and K. It makes the
*> appropriate calls to DLAED4 and then updates the eigenvectors by
*> multiplying the matrix of eigenvectors of the pair of eigensystems
*> being combined by the matrix of eigenvectors of the K-by-K system
*> which is solved here.
*>
*> This code makes very mild assumptions about floating point
*> arithmetic. It will work on machines with a guard digit in
*> add/subtract, or on those binary machines without guard digits
*> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
*> It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of terms in the rational function to be solved by
*> DLAED4. K >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the Q matrix.
*> N >= K (deflation may result in N>K).
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> The location of the last eigenvalue in the leading submatrix.
*> min(1,N) <= N1 <= N/2.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> D(I) contains the updated eigenvalues for
*> 1 <= I <= K.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> Initially the first K columns are used as workspace.
*> On output the columns 1 to K contain
*> the updated eigenvectors.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The value of the parameter in the rank one update equation.
*> RHO >= 0 required.
*> \endverbatim
*>
*> \param[in,out] DLAMDA
*> \verbatim
*> DLAMDA is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the old roots
*> of the deflated updating problem. These are the poles
*> of the secular equation. May be changed on output by
*> having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
*> Cray-2, or Cray C-90, as described above.
*> \endverbatim
*>
*> \param[in] Q2
*> \verbatim
*> Q2 is DOUBLE PRECISION array, dimension (LDQ2, N)
*> The first K columns of this matrix contain the non-deflated
*> eigenvectors for the split problem.
*> \endverbatim
*>
*> \param[in] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> The permutation used to arrange the columns of the deflated
*> Q matrix into three groups (see DLAED2).
*> The rows of the eigenvectors found by DLAED4 must be likewise
*> permuted before the matrix multiply can take place.
*> \endverbatim
*>
*> \param[in] CTOT
*> \verbatim
*> CTOT is INTEGER array, dimension (4)
*> A count of the total number of the various types of columns
*> in Q, as described in INDX. The fourth column type is any
*> column which has been deflated.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the components
*> of the deflation-adjusted updating vector. Destroyed on
*> output.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N1 + 1)*K
*> Will contain the eigenvectors of the repaired matrix which
*> will be multiplied by the previously accumulated eigenvectors
*> to update the system.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
$ CTOT, W, S, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDQ, N, N1
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER CTOT( * ), INDX( * )
DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
$ S( * ), W( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, II, IQ2, J, N12, N2, N23
DOUBLE PRECISION TEMP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3, DNRM2
EXTERNAL DLAMC3, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLACPY, DLAED4, DLASET, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( K.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.K ) THEN
INFO = -2
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED3', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 )
$ RETURN
*
* Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
* be computed with high relative accuracy (barring over/underflow).
* This is a problem on machines without a guard digit in
* add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
* The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
* which on any of these machines zeros out the bottommost
* bit of DLAMDA(I) if it is 1; this makes the subsequent
* subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
* occurs. On binary machines with a guard digit (almost all
* machines) it does not change DLAMDA(I) at all. On hexadecimal
* and decimal machines with a guard digit, it slightly
* changes the bottommost bits of DLAMDA(I). It does not account
* for hexadecimal or decimal machines without guard digits
* (we know of none). We use a subroutine call to compute
* 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
* this code.
*
DO 10 I = 1, K
DLAMDA( I ) = DLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
10 CONTINUE
*
DO 20 J = 1, K
CALL DLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO )
*
* If the zero finder fails, the computation is terminated.
*
IF( INFO.NE.0 )
$ GO TO 120
20 CONTINUE
*
IF( K.EQ.1 )
$ GO TO 110
IF( K.EQ.2 ) THEN
DO 30 J = 1, K
W( 1 ) = Q( 1, J )
W( 2 ) = Q( 2, J )
II = INDX( 1 )
Q( 1, J ) = W( II )
II = INDX( 2 )
Q( 2, J ) = W( II )
30 CONTINUE
GO TO 110
END IF
*
* Compute updated W.
*
CALL DCOPY( K, W, 1, S, 1 )
*
* Initialize W(I) = Q(I,I)
*
CALL DCOPY( K, Q, LDQ+1, W, 1 )
DO 60 J = 1, K
DO 40 I = 1, J - 1
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
40 CONTINUE
DO 50 I = J + 1, K
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
50 CONTINUE
60 CONTINUE
DO 70 I = 1, K
W( I ) = SIGN( SQRT( -W( I ) ), S( I ) )
70 CONTINUE
*
* Compute eigenvectors of the modified rank-1 modification.
*
DO 100 J = 1, K
DO 80 I = 1, K
S( I ) = W( I ) / Q( I, J )
80 CONTINUE
TEMP = DNRM2( K, S, 1 )
DO 90 I = 1, K
II = INDX( I )
Q( I, J ) = S( II ) / TEMP
90 CONTINUE
100 CONTINUE
*
* Compute the updated eigenvectors.
*
110 CONTINUE
*
N2 = N - N1
N12 = CTOT( 1 ) + CTOT( 2 )
N23 = CTOT( 2 ) + CTOT( 3 )
*
CALL DLACPY( 'A', N23, K, Q( CTOT( 1 )+1, 1 ), LDQ, S, N23 )
IQ2 = N1*N12 + 1
IF( N23.NE.0 ) THEN
CALL DGEMM( 'N', 'N', N2, K, N23, ONE, Q2( IQ2 ), N2, S, N23,
$ ZERO, Q( N1+1, 1 ), LDQ )
ELSE
CALL DLASET( 'A', N2, K, ZERO, ZERO, Q( N1+1, 1 ), LDQ )
END IF
*
CALL DLACPY( 'A', N12, K, Q, LDQ, S, N12 )
IF( N12.NE.0 ) THEN
CALL DGEMM( 'N', 'N', N1, K, N12, ONE, Q2, N1, S, N12, ZERO, Q,
$ LDQ )
ELSE
CALL DLASET( 'A', N1, K, ZERO, ZERO, Q( 1, 1 ), LDQ )
END IF
*
*
120 CONTINUE
RETURN
*
* End of DLAED3
*
END

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*> \brief \b DLAED4 used by sstedc. Finds a single root of the secular equation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed4.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
*
* .. Scalar Arguments ..
* INTEGER I, INFO, N
* DOUBLE PRECISION DLAM, RHO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DELTA( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine computes the I-th updated eigenvalue of a symmetric
*> rank-one modification to a diagonal matrix whose elements are
*> given in the array d, and that
*>
*> D(i) < D(j) for i < j
*>
*> and that RHO > 0. This is arranged by the calling routine, and is
*> no loss in generality. The rank-one modified system is thus
*>
*> diag( D ) + RHO * Z * Z_transpose.
*>
*> where we assume the Euclidean norm of Z is 1.
*>
*> The method consists of approximating the rational functions in the
*> secular equation by simpler interpolating rational functions.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The length of all arrays.
*> \endverbatim
*>
*> \param[in] I
*> \verbatim
*> I is INTEGER
*> The index of the eigenvalue to be computed. 1 <= I <= N.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The original eigenvalues. It is assumed that they are in
*> order, D(I) < D(J) for I < J.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (N)
*> The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*> DELTA is DOUBLE PRECISION array, dimension (N)
*> If N .GT. 2, DELTA contains (D(j) - lambda_I) in its j-th
*> component. If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
*> for detail. The vector DELTA contains the information necessary
*> to construct the eigenvectors by DLAED3 and DLAED9.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] DLAM
*> \verbatim
*> DLAM is DOUBLE PRECISION
*> The computed lambda_I, the I-th updated eigenvalue.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = 1, the updating process failed.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> Logical variable ORGATI (origin-at-i?) is used for distinguishing
*> whether D(i) or D(i+1) is treated as the origin.
*>
*> ORGATI = .true. origin at i
*> ORGATI = .false. origin at i+1
*>
*> Logical variable SWTCH3 (switch-for-3-poles?) is for noting
*> if we are working with THREE poles!
*>
*> MAXIT is the maximum number of iterations allowed for each
*> eigenvalue.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER I, INFO, N
DOUBLE PRECISION DLAM, RHO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DELTA( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 30 )
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0,
$ TEN = 10.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ORGATI, SWTCH, SWTCH3
INTEGER II, IIM1, IIP1, IP1, ITER, J, NITER
DOUBLE PRECISION A, B, C, DEL, DLTLB, DLTUB, DPHI, DPSI, DW,
$ EPS, ERRETM, ETA, MIDPT, PHI, PREW, PSI,
$ RHOINV, TAU, TEMP, TEMP1, W
* ..
* .. Local Arrays ..
DOUBLE PRECISION ZZ( 3 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLAED5, DLAED6
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Since this routine is called in an inner loop, we do no argument
* checking.
*
* Quick return for N=1 and 2.
*
INFO = 0
IF( N.EQ.1 ) THEN
*
* Presumably, I=1 upon entry
*
DLAM = D( 1 ) + RHO*Z( 1 )*Z( 1 )
DELTA( 1 ) = ONE
RETURN
END IF
IF( N.EQ.2 ) THEN
CALL DLAED5( I, D, Z, DELTA, RHO, DLAM )
RETURN
END IF
*
* Compute machine epsilon
*
EPS = DLAMCH( 'Epsilon' )
RHOINV = ONE / RHO
*
* The case I = N
*
IF( I.EQ.N ) THEN
*
* Initialize some basic variables
*
II = N - 1
NITER = 1
*
* Calculate initial guess
*
MIDPT = RHO / TWO
*
* If ||Z||_2 is not one, then TEMP should be set to
* RHO * ||Z||_2^2 / TWO
*
DO 10 J = 1, N
DELTA( J ) = ( D( J )-D( I ) ) - MIDPT
10 CONTINUE
*
PSI = ZERO
DO 20 J = 1, N - 2
PSI = PSI + Z( J )*Z( J ) / DELTA( J )
20 CONTINUE
*
C = RHOINV + PSI
W = C + Z( II )*Z( II ) / DELTA( II ) +
$ Z( N )*Z( N ) / DELTA( N )
*
IF( W.LE.ZERO ) THEN
TEMP = Z( N-1 )*Z( N-1 ) / ( D( N )-D( N-1 )+RHO ) +
$ Z( N )*Z( N ) / RHO
IF( C.LE.TEMP ) THEN
TAU = RHO
ELSE
DEL = D( N ) - D( N-1 )
A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
B = Z( N )*Z( N )*DEL
IF( A.LT.ZERO ) THEN
TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
END IF
*
* It can be proved that
* D(N)+RHO/2 <= LAMBDA(N) < D(N)+TAU <= D(N)+RHO
*
DLTLB = MIDPT
DLTUB = RHO
ELSE
DEL = D( N ) - D( N-1 )
A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
B = Z( N )*Z( N )*DEL
IF( A.LT.ZERO ) THEN
TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
*
* It can be proved that
* D(N) < D(N)+TAU < LAMBDA(N) < D(N)+RHO/2
*
DLTLB = ZERO
DLTUB = MIDPT
END IF
*
DO 30 J = 1, N
DELTA( J ) = ( D( J )-D( I ) ) - TAU
30 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 40 J = 1, II
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
40 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TEMP = Z( N ) / DELTA( N )
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
$ ABS( TAU )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
DLAM = D( I ) + TAU
GO TO 250
END IF
*
IF( W.LE.ZERO ) THEN
DLTLB = MAX( DLTLB, TAU )
ELSE
DLTUB = MIN( DLTUB, TAU )
END IF
*
* Calculate the new step
*
NITER = NITER + 1
C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI
A = ( DELTA( N-1 )+DELTA( N ) )*W -
$ DELTA( N-1 )*DELTA( N )*( DPSI+DPHI )
B = DELTA( N-1 )*DELTA( N )*W
IF( C.LT.ZERO )
$ C = ABS( C )
IF( C.EQ.ZERO ) THEN
* ETA = B/A
* ETA = RHO - TAU
ETA = DLTUB - TAU
ELSE IF( A.GE.ZERO ) THEN
ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GT.ZERO )
$ ETA = -W / ( DPSI+DPHI )
TEMP = TAU + ETA
IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( DLTUB-TAU ) / TWO
ELSE
ETA = ( DLTLB-TAU ) / TWO
END IF
END IF
DO 50 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
50 CONTINUE
*
TAU = TAU + ETA
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 60 J = 1, II
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
60 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TEMP = Z( N ) / DELTA( N )
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
$ ABS( TAU )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
* Main loop to update the values of the array DELTA
*
ITER = NITER + 1
*
DO 90 NITER = ITER, MAXIT
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
DLAM = D( I ) + TAU
GO TO 250
END IF
*
IF( W.LE.ZERO ) THEN
DLTLB = MAX( DLTLB, TAU )
ELSE
DLTUB = MIN( DLTUB, TAU )
END IF
*
* Calculate the new step
*
C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI
A = ( DELTA( N-1 )+DELTA( N ) )*W -
$ DELTA( N-1 )*DELTA( N )*( DPSI+DPHI )
B = DELTA( N-1 )*DELTA( N )*W
IF( A.GE.ZERO ) THEN
ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GT.ZERO )
$ ETA = -W / ( DPSI+DPHI )
TEMP = TAU + ETA
IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( DLTUB-TAU ) / TWO
ELSE
ETA = ( DLTLB-TAU ) / TWO
END IF
END IF
DO 70 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
70 CONTINUE
*
TAU = TAU + ETA
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 80 J = 1, II
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
80 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TEMP = Z( N ) / DELTA( N )
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
$ ABS( TAU )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
90 CONTINUE
*
* Return with INFO = 1, NITER = MAXIT and not converged
*
INFO = 1
DLAM = D( I ) + TAU
GO TO 250
*
* End for the case I = N
*
ELSE
*
* The case for I < N
*
NITER = 1
IP1 = I + 1
*
* Calculate initial guess
*
DEL = D( IP1 ) - D( I )
MIDPT = DEL / TWO
DO 100 J = 1, N
DELTA( J ) = ( D( J )-D( I ) ) - MIDPT
100 CONTINUE
*
PSI = ZERO
DO 110 J = 1, I - 1
PSI = PSI + Z( J )*Z( J ) / DELTA( J )
110 CONTINUE
*
PHI = ZERO
DO 120 J = N, I + 2, -1
PHI = PHI + Z( J )*Z( J ) / DELTA( J )
120 CONTINUE
C = RHOINV + PSI + PHI
W = C + Z( I )*Z( I ) / DELTA( I ) +
$ Z( IP1 )*Z( IP1 ) / DELTA( IP1 )
*
IF( W.GT.ZERO ) THEN
*
* d(i)< the ith eigenvalue < (d(i)+d(i+1))/2
*
* We choose d(i) as origin.
*
ORGATI = .TRUE.
A = C*DEL + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 )
B = Z( I )*Z( I )*DEL
IF( A.GT.ZERO ) THEN
TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
ELSE
TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
END IF
DLTLB = ZERO
DLTUB = MIDPT
ELSE
*
* (d(i)+d(i+1))/2 <= the ith eigenvalue < d(i+1)
*
* We choose d(i+1) as origin.
*
ORGATI = .FALSE.
A = C*DEL - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 )
B = Z( IP1 )*Z( IP1 )*DEL
IF( A.LT.ZERO ) THEN
TAU = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) )
ELSE
TAU = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
END IF
DLTLB = -MIDPT
DLTUB = ZERO
END IF
*
IF( ORGATI ) THEN
DO 130 J = 1, N
DELTA( J ) = ( D( J )-D( I ) ) - TAU
130 CONTINUE
ELSE
DO 140 J = 1, N
DELTA( J ) = ( D( J )-D( IP1 ) ) - TAU
140 CONTINUE
END IF
IF( ORGATI ) THEN
II = I
ELSE
II = I + 1
END IF
IIM1 = II - 1
IIP1 = II + 1
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 150 J = 1, IIM1
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
150 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 160 J = N, IIP1, -1
TEMP = Z( J ) / DELTA( J )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
160 CONTINUE
*
W = RHOINV + PHI + PSI
*
* W is the value of the secular function with
* its ii-th element removed.
*
SWTCH3 = .FALSE.
IF( ORGATI ) THEN
IF( W.LT.ZERO )
$ SWTCH3 = .TRUE.
ELSE
IF( W.GT.ZERO )
$ SWTCH3 = .TRUE.
END IF
IF( II.EQ.1 .OR. II.EQ.N )
$ SWTCH3 = .FALSE.
*
TEMP = Z( II ) / DELTA( II )
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = W + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV +
$ THREE*ABS( TEMP ) + ABS( TAU )*DW
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
IF( ORGATI ) THEN
DLAM = D( I ) + TAU
ELSE
DLAM = D( IP1 ) + TAU
END IF
GO TO 250
END IF
*
IF( W.LE.ZERO ) THEN
DLTLB = MAX( DLTLB, TAU )
ELSE
DLTUB = MIN( DLTUB, TAU )
END IF
*
* Calculate the new step
*
NITER = NITER + 1
IF( .NOT.SWTCH3 ) THEN
IF( ORGATI ) THEN
C = W - DELTA( IP1 )*DW - ( D( I )-D( IP1 ) )*
$ ( Z( I ) / DELTA( I ) )**2
ELSE
C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )*
$ ( Z( IP1 ) / DELTA( IP1 ) )**2
END IF
A = ( DELTA( I )+DELTA( IP1 ) )*W -
$ DELTA( I )*DELTA( IP1 )*DW
B = DELTA( I )*DELTA( IP1 )*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DELTA( IP1 )*DELTA( IP1 )*
$ ( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) + DELTA( I )*DELTA( I )*
$ ( DPSI+DPHI )
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
ELSE
*
* Interpolation using THREE most relevant poles
*
TEMP = RHOINV + PSI + PHI
IF( ORGATI ) THEN
TEMP1 = Z( IIM1 ) / DELTA( IIM1 )
TEMP1 = TEMP1*TEMP1
C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) -
$ ( D( IIM1 )-D( IIP1 ) )*TEMP1
ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*
$ ( ( DPSI-TEMP1 )+DPHI )
ELSE
TEMP1 = Z( IIP1 ) / DELTA( IIP1 )
TEMP1 = TEMP1*TEMP1
C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) -
$ ( D( IIP1 )-D( IIM1 ) )*TEMP1
ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*
$ ( DPSI+( DPHI-TEMP1 ) )
ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
END IF
ZZ( 2 ) = Z( II )*Z( II )
CALL DLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W, ETA,
$ INFO )
IF( INFO.NE.0 )
$ GO TO 250
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GE.ZERO )
$ ETA = -W / DW
TEMP = TAU + ETA
IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( DLTUB-TAU ) / TWO
ELSE
ETA = ( DLTLB-TAU ) / TWO
END IF
END IF
*
PREW = W
*
DO 180 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
180 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 190 J = 1, IIM1
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
190 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 200 J = N, IIP1, -1
TEMP = Z( J ) / DELTA( J )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
200 CONTINUE
*
TEMP = Z( II ) / DELTA( II )
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = RHOINV + PHI + PSI + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV +
$ THREE*ABS( TEMP ) + ABS( TAU+ETA )*DW
*
SWTCH = .FALSE.
IF( ORGATI ) THEN
IF( -W.GT.ABS( PREW ) / TEN )
$ SWTCH = .TRUE.
ELSE
IF( W.GT.ABS( PREW ) / TEN )
$ SWTCH = .TRUE.
END IF
*
TAU = TAU + ETA
*
* Main loop to update the values of the array DELTA
*
ITER = NITER + 1
*
DO 240 NITER = ITER, MAXIT
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
IF( ORGATI ) THEN
DLAM = D( I ) + TAU
ELSE
DLAM = D( IP1 ) + TAU
END IF
GO TO 250
END IF
*
IF( W.LE.ZERO ) THEN
DLTLB = MAX( DLTLB, TAU )
ELSE
DLTUB = MIN( DLTUB, TAU )
END IF
*
* Calculate the new step
*
IF( .NOT.SWTCH3 ) THEN
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
C = W - DELTA( IP1 )*DW -
$ ( D( I )-D( IP1 ) )*( Z( I ) / DELTA( I ) )**2
ELSE
C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )*
$ ( Z( IP1 ) / DELTA( IP1 ) )**2
END IF
ELSE
TEMP = Z( II ) / DELTA( II )
IF( ORGATI ) THEN
DPSI = DPSI + TEMP*TEMP
ELSE
DPHI = DPHI + TEMP*TEMP
END IF
C = W - DELTA( I )*DPSI - DELTA( IP1 )*DPHI
END IF
A = ( DELTA( I )+DELTA( IP1 ) )*W -
$ DELTA( I )*DELTA( IP1 )*DW
B = DELTA( I )*DELTA( IP1 )*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DELTA( IP1 )*
$ DELTA( IP1 )*( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) +
$ DELTA( I )*DELTA( I )*( DPSI+DPHI )
END IF
ELSE
A = DELTA( I )*DELTA( I )*DPSI +
$ DELTA( IP1 )*DELTA( IP1 )*DPHI
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
ELSE
*
* Interpolation using THREE most relevant poles
*
TEMP = RHOINV + PSI + PHI
IF( SWTCH ) THEN
C = TEMP - DELTA( IIM1 )*DPSI - DELTA( IIP1 )*DPHI
ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*DPSI
ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*DPHI
ELSE
IF( ORGATI ) THEN
TEMP1 = Z( IIM1 ) / DELTA( IIM1 )
TEMP1 = TEMP1*TEMP1
C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) -
$ ( D( IIM1 )-D( IIP1 ) )*TEMP1
ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*
$ ( ( DPSI-TEMP1 )+DPHI )
ELSE
TEMP1 = Z( IIP1 ) / DELTA( IIP1 )
TEMP1 = TEMP1*TEMP1
C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) -
$ ( D( IIP1 )-D( IIM1 ) )*TEMP1
ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*
$ ( DPSI+( DPHI-TEMP1 ) )
ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
END IF
END IF
CALL DLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W, ETA,
$ INFO )
IF( INFO.NE.0 )
$ GO TO 250
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GE.ZERO )
$ ETA = -W / DW
TEMP = TAU + ETA
IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( DLTUB-TAU ) / TWO
ELSE
ETA = ( DLTLB-TAU ) / TWO
END IF
END IF
*
DO 210 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
210 CONTINUE
*
TAU = TAU + ETA
PREW = W
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 220 J = 1, IIM1
TEMP = Z( J ) / DELTA( J )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
220 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 230 J = N, IIP1, -1
TEMP = Z( J ) / DELTA( J )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
230 CONTINUE
*
TEMP = Z( II ) / DELTA( II )
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = RHOINV + PHI + PSI + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV +
$ THREE*ABS( TEMP ) + ABS( TAU )*DW
IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN )
$ SWTCH = .NOT.SWTCH
*
240 CONTINUE
*
* Return with INFO = 1, NITER = MAXIT and not converged
*
INFO = 1
IF( ORGATI ) THEN
DLAM = D( I ) + TAU
ELSE
DLAM = D( IP1 ) + TAU
END IF
*
END IF
*
250 CONTINUE
*
RETURN
*
* End of DLAED4
*
END

189
lib/linalg/dlaed5.f Normal file
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@ -0,0 +1,189 @@
*> \brief \b DLAED5 used by sstedc. Solves the 2-by-2 secular equation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed5.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
*
* .. Scalar Arguments ..
* INTEGER I
* DOUBLE PRECISION DLAM, RHO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( 2 ), DELTA( 2 ), Z( 2 )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine computes the I-th eigenvalue of a symmetric rank-one
*> modification of a 2-by-2 diagonal matrix
*>
*> diag( D ) + RHO * Z * transpose(Z) .
*>
*> The diagonal elements in the array D are assumed to satisfy
*>
*> D(i) < D(j) for i < j .
*>
*> We also assume RHO > 0 and that the Euclidean norm of the vector
*> Z is one.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I
*> \verbatim
*> I is INTEGER
*> The index of the eigenvalue to be computed. I = 1 or I = 2.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (2)
*> The original eigenvalues. We assume D(1) < D(2).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (2)
*> The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*> DELTA is DOUBLE PRECISION array, dimension (2)
*> The vector DELTA contains the information necessary
*> to construct the eigenvectors.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] DLAM
*> \verbatim
*> DLAM is DOUBLE PRECISION
*> The computed lambda_I, the I-th updated eigenvalue.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER I
DOUBLE PRECISION DLAM, RHO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( 2 ), DELTA( 2 ), Z( 2 )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, FOUR
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ FOUR = 4.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION B, C, DEL, TAU, TEMP, W
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
DEL = D( 2 ) - D( 1 )
IF( I.EQ.1 ) THEN
W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
IF( W.GT.ZERO ) THEN
B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 1 )*Z( 1 )*DEL
*
* B > ZERO, always
*
TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
DLAM = D( 1 ) + TAU
DELTA( 1 ) = -Z( 1 ) / TAU
DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
ELSE
B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DEL
IF( B.GT.ZERO ) THEN
TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
ELSE
TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
END IF
DLAM = D( 2 ) + TAU
DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
DELTA( 2 ) = -Z( 2 ) / TAU
END IF
TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
DELTA( 1 ) = DELTA( 1 ) / TEMP
DELTA( 2 ) = DELTA( 2 ) / TEMP
ELSE
*
* Now I=2
*
B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DEL
IF( B.GT.ZERO ) THEN
TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
ELSE
TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
END IF
DLAM = D( 2 ) + TAU
DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
DELTA( 2 ) = -Z( 2 ) / TAU
TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
DELTA( 1 ) = DELTA( 1 ) / TEMP
DELTA( 2 ) = DELTA( 2 ) / TEMP
END IF
RETURN
*
* End OF DLAED5
*
END

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*> \brief \b DLAED6 used by sstedc. Computes one Newton step in solution of the secular equation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed6.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
* .. Scalar Arguments ..
* LOGICAL ORGATI
* INTEGER INFO, KNITER
* DOUBLE PRECISION FINIT, RHO, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( 3 ), Z( 3 )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED6 computes the positive or negative root (closest to the origin)
*> of
*> z(1) z(2) z(3)
*> f(x) = rho + --------- + ---------- + ---------
*> d(1)-x d(2)-x d(3)-x
*>
*> It is assumed that
*>
*> if ORGATI = .true. the root is between d(2) and d(3);
*> otherwise it is between d(1) and d(2)
*>
*> This routine will be called by DLAED4 when necessary. In most cases,
*> the root sought is the smallest in magnitude, though it might not be
*> in some extremely rare situations.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] KNITER
*> \verbatim
*> KNITER is INTEGER
*> Refer to DLAED4 for its significance.
*> \endverbatim
*>
*> \param[in] ORGATI
*> \verbatim
*> ORGATI is LOGICAL
*> If ORGATI is true, the needed root is between d(2) and
*> d(3); otherwise it is between d(1) and d(2). See
*> DLAED4 for further details.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> Refer to the equation f(x) above.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (3)
*> D satisfies d(1) < d(2) < d(3).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (3)
*> Each of the elements in z must be positive.
*> \endverbatim
*>
*> \param[in] FINIT
*> \verbatim
*> FINIT is DOUBLE PRECISION
*> The value of f at 0. It is more accurate than the one
*> evaluated inside this routine (if someone wants to do
*> so).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The root of the equation f(x).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = 1, failure to converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 10/02/03: This version has a few statements commented out for thread
*> safety (machine parameters are computed on each entry). SJH.
*>
*> 05/10/06: Modified from a new version of Ren-Cang Li, use
*> Gragg-Thornton-Warner cubic convergent scheme for better stability.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
LOGICAL ORGATI
INTEGER INFO, KNITER
DOUBLE PRECISION FINIT, RHO, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( 3 ), Z( 3 )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 40 )
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Local Arrays ..
DOUBLE PRECISION DSCALE( 3 ), ZSCALE( 3 )
* ..
* .. Local Scalars ..
LOGICAL SCALE
INTEGER I, ITER, NITER
DOUBLE PRECISION A, B, BASE, C, DDF, DF, EPS, ERRETM, ETA, F,
$ FC, SCLFAC, SCLINV, SMALL1, SMALL2, SMINV1,
$ SMINV2, TEMP, TEMP1, TEMP2, TEMP3, TEMP4,
$ LBD, UBD
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
IF( ORGATI ) THEN
LBD = D(2)
UBD = D(3)
ELSE
LBD = D(1)
UBD = D(2)
END IF
IF( FINIT .LT. ZERO )THEN
LBD = ZERO
ELSE
UBD = ZERO
END IF
*
NITER = 1
TAU = ZERO
IF( KNITER.EQ.2 ) THEN
IF( ORGATI ) THEN
TEMP = ( D( 3 )-D( 2 ) ) / TWO
C = RHO + Z( 1 ) / ( ( D( 1 )-D( 2 ) )-TEMP )
A = C*( D( 2 )+D( 3 ) ) + Z( 2 ) + Z( 3 )
B = C*D( 2 )*D( 3 ) + Z( 2 )*D( 3 ) + Z( 3 )*D( 2 )
ELSE
TEMP = ( D( 1 )-D( 2 ) ) / TWO
C = RHO + Z( 3 ) / ( ( D( 3 )-D( 2 ) )-TEMP )
A = C*( D( 1 )+D( 2 ) ) + Z( 1 ) + Z( 2 )
B = C*D( 1 )*D( 2 ) + Z( 1 )*D( 2 ) + Z( 2 )*D( 1 )
END IF
TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
A = A / TEMP
B = B / TEMP
C = C / TEMP
IF( C.EQ.ZERO ) THEN
TAU = B / A
ELSE IF( A.LE.ZERO ) THEN
TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
IF( TAU .LT. LBD .OR. TAU .GT. UBD )
$ TAU = ( LBD+UBD )/TWO
IF( D(1).EQ.TAU .OR. D(2).EQ.TAU .OR. D(3).EQ.TAU ) THEN
TAU = ZERO
ELSE
TEMP = FINIT + TAU*Z(1)/( D(1)*( D( 1 )-TAU ) ) +
$ TAU*Z(2)/( D(2)*( D( 2 )-TAU ) ) +
$ TAU*Z(3)/( D(3)*( D( 3 )-TAU ) )
IF( TEMP .LE. ZERO )THEN
LBD = TAU
ELSE
UBD = TAU
END IF
IF( ABS( FINIT ).LE.ABS( TEMP ) )
$ TAU = ZERO
END IF
END IF
*
* get machine parameters for possible scaling to avoid overflow
*
* modified by Sven: parameters SMALL1, SMINV1, SMALL2,
* SMINV2, EPS are not SAVEd anymore between one call to the
* others but recomputed at each call
*
EPS = DLAMCH( 'Epsilon' )
BASE = DLAMCH( 'Base' )
SMALL1 = BASE**( INT( LOG( DLAMCH( 'SafMin' ) ) / LOG( BASE ) /
$ THREE ) )
SMINV1 = ONE / SMALL1
SMALL2 = SMALL1*SMALL1
SMINV2 = SMINV1*SMINV1
*
* Determine if scaling of inputs necessary to avoid overflow
* when computing 1/TEMP**3
*
IF( ORGATI ) THEN
TEMP = MIN( ABS( D( 2 )-TAU ), ABS( D( 3 )-TAU ) )
ELSE
TEMP = MIN( ABS( D( 1 )-TAU ), ABS( D( 2 )-TAU ) )
END IF
SCALE = .FALSE.
IF( TEMP.LE.SMALL1 ) THEN
SCALE = .TRUE.
IF( TEMP.LE.SMALL2 ) THEN
*
* Scale up by power of radix nearest 1/SAFMIN**(2/3)
*
SCLFAC = SMINV2
SCLINV = SMALL2
ELSE
*
* Scale up by power of radix nearest 1/SAFMIN**(1/3)
*
SCLFAC = SMINV1
SCLINV = SMALL1
END IF
*
* Scaling up safe because D, Z, TAU scaled elsewhere to be O(1)
*
DO 10 I = 1, 3
DSCALE( I ) = D( I )*SCLFAC
ZSCALE( I ) = Z( I )*SCLFAC
10 CONTINUE
TAU = TAU*SCLFAC
LBD = LBD*SCLFAC
UBD = UBD*SCLFAC
ELSE
*
* Copy D and Z to DSCALE and ZSCALE
*
DO 20 I = 1, 3
DSCALE( I ) = D( I )
ZSCALE( I ) = Z( I )
20 CONTINUE
END IF
*
FC = ZERO
DF = ZERO
DDF = ZERO
DO 30 I = 1, 3
TEMP = ONE / ( DSCALE( I )-TAU )
TEMP1 = ZSCALE( I )*TEMP
TEMP2 = TEMP1*TEMP
TEMP3 = TEMP2*TEMP
FC = FC + TEMP1 / DSCALE( I )
DF = DF + TEMP2
DDF = DDF + TEMP3
30 CONTINUE
F = FINIT + TAU*FC
*
IF( ABS( F ).LE.ZERO )
$ GO TO 60
IF( F .LE. ZERO )THEN
LBD = TAU
ELSE
UBD = TAU
END IF
*
* Iteration begins -- Use Gragg-Thornton-Warner cubic convergent
* scheme
*
* It is not hard to see that
*
* 1) Iterations will go up monotonically
* if FINIT < 0;
*
* 2) Iterations will go down monotonically
* if FINIT > 0.
*
ITER = NITER + 1
*
DO 50 NITER = ITER, MAXIT
*
IF( ORGATI ) THEN
TEMP1 = DSCALE( 2 ) - TAU
TEMP2 = DSCALE( 3 ) - TAU
ELSE
TEMP1 = DSCALE( 1 ) - TAU
TEMP2 = DSCALE( 2 ) - TAU
END IF
A = ( TEMP1+TEMP2 )*F - TEMP1*TEMP2*DF
B = TEMP1*TEMP2*F
C = F - ( TEMP1+TEMP2 )*DF + TEMP1*TEMP2*DDF
TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
A = A / TEMP
B = B / TEMP
C = C / TEMP
IF( C.EQ.ZERO ) THEN
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
IF( F*ETA.GE.ZERO ) THEN
ETA = -F / DF
END IF
*
TAU = TAU + ETA
IF( TAU .LT. LBD .OR. TAU .GT. UBD )
$ TAU = ( LBD + UBD )/TWO
*
FC = ZERO
ERRETM = ZERO
DF = ZERO
DDF = ZERO
DO 40 I = 1, 3
IF ( ( DSCALE( I )-TAU ).NE.ZERO ) THEN
TEMP = ONE / ( DSCALE( I )-TAU )
TEMP1 = ZSCALE( I )*TEMP
TEMP2 = TEMP1*TEMP
TEMP3 = TEMP2*TEMP
TEMP4 = TEMP1 / DSCALE( I )
FC = FC + TEMP4
ERRETM = ERRETM + ABS( TEMP4 )
DF = DF + TEMP2
DDF = DDF + TEMP3
ELSE
GO TO 60
END IF
40 CONTINUE
F = FINIT + TAU*FC
ERRETM = EIGHT*( ABS( FINIT )+ABS( TAU )*ERRETM ) +
$ ABS( TAU )*DF
IF( ABS( F ).LE.EPS*ERRETM )
$ GO TO 60
IF( F .LE. ZERO )THEN
LBD = TAU
ELSE
UBD = TAU
END IF
50 CONTINUE
INFO = 1
60 CONTINUE
*
* Undo scaling
*
IF( SCALE )
$ TAU = TAU*SCLINV
RETURN
*
* End of DLAED6
*
END

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*> \brief \b DLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED7 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed7.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed7.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed7.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
* LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
* PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
* $ QSIZ, TLVLS
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
* $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
* DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
* $ QSTORE( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED7 computes the updated eigensystem of a diagonal
*> matrix after modification by a rank-one symmetric matrix. This
*> routine is used only for the eigenproblem which requires all
*> eigenvalues and optionally eigenvectors of a dense symmetric matrix
*> that has been reduced to tridiagonal form. DLAED1 handles
*> the case in which all eigenvalues and eigenvectors of a symmetric
*> tridiagonal matrix are desired.
*>
*> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
*>
*> where Z = Q**Tu, u is a vector of length N with ones in the
*> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
*>
*> The eigenvectors of the original matrix are stored in Q, and the
*> eigenvalues are in D. The algorithm consists of three stages:
*>
*> The first stage consists of deflating the size of the problem
*> when there are multiple eigenvalues or if there is a zero in
*> the Z vector. For each such occurence the dimension of the
*> secular equation problem is reduced by one. This stage is
*> performed by the routine DLAED8.
*>
*> The second stage consists of calculating the updated
*> eigenvalues. This is done by finding the roots of the secular
*> equation via the routine DLAED4 (as called by DLAED9).
*> This routine also calculates the eigenvectors of the current
*> problem.
*>
*> The final stage consists of computing the updated eigenvectors
*> directly using the updated eigenvalues. The eigenvectors for
*> the current problem are multiplied with the eigenvectors from
*> the overall problem.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> = 0: Compute eigenvalues only.
*> = 1: Compute eigenvectors of original dense symmetric matrix
*> also. On entry, Q contains the orthogonal matrix used
*> to reduce the original matrix to tridiagonal form.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*> QSIZ is INTEGER
*> The dimension of the orthogonal matrix used to reduce
*> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in] TLVLS
*> \verbatim
*> TLVLS is INTEGER
*> The total number of merging levels in the overall divide and
*> conquer tree.
*> \endverbatim
*>
*> \param[in] CURLVL
*> \verbatim
*> CURLVL is INTEGER
*> The current level in the overall merge routine,
*> 0 <= CURLVL <= TLVLS.
*> \endverbatim
*>
*> \param[in] CURPBM
*> \verbatim
*> CURPBM is INTEGER
*> The current problem in the current level in the overall
*> merge routine (counting from upper left to lower right).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the eigenvalues of the rank-1-perturbed matrix.
*> On exit, the eigenvalues of the repaired matrix.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, the eigenvectors of the rank-1-perturbed matrix.
*> On exit, the eigenvectors of the repaired tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[out] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> The permutation which will reintegrate the subproblem just
*> solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
*> will be in ascending order.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The subdiagonal element used to create the rank-1
*> modification.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*> CUTPNT is INTEGER
*> Contains the location of the last eigenvalue in the leading
*> sub-matrix. min(1,N) <= CUTPNT <= N.
*> \endverbatim
*>
*> \param[in,out] QSTORE
*> \verbatim
*> QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
*> Stores eigenvectors of submatrices encountered during
*> divide and conquer, packed together. QPTR points to
*> beginning of the submatrices.
*> \endverbatim
*>
*> \param[in,out] QPTR
*> \verbatim
*> QPTR is INTEGER array, dimension (N+2)
*> List of indices pointing to beginning of submatrices stored
*> in QSTORE. The submatrices are numbered starting at the
*> bottom left of the divide and conquer tree, from left to
*> right and bottom to top.
*> \endverbatim
*>
*> \param[in] PRMPTR
*> \verbatim
*> PRMPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in PERM a
*> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
*> indicates the size of the permutation and also the size of
*> the full, non-deflated problem.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*> PERM is INTEGER array, dimension (N lg N)
*> Contains the permutations (from deflation and sorting) to be
*> applied to each eigenblock.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in GIVCOL a
*> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
*> indicates the number of Givens rotations.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension (2, N lg N)
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
*> Each number indicates the S value to be used in the
*> corresponding Givens rotation.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
$ LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
$ PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
$ QSIZ, TLVLS
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
$ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
$ QSTORE( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP,
$ IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLAED8, DLAED9, DLAEDA, DLAMRG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
INFO = -4
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED7', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* The following values are for bookkeeping purposes only. They are
* integer pointers which indicate the portion of the workspace
* used by a particular array in DLAED8 and DLAED9.
*
IF( ICOMPQ.EQ.1 ) THEN
LDQ2 = QSIZ
ELSE
LDQ2 = N
END IF
*
IZ = 1
IDLMDA = IZ + N
IW = IDLMDA + N
IQ2 = IW + N
IS = IQ2 + N*LDQ2
*
INDX = 1
INDXC = INDX + N
COLTYP = INDXC + N
INDXP = COLTYP + N
*
* Form the z-vector which consists of the last row of Q_1 and the
* first row of Q_2.
*
PTR = 1 + 2**TLVLS
DO 10 I = 1, CURLVL - 1
PTR = PTR + 2**( TLVLS-I )
10 CONTINUE
CURR = PTR + CURPBM
CALL DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
$ GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ),
$ WORK( IZ+N ), INFO )
*
* When solving the final problem, we no longer need the stored data,
* so we will overwrite the data from this level onto the previously
* used storage space.
*
IF( CURLVL.EQ.TLVLS ) THEN
QPTR( CURR ) = 1
PRMPTR( CURR ) = 1
GIVPTR( CURR ) = 1
END IF
*
* Sort and Deflate eigenvalues.
*
CALL DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
$ WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2,
$ WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ),
$ GIVCOL( 1, GIVPTR( CURR ) ),
$ GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ),
$ IWORK( INDX ), INFO )
PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N
GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR )
*
* Solve Secular Equation.
*
IF( K.NE.0 ) THEN
CALL DLAED9( K, 1, K, N, D, WORK( IS ), K, RHO, WORK( IDLMDA ),
$ WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO )
IF( INFO.NE.0 )
$ GO TO 30
IF( ICOMPQ.EQ.1 ) THEN
CALL DGEMM( 'N', 'N', QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2,
$ QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ )
END IF
QPTR( CURR+1 ) = QPTR( CURR ) + K**2
*
* Prepare the INDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
ELSE
QPTR( CURR+1 ) = QPTR( CURR )
DO 20 I = 1, N
INDXQ( I ) = I
20 CONTINUE
END IF
*
30 CONTINUE
RETURN
*
* End of DLAED7
*
END

524
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*> \brief \b DLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED8 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed8.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed8.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed8.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
* CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
* GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
* .. Scalar Arguments ..
* INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
* $ QSIZ
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
* $ INDXQ( * ), PERM( * )
* DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ),
* $ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED8 merges the two sets of eigenvalues together into a single
*> sorted set. Then it tries to deflate the size of the problem.
*> There are two ways in which deflation can occur: when two or more
*> eigenvalues are close together or if there is a tiny element in the
*> Z vector. For each such occurrence the order of the related secular
*> equation problem is reduced by one.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> = 0: Compute eigenvalues only.
*> = 1: Compute eigenvectors of original dense symmetric matrix
*> also. On entry, Q contains the orthogonal matrix used
*> to reduce the original matrix to tridiagonal form.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> The number of non-deflated eigenvalues, and the order of the
*> related secular equation.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*> QSIZ is INTEGER
*> The dimension of the orthogonal matrix used to reduce
*> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the eigenvalues of the two submatrices to be
*> combined. On exit, the trailing (N-K) updated eigenvalues
*> (those which were deflated) sorted into increasing order.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> If ICOMPQ = 0, Q is not referenced. Otherwise,
*> on entry, Q contains the eigenvectors of the partially solved
*> system which has been previously updated in matrix
*> multiplies with other partially solved eigensystems.
*> On exit, Q contains the trailing (N-K) updated eigenvectors
*> (those which were deflated) in its last N-K columns.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> The permutation which separately sorts the two sub-problems
*> in D into ascending order. Note that elements in the second
*> half of this permutation must first have CUTPNT added to
*> their values in order to be accurate.
*> \endverbatim
*>
*> \param[in,out] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> On entry, the off-diagonal element associated with the rank-1
*> cut which originally split the two submatrices which are now
*> being recombined.
*> On exit, RHO has been modified to the value required by
*> DLAED3.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*> CUTPNT is INTEGER
*> The location of the last eigenvalue in the leading
*> sub-matrix. min(1,N) <= CUTPNT <= N.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (N)
*> On entry, Z contains the updating vector (the last row of
*> the first sub-eigenvector matrix and the first row of the
*> second sub-eigenvector matrix).
*> On exit, the contents of Z are destroyed by the updating
*> process.
*> \endverbatim
*>
*> \param[out] DLAMDA
*> \verbatim
*> DLAMDA is DOUBLE PRECISION array, dimension (N)
*> A copy of the first K eigenvalues which will be used by
*> DLAED3 to form the secular equation.
*> \endverbatim
*>
*> \param[out] Q2
*> \verbatim
*> Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
*> If ICOMPQ = 0, Q2 is not referenced. Otherwise,
*> a copy of the first K eigenvectors which will be used by
*> DLAED7 in a matrix multiply (DGEMM) to update the new
*> eigenvectors.
*> \endverbatim
*>
*> \param[in] LDQ2
*> \verbatim
*> LDQ2 is INTEGER
*> The leading dimension of the array Q2. LDQ2 >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first k values of the final deflation-altered z-vector and
*> will be passed to DLAED3.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*> PERM is INTEGER array, dimension (N)
*> The permutations (from deflation and sorting) to be applied
*> to each eigenblock.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER
*> The number of Givens rotations which took place in this
*> subproblem.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension (2, N)
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension (2, N)
*> Each number indicates the S value to be used in the
*> corresponding Givens rotation.
*> \endverbatim
*>
*> \param[out] INDXP
*> \verbatim
*> INDXP is INTEGER array, dimension (N)
*> The permutation used to place deflated values of D at the end
*> of the array. INDXP(1:K) points to the nondeflated D-values
*> and INDXP(K+1:N) points to the deflated eigenvalues.
*> \endverbatim
*>
*> \param[out] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> The permutation used to sort the contents of D into ascending
*> order.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
$ CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
$ GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
$ QSIZ
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
$ INDXQ( * ), PERM( * )
DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ),
$ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, EIGHT = 8.0D0 )
* ..
* .. Local Scalars ..
*
INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
DOUBLE PRECISION C, EPS, S, T, TAU, TOL
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL IDAMAX, DLAMCH, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DLAMRG, DROT, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
INFO = -4
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
INFO = -10
ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
INFO = -14
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED8', -INFO )
RETURN
END IF
*
* Need to initialize GIVPTR to O here in case of quick exit
* to prevent an unspecified code behavior (usually sigfault)
* when IWORK array on entry to *stedc is not zeroed
* (or at least some IWORK entries which used in *laed7 for GIVPTR).
*
GIVPTR = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
N1 = CUTPNT
N2 = N - N1
N1P1 = N1 + 1
*
IF( RHO.LT.ZERO ) THEN
CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
END IF
*
* Normalize z so that norm(z) = 1
*
T = ONE / SQRT( TWO )
DO 10 J = 1, N
INDX( J ) = J
10 CONTINUE
CALL DSCAL( N, T, Z, 1 )
RHO = ABS( TWO*RHO )
*
* Sort the eigenvalues into increasing order
*
DO 20 I = CUTPNT + 1, N
INDXQ( I ) = INDXQ( I ) + CUTPNT
20 CONTINUE
DO 30 I = 1, N
DLAMDA( I ) = D( INDXQ( I ) )
W( I ) = Z( INDXQ( I ) )
30 CONTINUE
I = 1
J = CUTPNT + 1
CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDX )
DO 40 I = 1, N
D( I ) = DLAMDA( INDX( I ) )
Z( I ) = W( INDX( I ) )
40 CONTINUE
*
* Calculate the allowable deflation tolerence
*
IMAX = IDAMAX( N, Z, 1 )
JMAX = IDAMAX( N, D, 1 )
EPS = DLAMCH( 'Epsilon' )
TOL = EIGHT*EPS*ABS( D( JMAX ) )
*
* If the rank-1 modifier is small enough, no more needs to be done
* except to reorganize Q so that its columns correspond with the
* elements in D.
*
IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
K = 0
IF( ICOMPQ.EQ.0 ) THEN
DO 50 J = 1, N
PERM( J ) = INDXQ( INDX( J ) )
50 CONTINUE
ELSE
DO 60 J = 1, N
PERM( J ) = INDXQ( INDX( J ) )
CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
60 CONTINUE
CALL DLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ),
$ LDQ )
END IF
RETURN
END IF
*
* If there are multiple eigenvalues then the problem deflates. Here
* the number of equal eigenvalues are found. As each equal
* eigenvalue is found, an elementary reflector is computed to rotate
* the corresponding eigensubspace so that the corresponding
* components of Z are zero in this new basis.
*
K = 0
K2 = N + 1
DO 70 J = 1, N
IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
INDXP( K2 ) = J
IF( J.EQ.N )
$ GO TO 110
ELSE
JLAM = J
GO TO 80
END IF
70 CONTINUE
80 CONTINUE
J = J + 1
IF( J.GT.N )
$ GO TO 100
IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
INDXP( K2 ) = J
ELSE
*
* Check if eigenvalues are close enough to allow deflation.
*
S = Z( JLAM )
C = Z( J )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = DLAPY2( C, S )
T = D( J ) - D( JLAM )
C = C / TAU
S = -S / TAU
IF( ABS( T*C*S ).LE.TOL ) THEN
*
* Deflation is possible.
*
Z( J ) = TAU
Z( JLAM ) = ZERO
*
* Record the appropriate Givens rotation
*
GIVPTR = GIVPTR + 1
GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) )
GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) )
GIVNUM( 1, GIVPTR ) = C
GIVNUM( 2, GIVPTR ) = S
IF( ICOMPQ.EQ.1 ) THEN
CALL DROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
$ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
END IF
T = D( JLAM )*C*C + D( J )*S*S
D( J ) = D( JLAM )*S*S + D( J )*C*C
D( JLAM ) = T
K2 = K2 - 1
I = 1
90 CONTINUE
IF( K2+I.LE.N ) THEN
IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN
INDXP( K2+I-1 ) = INDXP( K2+I )
INDXP( K2+I ) = JLAM
I = I + 1
GO TO 90
ELSE
INDXP( K2+I-1 ) = JLAM
END IF
ELSE
INDXP( K2+I-1 ) = JLAM
END IF
JLAM = J
ELSE
K = K + 1
W( K ) = Z( JLAM )
DLAMDA( K ) = D( JLAM )
INDXP( K ) = JLAM
JLAM = J
END IF
END IF
GO TO 80
100 CONTINUE
*
* Record the last eigenvalue.
*
K = K + 1
W( K ) = Z( JLAM )
DLAMDA( K ) = D( JLAM )
INDXP( K ) = JLAM
*
110 CONTINUE
*
* Sort the eigenvalues and corresponding eigenvectors into DLAMDA
* and Q2 respectively. The eigenvalues/vectors which were not
* deflated go into the first K slots of DLAMDA and Q2 respectively,
* while those which were deflated go into the last N - K slots.
*
IF( ICOMPQ.EQ.0 ) THEN
DO 120 J = 1, N
JP = INDXP( J )
DLAMDA( J ) = D( JP )
PERM( J ) = INDXQ( INDX( JP ) )
120 CONTINUE
ELSE
DO 130 J = 1, N
JP = INDXP( J )
DLAMDA( J ) = D( JP )
PERM( J ) = INDXQ( INDX( JP ) )
CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
130 CONTINUE
END IF
*
* The deflated eigenvalues and their corresponding vectors go back
* into the last N - K slots of D and Q respectively.
*
IF( K.LT.N ) THEN
IF( ICOMPQ.EQ.0 ) THEN
CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
ELSE
CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
CALL DLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2,
$ Q( 1, K+1 ), LDQ )
END IF
END IF
*
RETURN
*
* End of DLAED8
*
END

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*> \brief \b DLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED9 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed9.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed9.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed9.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
* S, LDS, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
* $ W( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED9 finds the roots of the secular equation, as defined by the
*> values in D, Z, and RHO, between KSTART and KSTOP. It makes the
*> appropriate calls to DLAED4 and then stores the new matrix of
*> eigenvectors for use in calculating the next level of Z vectors.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of terms in the rational function to be solved by
*> DLAED4. K >= 0.
*> \endverbatim
*>
*> \param[in] KSTART
*> \verbatim
*> KSTART is INTEGER
*> \endverbatim
*>
*> \param[in] KSTOP
*> \verbatim
*> KSTOP is INTEGER
*> The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
*> are to be computed. 1 <= KSTART <= KSTOP <= K.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the Q matrix.
*> N >= K (delation may result in N > K).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> D(I) contains the updated eigenvalues
*> for KSTART <= I <= KSTOP.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max( 1, N ).
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The value of the parameter in the rank one update equation.
*> RHO >= 0 required.
*> \endverbatim
*>
*> \param[in] DLAMDA
*> \verbatim
*> DLAMDA is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the old roots
*> of the deflated updating problem. These are the poles
*> of the secular equation.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the components
*> of the deflation-adjusted updating vector.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (LDS, K)
*> Will contain the eigenvectors of the repaired matrix which
*> will be stored for subsequent Z vector calculation and
*> multiplied by the previously accumulated eigenvectors
*> to update the system.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*> LDS is INTEGER
*> The leading dimension of S. LDS >= max( 1, K ).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
$ S, LDS, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
$ W( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION TEMP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3, DNRM2
EXTERNAL DLAMC3, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAED4, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( K.LT.0 ) THEN
INFO = -1
ELSE IF( KSTART.LT.1 .OR. KSTART.GT.MAX( 1, K ) ) THEN
INFO = -2
ELSE IF( MAX( 1, KSTOP ).LT.KSTART .OR. KSTOP.GT.MAX( 1, K ) )
$ THEN
INFO = -3
ELSE IF( N.LT.K ) THEN
INFO = -4
ELSE IF( LDQ.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDS.LT.MAX( 1, K ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED9', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 )
$ RETURN
*
* Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
* be computed with high relative accuracy (barring over/underflow).
* This is a problem on machines without a guard digit in
* add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
* The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
* which on any of these machines zeros out the bottommost
* bit of DLAMDA(I) if it is 1; this makes the subsequent
* subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
* occurs. On binary machines with a guard digit (almost all
* machines) it does not change DLAMDA(I) at all. On hexadecimal
* and decimal machines with a guard digit, it slightly
* changes the bottommost bits of DLAMDA(I). It does not account
* for hexadecimal or decimal machines without guard digits
* (we know of none). We use a subroutine call to compute
* 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
* this code.
*
DO 10 I = 1, N
DLAMDA( I ) = DLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
10 CONTINUE
*
DO 20 J = KSTART, KSTOP
CALL DLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO )
*
* If the zero finder fails, the computation is terminated.
*
IF( INFO.NE.0 )
$ GO TO 120
20 CONTINUE
*
IF( K.EQ.1 .OR. K.EQ.2 ) THEN
DO 40 I = 1, K
DO 30 J = 1, K
S( J, I ) = Q( J, I )
30 CONTINUE
40 CONTINUE
GO TO 120
END IF
*
* Compute updated W.
*
CALL DCOPY( K, W, 1, S, 1 )
*
* Initialize W(I) = Q(I,I)
*
CALL DCOPY( K, Q, LDQ+1, W, 1 )
DO 70 J = 1, K
DO 50 I = 1, J - 1
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
50 CONTINUE
DO 60 I = J + 1, K
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
60 CONTINUE
70 CONTINUE
DO 80 I = 1, K
W( I ) = SIGN( SQRT( -W( I ) ), S( I, 1 ) )
80 CONTINUE
*
* Compute eigenvectors of the modified rank-1 modification.
*
DO 110 J = 1, K
DO 90 I = 1, K
Q( I, J ) = W( I ) / Q( I, J )
90 CONTINUE
TEMP = DNRM2( K, Q( 1, J ), 1 )
DO 100 I = 1, K
S( I, J ) = Q( I, J ) / TEMP
100 CONTINUE
110 CONTINUE
*
120 CONTINUE
RETURN
*
* End of DLAED9
*
END

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*> \brief \b DLAEDA used by sstedc. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEDA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaeda.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaeda.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaeda.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
* GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO )
*
* .. Scalar Arguments ..
* INTEGER CURLVL, CURPBM, INFO, N, TLVLS
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( 2, * ), GIVPTR( * ), PERM( * ),
* $ PRMPTR( * ), QPTR( * )
* DOUBLE PRECISION GIVNUM( 2, * ), Q( * ), Z( * ), ZTEMP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAEDA computes the Z vector corresponding to the merge step in the
*> CURLVLth step of the merge process with TLVLS steps for the CURPBMth
*> problem.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] TLVLS
*> \verbatim
*> TLVLS is INTEGER
*> The total number of merging levels in the overall divide and
*> conquer tree.
*> \endverbatim
*>
*> \param[in] CURLVL
*> \verbatim
*> CURLVL is INTEGER
*> The current level in the overall merge routine,
*> 0 <= curlvl <= tlvls.
*> \endverbatim
*>
*> \param[in] CURPBM
*> \verbatim
*> CURPBM is INTEGER
*> The current problem in the current level in the overall
*> merge routine (counting from upper left to lower right).
*> \endverbatim
*>
*> \param[in] PRMPTR
*> \verbatim
*> PRMPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in PERM a
*> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
*> indicates the size of the permutation and incidentally the
*> size of the full, non-deflated problem.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*> PERM is INTEGER array, dimension (N lg N)
*> Contains the permutations (from deflation and sorting) to be
*> applied to each eigenblock.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in GIVCOL a
*> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
*> indicates the number of Givens rotations.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension (2, N lg N)
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
*> Each number indicates the S value to be used in the
*> corresponding Givens rotation.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (N**2)
*> Contains the square eigenblocks from previous levels, the
*> starting positions for blocks are given by QPTR.
*> \endverbatim
*>
*> \param[in] QPTR
*> \verbatim
*> QPTR is INTEGER array, dimension (N+2)
*> Contains a list of pointers which indicate where in Q an
*> eigenblock is stored. SQRT( QPTR(i+1) - QPTR(i) ) indicates
*> the size of the block.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (N)
*> On output this vector contains the updating vector (the last
*> row of the first sub-eigenvector matrix and the first row of
*> the second sub-eigenvector matrix).
*> \endverbatim
*>
*> \param[out] ZTEMP
*> \verbatim
*> ZTEMP is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
$ GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER CURLVL, CURPBM, INFO, N, TLVLS
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), GIVPTR( * ), PERM( * ),
$ PRMPTR( * ), QPTR( * )
DOUBLE PRECISION GIVNUM( 2, * ), Q( * ), Z( * ), ZTEMP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER BSIZ1, BSIZ2, CURR, I, K, MID, PSIZ1, PSIZ2,
$ PTR, ZPTR1
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMV, DROT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, INT, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -1
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAEDA', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine location of first number in second half.
*
MID = N / 2 + 1
*
* Gather last/first rows of appropriate eigenblocks into center of Z
*
PTR = 1
*
* Determine location of lowest level subproblem in the full storage
* scheme
*
CURR = PTR + CURPBM*2**CURLVL + 2**( CURLVL-1 ) - 1
*
* Determine size of these matrices. We add HALF to the value of
* the SQRT in case the machine underestimates one of these square
* roots.
*
BSIZ1 = INT( HALF+SQRT( DBLE( QPTR( CURR+1 )-QPTR( CURR ) ) ) )
BSIZ2 = INT( HALF+SQRT( DBLE( QPTR( CURR+2 )-QPTR( CURR+1 ) ) ) )
DO 10 K = 1, MID - BSIZ1 - 1
Z( K ) = ZERO
10 CONTINUE
CALL DCOPY( BSIZ1, Q( QPTR( CURR )+BSIZ1-1 ), BSIZ1,
$ Z( MID-BSIZ1 ), 1 )
CALL DCOPY( BSIZ2, Q( QPTR( CURR+1 ) ), BSIZ2, Z( MID ), 1 )
DO 20 K = MID + BSIZ2, N
Z( K ) = ZERO
20 CONTINUE
*
* Loop through remaining levels 1 -> CURLVL applying the Givens
* rotations and permutation and then multiplying the center matrices
* against the current Z.
*
PTR = 2**TLVLS + 1
DO 70 K = 1, CURLVL - 1
CURR = PTR + CURPBM*2**( CURLVL-K ) + 2**( CURLVL-K-1 ) - 1
PSIZ1 = PRMPTR( CURR+1 ) - PRMPTR( CURR )
PSIZ2 = PRMPTR( CURR+2 ) - PRMPTR( CURR+1 )
ZPTR1 = MID - PSIZ1
*
* Apply Givens at CURR and CURR+1
*
DO 30 I = GIVPTR( CURR ), GIVPTR( CURR+1 ) - 1
CALL DROT( 1, Z( ZPTR1+GIVCOL( 1, I )-1 ), 1,
$ Z( ZPTR1+GIVCOL( 2, I )-1 ), 1, GIVNUM( 1, I ),
$ GIVNUM( 2, I ) )
30 CONTINUE
DO 40 I = GIVPTR( CURR+1 ), GIVPTR( CURR+2 ) - 1
CALL DROT( 1, Z( MID-1+GIVCOL( 1, I ) ), 1,
$ Z( MID-1+GIVCOL( 2, I ) ), 1, GIVNUM( 1, I ),
$ GIVNUM( 2, I ) )
40 CONTINUE
PSIZ1 = PRMPTR( CURR+1 ) - PRMPTR( CURR )
PSIZ2 = PRMPTR( CURR+2 ) - PRMPTR( CURR+1 )
DO 50 I = 0, PSIZ1 - 1
ZTEMP( I+1 ) = Z( ZPTR1+PERM( PRMPTR( CURR )+I )-1 )
50 CONTINUE
DO 60 I = 0, PSIZ2 - 1
ZTEMP( PSIZ1+I+1 ) = Z( MID+PERM( PRMPTR( CURR+1 )+I )-1 )
60 CONTINUE
*
* Multiply Blocks at CURR and CURR+1
*
* Determine size of these matrices. We add HALF to the value of
* the SQRT in case the machine underestimates one of these
* square roots.
*
BSIZ1 = INT( HALF+SQRT( DBLE( QPTR( CURR+1 )-QPTR( CURR ) ) ) )
BSIZ2 = INT( HALF+SQRT( DBLE( QPTR( CURR+2 )-QPTR( CURR+
$ 1 ) ) ) )
IF( BSIZ1.GT.0 ) THEN
CALL DGEMV( 'T', BSIZ1, BSIZ1, ONE, Q( QPTR( CURR ) ),
$ BSIZ1, ZTEMP( 1 ), 1, ZERO, Z( ZPTR1 ), 1 )
END IF
CALL DCOPY( PSIZ1-BSIZ1, ZTEMP( BSIZ1+1 ), 1, Z( ZPTR1+BSIZ1 ),
$ 1 )
IF( BSIZ2.GT.0 ) THEN
CALL DGEMV( 'T', BSIZ2, BSIZ2, ONE, Q( QPTR( CURR+1 ) ),
$ BSIZ2, ZTEMP( PSIZ1+1 ), 1, ZERO, Z( MID ), 1 )
END IF
CALL DCOPY( PSIZ2-BSIZ2, ZTEMP( PSIZ1+BSIZ2+1 ), 1,
$ Z( MID+BSIZ2 ), 1 )
*
PTR = PTR + 2**( TLVLS-K )
70 CONTINUE
*
RETURN
*
* End of DLAEDA
*
END

238
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*> \brief \b DLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAEV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaev2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaev2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaev2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, CS1, RT1, RT2, SN1
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
*> [ A B ]
*> [ B C ].
*> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
*> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
*> eigenvector for RT1, giving the decomposition
*>
*> [ CS1 SN1 ] [ A B ] [ CS1 -SN1 ] = [ RT1 0 ]
*> [-SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ].
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION
*> The (1,2) element and the conjugate of the (2,1) element of
*> the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] RT1
*> \verbatim
*> RT1 is DOUBLE PRECISION
*> The eigenvalue of larger absolute value.
*> \endverbatim
*>
*> \param[out] RT2
*> \verbatim
*> RT2 is DOUBLE PRECISION
*> The eigenvalue of smaller absolute value.
*> \endverbatim
*>
*> \param[out] CS1
*> \verbatim
*> CS1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SN1
*> \verbatim
*> SN1 is DOUBLE PRECISION
*> The vector (CS1, SN1) is a unit right eigenvector for RT1.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> RT1 is accurate to a few ulps barring over/underflow.
*>
*> RT2 may be inaccurate if there is massive cancellation in the
*> determinant A*C-B*B; higher precision or correctly rounded or
*> correctly truncated arithmetic would be needed to compute RT2
*> accurately in all cases.
*>
*> CS1 and SN1 are accurate to a few ulps barring over/underflow.
*>
*> Overflow is possible only if RT1 is within a factor of 5 of overflow.
*> Underflow is harmless if the input data is 0 or exceeds
*> underflow_threshold / macheps.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, CS1, RT1, RT2, SN1
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
* ..
* .. Local Scalars ..
INTEGER SGN1, SGN2
DOUBLE PRECISION AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
$ TB, TN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
* Compute the eigenvalues
*
SM = A + C
DF = A - C
ADF = ABS( DF )
TB = B + B
AB = ABS( TB )
IF( ABS( A ).GT.ABS( C ) ) THEN
ACMX = A
ACMN = C
ELSE
ACMX = C
ACMN = A
END IF
IF( ADF.GT.AB ) THEN
RT = ADF*SQRT( ONE+( AB / ADF )**2 )
ELSE IF( ADF.LT.AB ) THEN
RT = AB*SQRT( ONE+( ADF / AB )**2 )
ELSE
*
* Includes case AB=ADF=0
*
RT = AB*SQRT( TWO )
END IF
IF( SM.LT.ZERO ) THEN
RT1 = HALF*( SM-RT )
SGN1 = -1
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE IF( SM.GT.ZERO ) THEN
RT1 = HALF*( SM+RT )
SGN1 = 1
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE
*
* Includes case RT1 = RT2 = 0
*
RT1 = HALF*RT
RT2 = -HALF*RT
SGN1 = 1
END IF
*
* Compute the eigenvector
*
IF( DF.GE.ZERO ) THEN
CS = DF + RT
SGN2 = 1
ELSE
CS = DF - RT
SGN2 = -1
END IF
ACS = ABS( CS )
IF( ACS.GT.AB ) THEN
CT = -TB / CS
SN1 = ONE / SQRT( ONE+CT*CT )
CS1 = CT*SN1
ELSE
IF( AB.EQ.ZERO ) THEN
CS1 = ONE
SN1 = ZERO
ELSE
TN = -CS / TB
CS1 = ONE / SQRT( ONE+TN*TN )
SN1 = TN*CS1
END IF
END IF
IF( SGN1.EQ.SGN2 ) THEN
TN = CS1
CS1 = -SN1
SN1 = TN
END IF
RETURN
*
* End of DLAEV2
*
END

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*> \brief \b DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAMRG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlamrg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlamrg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlamrg.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAMRG( N1, N2, A, DTRD1, DTRD2, INDEX )
*
* .. Scalar Arguments ..
* INTEGER DTRD1, DTRD2, N1, N2
* ..
* .. Array Arguments ..
* INTEGER INDEX( * )
* DOUBLE PRECISION A( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAMRG will create a permutation list which will merge the elements
*> of A (which is composed of two independently sorted sets) into a
*> single set which is sorted in ascending order.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> These arguements contain the respective lengths of the two
*> sorted lists to be merged.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (N1+N2)
*> The first N1 elements of A contain a list of numbers which
*> are sorted in either ascending or descending order. Likewise
*> for the final N2 elements.
*> \endverbatim
*>
*> \param[in] DTRD1
*> \verbatim
*> DTRD1 is INTEGER
*> \endverbatim
*>
*> \param[in] DTRD2
*> \verbatim
*> DTRD2 is INTEGER
*> These are the strides to be taken through the array A.
*> Allowable strides are 1 and -1. They indicate whether a
*> subset of A is sorted in ascending (DTRDx = 1) or descending
*> (DTRDx = -1) order.
*> \endverbatim
*>
*> \param[out] INDEX
*> \verbatim
*> INDEX is INTEGER array, dimension (N1+N2)
*> On exit this array will contain a permutation such that
*> if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
*> sorted in ascending order.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DLAMRG( N1, N2, A, DTRD1, DTRD2, INDEX )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER DTRD1, DTRD2, N1, N2
* ..
* .. Array Arguments ..
INTEGER INDEX( * )
DOUBLE PRECISION A( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IND1, IND2, N1SV, N2SV
* ..
* .. Executable Statements ..
*
N1SV = N1
N2SV = N2
IF( DTRD1.GT.0 ) THEN
IND1 = 1
ELSE
IND1 = N1
END IF
IF( DTRD2.GT.0 ) THEN
IND2 = 1 + N1
ELSE
IND2 = N1 + N2
END IF
I = 1
* while ( (N1SV > 0) & (N2SV > 0) )
10 CONTINUE
IF( N1SV.GT.0 .AND. N2SV.GT.0 ) THEN
IF( A( IND1 ).LE.A( IND2 ) ) THEN
INDEX( I ) = IND1
I = I + 1
IND1 = IND1 + DTRD1
N1SV = N1SV - 1
ELSE
INDEX( I ) = IND2
I = I + 1
IND2 = IND2 + DTRD2
N2SV = N2SV - 1
END IF
GO TO 10
END IF
* end while
IF( N1SV.EQ.0 ) THEN
DO 20 N1SV = 1, N2SV
INDEX( I ) = IND2
I = I + 1
IND2 = IND2 + DTRD2
20 CONTINUE
ELSE
* N2SV .EQ. 0
DO 30 N2SV = 1, N1SV
INDEX( I ) = IND1
I = I + 1
IND1 = IND1 + DTRD1
30 CONTINUE
END IF
*
RETURN
*
* End of DLAMRG
*
END

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*> \brief \b DLANST returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlanst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlanst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlanst.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANST( NORM, N, D, E )
*
* .. Scalar Arguments ..
* CHARACTER NORM
* INTEGER N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANST returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real symmetric tridiagonal matrix A.
*> \endverbatim
*>
*> \return DLANST
*> \verbatim
*>
*> DLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANST as described
*> above.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANST is
*> set to zero.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) sub-diagonal or super-diagonal elements of A.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANST( NORM, N, D, E )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION ANORM, SCALE, SUM
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( N.LE.0 ) THEN
ANORM = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
ANORM = ABS( D( N ) )
DO 10 I = 1, N - 1
SUM = ABS( D( I ) )
IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
SUM = ABS( E( I ) )
IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
10 CONTINUE
ELSE IF( LSAME( NORM, 'O' ) .OR. NORM.EQ.'1' .OR.
$ LSAME( NORM, 'I' ) ) THEN
*
* Find norm1(A).
*
IF( N.EQ.1 ) THEN
ANORM = ABS( D( 1 ) )
ELSE
ANORM = ABS( D( 1 ) )+ABS( E( 1 ) )
SUM = ABS( E( N-1 ) )+ABS( D( N ) )
IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
DO 20 I = 2, N - 1
SUM = ABS( D( I ) )+ABS( E( I ) )+ABS( E( I-1 ) )
IF( ANORM .LT. SUM .OR. DISNAN( SUM ) ) ANORM = SUM
20 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
IF( N.GT.1 ) THEN
CALL DLASSQ( N-1, E, 1, SCALE, SUM )
SUM = 2*SUM
END IF
CALL DLASSQ( N, D, 1, SCALE, SUM )
ANORM = SCALE*SQRT( SUM )
END IF
*
DLANST = ANORM
RETURN
*
* End of DLANST
*
END

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*> \brief \b DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANSY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansy.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLANSY( NORM, UPLO, N, A, LDA, WORK )
*
* .. Scalar Arguments ..
* CHARACTER NORM, UPLO
* INTEGER LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANSY returns the value of the one norm, or the Frobenius norm, or
*> the infinity norm, or the element of largest absolute value of a
*> real symmetric matrix A.
*> \endverbatim
*>
*> \return DLANSY
*> \verbatim
*>
*> DLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*> (
*> ( norm1(A), NORM = '1', 'O' or 'o'
*> (
*> ( normI(A), NORM = 'I' or 'i'
*> (
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*>
*> where norm1 denotes the one norm of a matrix (maximum column sum),
*> normI denotes the infinity norm of a matrix (maximum row sum) and
*> normF denotes the Frobenius norm of a matrix (square root of sum of
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NORM
*> \verbatim
*> NORM is CHARACTER*1
*> Specifies the value to be returned in DLANSY as described
*> above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is to be referenced.
*> = 'U': Upper triangular part of A is referenced
*> = 'L': Lower triangular part of A is referenced
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0. When N = 0, DLANSY is
*> set to zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The symmetric matrix A. If UPLO = 'U', the leading n by n
*> upper triangular part of A contains the upper triangular part
*> of the matrix A, and the strictly lower triangular part of A
*> is not referenced. If UPLO = 'L', the leading n by n lower
*> triangular part of A contains the lower triangular part of
*> the matrix A, and the strictly upper triangular part of A is
*> not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(N,1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*> WORK is not referenced.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleSYauxiliary
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLANSY( NORM, UPLO, N, A, LDA, WORK )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL DLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = 1, J
SUM = ABS( A( I, J ) )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1, N
DO 30 I = J, N
SUM = ABS( A( I, J ) )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
30 CONTINUE
40 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR.
$ ( NORM.EQ.'1' ) ) THEN
*
* Find normI(A) ( = norm1(A), since A is symmetric).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
SUM = ZERO
DO 50 I = 1, J - 1
ABSA = ABS( A( I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
50 CONTINUE
WORK( J ) = SUM + ABS( A( J, J ) )
60 CONTINUE
DO 70 I = 1, N
SUM = WORK( I )
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
70 CONTINUE
ELSE
DO 80 I = 1, N
WORK( I ) = ZERO
80 CONTINUE
DO 100 J = 1, N
SUM = WORK( J ) + ABS( A( J, J ) )
DO 90 I = J + 1, N
ABSA = ABS( A( I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
90 CONTINUE
IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
100 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 2, N
CALL DLASSQ( J-1, A( 1, J ), 1, SCALE, SUM )
110 CONTINUE
ELSE
DO 120 J = 1, N - 1
CALL DLASSQ( N-J, A( J+1, J ), 1, SCALE, SUM )
120 CONTINUE
END IF
SUM = 2*SUM
CALL DLASSQ( N, A, LDA+1, SCALE, SUM )
VALUE = SCALE*SQRT( SUM )
END IF
*
DLANSY = VALUE
RETURN
*
* End of DLANSY
*
END

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*> \brief \b DLAPY2 returns sqrt(x2+y2).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPY2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapy2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapy2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapy2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLAPY2( X, Y )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION X, Y
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary
*> overflow.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*> Y is DOUBLE PRECISION
*> X and Y specify the values x and y.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLAPY2( X, Y )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION X, Y
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION W, XABS, YABS, Z
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
XABS = ABS( X )
YABS = ABS( Y )
W = MAX( XABS, YABS )
Z = MIN( XABS, YABS )
IF( Z.EQ.ZERO ) THEN
DLAPY2 = W
ELSE
DLAPY2 = W*SQRT( ONE+( Z / W )**2 )
END IF
RETURN
*
* End of DLAPY2
*
END

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*> \brief \b DLARF applies an elementary reflector to a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* .. Scalar Arguments ..
* CHARACTER SIDE
* INTEGER INCV, LDC, M, N
* DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARF applies a real elementary reflector H to a real m by n matrix
*> C, from either the left or the right. H is represented in the form
*>
*> H = I - tau * v * v**T
*>
*> where tau is a real scalar and v is a real vector.
*>
*> If tau = 0, then H is taken to be the unit matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': form H * C
*> = 'R': form C * H
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (1 + (M-1)*abs(INCV)) if SIDE = 'L'
*> or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
*> The vector v in the representation of H. V is not used if
*> TAU = 0.
*> \endverbatim
*>
*> \param[in] INCV
*> \verbatim
*> INCV is INTEGER
*> The increment between elements of v. INCV <> 0.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau in the representation of H.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by the matrix H * C if SIDE = 'L',
*> or C * H if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L'
*> or (M) if SIDE = 'R'
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER INCV, LDC, M, N
DOUBLE PRECISION TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL APPLYLEFT
INTEGER I, LASTV, LASTC
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILADLR, ILADLC
EXTERNAL LSAME, ILADLR, ILADLC
* ..
* .. Executable Statements ..
*
APPLYLEFT = LSAME( SIDE, 'L' )
LASTV = 0
LASTC = 0
IF( TAU.NE.ZERO ) THEN
! Set up variables for scanning V. LASTV begins pointing to the end
! of V.
IF( APPLYLEFT ) THEN
LASTV = M
ELSE
LASTV = N
END IF
IF( INCV.GT.0 ) THEN
I = 1 + (LASTV-1) * INCV
ELSE
I = 1
END IF
! Look for the last non-zero row in V.
DO WHILE( LASTV.GT.0 .AND. V( I ).EQ.ZERO )
LASTV = LASTV - 1
I = I - INCV
END DO
IF( APPLYLEFT ) THEN
! Scan for the last non-zero column in C(1:lastv,:).
LASTC = ILADLC(LASTV, N, C, LDC)
ELSE
! Scan for the last non-zero row in C(:,1:lastv).
LASTC = ILADLR(M, LASTV, C, LDC)
END IF
END IF
! Note that lastc.eq.0 renders the BLAS operations null; no special
! case is needed at this level.
IF( APPLYLEFT ) THEN
*
* Form H * C
*
IF( LASTV.GT.0 ) THEN
*
* w(1:lastc,1) := C(1:lastv,1:lastc)**T * v(1:lastv,1)
*
CALL DGEMV( 'Transpose', LASTV, LASTC, ONE, C, LDC, V, INCV,
$ ZERO, WORK, 1 )
*
* C(1:lastv,1:lastc) := C(...) - v(1:lastv,1) * w(1:lastc,1)**T
*
CALL DGER( LASTV, LASTC, -TAU, V, INCV, WORK, 1, C, LDC )
END IF
ELSE
*
* Form C * H
*
IF( LASTV.GT.0 ) THEN
*
* w(1:lastc,1) := C(1:lastc,1:lastv) * v(1:lastv,1)
*
CALL DGEMV( 'No transpose', LASTC, LASTV, ONE, C, LDC,
$ V, INCV, ZERO, WORK, 1 )
*
* C(1:lastc,1:lastv) := C(...) - w(1:lastc,1) * v(1:lastv,1)**T
*
CALL DGER( LASTC, LASTV, -TAU, WORK, 1, V, INCV, C, LDC )
END IF
END IF
RETURN
*
* End of DLARF
*
END

758
lib/linalg/dlarfb.f Normal file
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@ -0,0 +1,758 @@
*> \brief \b DLARFB applies a block reflector or its transpose to a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV,
* T, LDT, C, LDC, WORK, LDWORK )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, SIDE, STOREV, TRANS
* INTEGER K, LDC, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ),
* $ WORK( LDWORK, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFB applies a real block reflector H or its transpose H**T to a
*> real m by n matrix C, from either the left or the right.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply H or H**T from the Left
*> = 'R': apply H or H**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply H (No transpose)
*> = 'T': apply H**T (Transpose)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Indicates how H is formed from a product of elementary
*> reflectors
*> = 'F': H = H(1) H(2) . . . H(k) (Forward)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Indicates how the vectors which define the elementary
*> reflectors are stored:
*> = 'C': Columnwise
*> = 'R': Rowwise
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the matrix T (= the number of elementary
*> reflectors whose product defines the block reflector).
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (LDV,K) if STOREV = 'C'
*> (LDV,M) if STOREV = 'R' and SIDE = 'L'
*> (LDV,N) if STOREV = 'R' and SIDE = 'R'
*> The matrix V. See Further Details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
*> if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
*> if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The triangular k by k matrix T in the representation of the
*> block reflector.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= K.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LDWORK,K)
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*> LDWORK is INTEGER
*> The leading dimension of the array WORK.
*> If SIDE = 'L', LDWORK >= max(1,N);
*> if SIDE = 'R', LDWORK >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The shape of the matrix V and the storage of the vectors which define
*> the H(i) is best illustrated by the following example with n = 5 and
*> k = 3. The elements equal to 1 are not stored; the corresponding
*> array elements are modified but restored on exit. The rest of the
*> array is not used.
*>
*> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
*>
*> V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
*> ( v1 1 ) ( 1 v2 v2 v2 )
*> ( v1 v2 1 ) ( 1 v3 v3 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
*>
*> V = ( v1 v2 v3 ) V = ( v1 v1 1 )
*> ( v1 v2 v3 ) ( v2 v2 v2 1 )
*> ( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
*> ( 1 v3 )
*> ( 1 )
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV,
$ T, LDT, C, LDC, WORK, LDWORK )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER DIRECT, SIDE, STOREV, TRANS
INTEGER K, LDC, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ),
$ WORK( LDWORK, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
CHARACTER TRANST
INTEGER I, J, LASTV, LASTC, lastv2
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILADLR, ILADLC
EXTERNAL LSAME, ILADLR, ILADLC
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DTRMM
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
IF( LSAME( TRANS, 'N' ) ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
IF( LSAME( STOREV, 'C' ) ) THEN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
*
* Let V = ( V1 ) (first K rows)
* ( V2 )
* where V1 is unit lower triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
LASTV = MAX( K, ILADLR( M, K, V, LDV ) )
LASTC = ILADLC( LASTV, N, C, LDC )
*
* W := C**T * V = (C1**T * V1 + C2**T * V2) (stored in WORK)
*
* W := C1**T
*
DO 10 J = 1, K
CALL DCOPY( LASTC, C( J, 1 ), LDC, WORK( 1, J ), 1 )
10 CONTINUE
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
$ LASTC, K, ONE, V, LDV, WORK, LDWORK )
IF( LASTV.GT.K ) THEN
*
* W := W + C2**T *V2
*
CALL DGEMM( 'Transpose', 'No transpose',
$ LASTC, K, LASTV-K,
$ ONE, C( K+1, 1 ), LDC, V( K+1, 1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit',
$ LASTC, K, ONE, T, LDT, WORK, LDWORK )
*
* C := C - V * W**T
*
IF( LASTV.GT.K ) THEN
*
* C2 := C2 - V2 * W**T
*
CALL DGEMM( 'No transpose', 'Transpose',
$ LASTV-K, LASTC, K,
$ -ONE, V( K+1, 1 ), LDV, WORK, LDWORK, ONE,
$ C( K+1, 1 ), LDC )
END IF
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit',
$ LASTC, K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W**T
*
DO 30 J = 1, K
DO 20 I = 1, LASTC
C( J, I ) = C( J, I ) - WORK( I, J )
20 CONTINUE
30 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
LASTV = MAX( K, ILADLR( N, K, V, LDV ) )
LASTC = ILADLR( M, LASTV, C, LDC )
*
* W := C * V = (C1*V1 + C2*V2) (stored in WORK)
*
* W := C1
*
DO 40 J = 1, K
CALL DCOPY( LASTC, C( 1, J ), 1, WORK( 1, J ), 1 )
40 CONTINUE
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
$ LASTC, K, ONE, V, LDV, WORK, LDWORK )
IF( LASTV.GT.K ) THEN
*
* W := W + C2 * V2
*
CALL DGEMM( 'No transpose', 'No transpose',
$ LASTC, K, LASTV-K,
$ ONE, C( 1, K+1 ), LDC, V( K+1, 1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit',
$ LASTC, K, ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V**T
*
IF( LASTV.GT.K ) THEN
*
* C2 := C2 - W * V2**T
*
CALL DGEMM( 'No transpose', 'Transpose',
$ LASTC, LASTV-K, K,
$ -ONE, WORK, LDWORK, V( K+1, 1 ), LDV, ONE,
$ C( 1, K+1 ), LDC )
END IF
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit',
$ LASTC, K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 60 J = 1, K
DO 50 I = 1, LASTC
C( I, J ) = C( I, J ) - WORK( I, J )
50 CONTINUE
60 CONTINUE
END IF
*
ELSE
*
* Let V = ( V1 )
* ( V2 ) (last K rows)
* where V2 is unit upper triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
LASTC = ILADLC( M, N, C, LDC )
*
* W := C**T * V = (C1**T * V1 + C2**T * V2) (stored in WORK)
*
* W := C2**T
*
DO 70 J = 1, K
CALL DCOPY( LASTC, C( M-K+J, 1 ), LDC,
$ WORK( 1, J ), 1 )
70 CONTINUE
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
$ LASTC, K, ONE, V( M-K+1, 1 ), LDV,
$ WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C1**T*V1
*
CALL DGEMM( 'Transpose', 'No transpose',
$ LASTC, K, M-K, ONE, C, LDC, V, LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit',
$ LASTC, K, ONE, T, LDT, WORK, LDWORK )
*
* C := C - V * W**T
*
IF( M.GT.K ) THEN
*
* C1 := C1 - V1 * W**T
*
CALL DGEMM( 'No transpose', 'Transpose',
$ M-K, LASTC, K, -ONE, V, LDV, WORK, LDWORK,
$ ONE, C, LDC )
END IF
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit',
$ LASTC, K, ONE, V( M-K+1, 1 ), LDV,
$ WORK, LDWORK )
*
* C2 := C2 - W**T
*
DO 90 J = 1, K
DO 80 I = 1, LASTC
C( M-K+J, I ) = C( M-K+J, I ) - WORK(I, J)
80 CONTINUE
90 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
LASTC = ILADLR( M, N, C, LDC )
*
* W := C * V = (C1*V1 + C2*V2) (stored in WORK)
*
* W := C2
*
DO 100 J = 1, K
CALL DCOPY( LASTC, C( 1, N-K+J ), 1, WORK( 1, J ), 1 )
100 CONTINUE
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
$ LASTC, K, ONE, V( N-K+1, 1 ), LDV,
$ WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C1 * V1
*
CALL DGEMM( 'No transpose', 'No transpose',
$ LASTC, K, N-K, ONE, C, LDC, V, LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit',
$ LASTC, K, ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V**T
*
IF( N.GT.K ) THEN
*
* C1 := C1 - W * V1**T
*
CALL DGEMM( 'No transpose', 'Transpose',
$ LASTC, N-K, K, -ONE, WORK, LDWORK, V, LDV,
$ ONE, C, LDC )
END IF
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit',
$ LASTC, K, ONE, V( N-K+1, 1 ), LDV,
$ WORK, LDWORK )
*
* C2 := C2 - W
*
DO 120 J = 1, K
DO 110 I = 1, LASTC
C( I, N-K+J ) = C( I, N-K+J ) - WORK(I, J)
110 CONTINUE
120 CONTINUE
END IF
END IF
*
ELSE IF( LSAME( STOREV, 'R' ) ) THEN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
*
* Let V = ( V1 V2 ) (V1: first K columns)
* where V1 is unit upper triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
LASTV = MAX( K, ILADLC( K, M, V, LDV ) )
LASTC = ILADLC( LASTV, N, C, LDC )
*
* W := C**T * V**T = (C1**T * V1**T + C2**T * V2**T) (stored in WORK)
*
* W := C1**T
*
DO 130 J = 1, K
CALL DCOPY( LASTC, C( J, 1 ), LDC, WORK( 1, J ), 1 )
130 CONTINUE
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit',
$ LASTC, K, ONE, V, LDV, WORK, LDWORK )
IF( LASTV.GT.K ) THEN
*
* W := W + C2**T*V2**T
*
CALL DGEMM( 'Transpose', 'Transpose',
$ LASTC, K, LASTV-K,
$ ONE, C( K+1, 1 ), LDC, V( 1, K+1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit',
$ LASTC, K, ONE, T, LDT, WORK, LDWORK )
*
* C := C - V**T * W**T
*
IF( LASTV.GT.K ) THEN
*
* C2 := C2 - V2**T * W**T
*
CALL DGEMM( 'Transpose', 'Transpose',
$ LASTV-K, LASTC, K,
$ -ONE, V( 1, K+1 ), LDV, WORK, LDWORK,
$ ONE, C( K+1, 1 ), LDC )
END IF
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
$ LASTC, K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W**T
*
DO 150 J = 1, K
DO 140 I = 1, LASTC
C( J, I ) = C( J, I ) - WORK( I, J )
140 CONTINUE
150 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
LASTV = MAX( K, ILADLC( K, N, V, LDV ) )
LASTC = ILADLR( M, LASTV, C, LDC )
*
* W := C * V**T = (C1*V1**T + C2*V2**T) (stored in WORK)
*
* W := C1
*
DO 160 J = 1, K
CALL DCOPY( LASTC, C( 1, J ), 1, WORK( 1, J ), 1 )
160 CONTINUE
*
* W := W * V1**T
*
CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit',
$ LASTC, K, ONE, V, LDV, WORK, LDWORK )
IF( LASTV.GT.K ) THEN
*
* W := W + C2 * V2**T
*
CALL DGEMM( 'No transpose', 'Transpose',
$ LASTC, K, LASTV-K,
$ ONE, C( 1, K+1 ), LDC, V( 1, K+1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit',
$ LASTC, K, ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V
*
IF( LASTV.GT.K ) THEN
*
* C2 := C2 - W * V2
*
CALL DGEMM( 'No transpose', 'No transpose',
$ LASTC, LASTV-K, K,
$ -ONE, WORK, LDWORK, V( 1, K+1 ), LDV,
$ ONE, C( 1, K+1 ), LDC )
END IF
*
* W := W * V1
*
CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit',
$ LASTC, K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 180 J = 1, K
DO 170 I = 1, LASTC
C( I, J ) = C( I, J ) - WORK( I, J )
170 CONTINUE
180 CONTINUE
*
END IF
*
ELSE
*
* Let V = ( V1 V2 ) (V2: last K columns)
* where V2 is unit lower triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H**T * C where C = ( C1 )
* ( C2 )
*
LASTC = ILADLC( M, N, C, LDC )
*
* W := C**T * V**T = (C1**T * V1**T + C2**T * V2**T) (stored in WORK)
*
* W := C2**T
*
DO 190 J = 1, K
CALL DCOPY( LASTC, C( M-K+J, 1 ), LDC,
$ WORK( 1, J ), 1 )
190 CONTINUE
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit',
$ LASTC, K, ONE, V( 1, M-K+1 ), LDV,
$ WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C1**T * V1**T
*
CALL DGEMM( 'Transpose', 'Transpose',
$ LASTC, K, M-K, ONE, C, LDC, V, LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T**T or W * T
*
CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit',
$ LASTC, K, ONE, T, LDT, WORK, LDWORK )
*
* C := C - V**T * W**T
*
IF( M.GT.K ) THEN
*
* C1 := C1 - V1**T * W**T
*
CALL DGEMM( 'Transpose', 'Transpose',
$ M-K, LASTC, K, -ONE, V, LDV, WORK, LDWORK,
$ ONE, C, LDC )
END IF
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
$ LASTC, K, ONE, V( 1, M-K+1 ), LDV,
$ WORK, LDWORK )
*
* C2 := C2 - W**T
*
DO 210 J = 1, K
DO 200 I = 1, LASTC
C( M-K+J, I ) = C( M-K+J, I ) - WORK(I, J)
200 CONTINUE
210 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H**T where C = ( C1 C2 )
*
LASTC = ILADLR( M, N, C, LDC )
*
* W := C * V**T = (C1*V1**T + C2*V2**T) (stored in WORK)
*
* W := C2
*
DO 220 J = 1, K
CALL DCOPY( LASTC, C( 1, N-K+J ), 1,
$ WORK( 1, J ), 1 )
220 CONTINUE
*
* W := W * V2**T
*
CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit',
$ LASTC, K, ONE, V( 1, N-K+1 ), LDV,
$ WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C1 * V1**T
*
CALL DGEMM( 'No transpose', 'Transpose',
$ LASTC, K, N-K, ONE, C, LDC, V, LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T**T
*
CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit',
$ LASTC, K, ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V
*
IF( N.GT.K ) THEN
*
* C1 := C1 - W * V1
*
CALL DGEMM( 'No transpose', 'No transpose',
$ LASTC, N-K, K, -ONE, WORK, LDWORK, V, LDV,
$ ONE, C, LDC )
END IF
*
* W := W * V2
*
CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit',
$ LASTC, K, ONE, V( 1, N-K+1 ), LDV,
$ WORK, LDWORK )
*
* C1 := C1 - W
*
DO 240 J = 1, K
DO 230 I = 1, LASTC
C( I, N-K+J ) = C( I, N-K+J ) - WORK(I, J)
230 CONTINUE
240 CONTINUE
*
END IF
*
END IF
END IF
*
RETURN
*
* End of DLARFB
*
END

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*> \brief \b DLARFG generates an elementary reflector (Householder matrix).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfg.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
* .. Scalar Arguments ..
* INTEGER INCX, N
* DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFG generates a real elementary reflector H of order n, such
*> that
*>
*> H * ( alpha ) = ( beta ), H**T * H = I.
*> ( x ) ( 0 )
*>
*> where alpha and beta are scalars, and x is an (n-1)-element real
*> vector. H is represented in the form
*>
*> H = I - tau * ( 1 ) * ( 1 v**T ) ,
*> ( v )
*>
*> where tau is a real scalar and v is a real (n-1)-element
*> vector.
*>
*> If the elements of x are all zero, then tau = 0 and H is taken to be
*> the unit matrix.
*>
*> Otherwise 1 <= tau <= 2.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the elementary reflector.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> On entry, the value alpha.
*> On exit, it is overwritten with the value beta.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension
*> (1+(N-2)*abs(INCX))
*> On entry, the vector x.
*> On exit, it is overwritten with the vector v.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INCX, N
DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER J, KNT
DOUBLE PRECISION BETA, RSAFMN, SAFMIN, XNORM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
EXTERNAL DLAMCH, DLAPY2, DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN
* ..
* .. External Subroutines ..
EXTERNAL DSCAL
* ..
* .. Executable Statements ..
*
IF( N.LE.1 ) THEN
TAU = ZERO
RETURN
END IF
*
XNORM = DNRM2( N-1, X, INCX )
*
IF( XNORM.EQ.ZERO ) THEN
*
* H = I
*
TAU = ZERO
ELSE
*
* general case
*
BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
KNT = 0
IF( ABS( BETA ).LT.SAFMIN ) THEN
*
* XNORM, BETA may be inaccurate; scale X and recompute them
*
RSAFMN = ONE / SAFMIN
10 CONTINUE
KNT = KNT + 1
CALL DSCAL( N-1, RSAFMN, X, INCX )
BETA = BETA*RSAFMN
ALPHA = ALPHA*RSAFMN
IF( ABS( BETA ).LT.SAFMIN )
$ GO TO 10
*
* New BETA is at most 1, at least SAFMIN
*
XNORM = DNRM2( N-1, X, INCX )
BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
END IF
TAU = ( BETA-ALPHA ) / BETA
CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
*
* If ALPHA is subnormal, it may lose relative accuracy
*
DO 20 J = 1, KNT
BETA = BETA*SAFMIN
20 CONTINUE
ALPHA = BETA
END IF
*
RETURN
*
* End of DLARFG
*
END

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*> \brief \b DLARFT forms the triangular factor T of a block reflector H = I - vtvH
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarft.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarft.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarft.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
* INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFT forms the triangular factor T of a real block reflector H
*> of order n, which is defined as a product of k elementary reflectors.
*>
*> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
*>
*> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
*>
*> If STOREV = 'C', the vector which defines the elementary reflector
*> H(i) is stored in the i-th column of the array V, and
*>
*> H = I - V * T * V**T
*>
*> If STOREV = 'R', the vector which defines the elementary reflector
*> H(i) is stored in the i-th row of the array V, and
*>
*> H = I - V**T * T * V
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Specifies the order in which the elementary reflectors are
*> multiplied to form the block reflector:
*> = 'F': H = H(1) H(2) . . . H(k) (Forward)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Specifies how the vectors which define the elementary
*> reflectors are stored (see also Further Details):
*> = 'C': columnwise
*> = 'R': rowwise
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the block reflector H. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the triangular factor T (= the number of
*> elementary reflectors). K >= 1.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (LDV,K) if STOREV = 'C'
*> (LDV,N) if STOREV = 'R'
*> The matrix V. See further details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The k by k triangular factor T of the block reflector.
*> If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
*> lower triangular. The rest of the array is not used.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= K.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The shape of the matrix V and the storage of the vectors which define
*> the H(i) is best illustrated by the following example with n = 5 and
*> k = 3. The elements equal to 1 are not stored.
*>
*> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
*>
*> V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
*> ( v1 1 ) ( 1 v2 v2 v2 )
*> ( v1 v2 1 ) ( 1 v3 v3 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
*>
*> V = ( v1 v2 v3 ) V = ( v1 v1 1 )
*> ( v1 v2 v3 ) ( v2 v2 v2 1 )
*> ( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
*> ( 1 v3 )
*> ( 1 )
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, PREVLASTV, LASTV
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DTRMV
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
PREVLASTV = N
DO I = 1, K
PREVLASTV = MAX( I, PREVLASTV )
IF( TAU( I ).EQ.ZERO ) THEN
*
* H(i) = I
*
DO J = 1, I
T( J, I ) = ZERO
END DO
ELSE
*
* general case
*
IF( LSAME( STOREV, 'C' ) ) THEN
* Skip any trailing zeros.
DO LASTV = N, I+1, -1
IF( V( LASTV, I ).NE.ZERO ) EXIT
END DO
DO J = 1, I-1
T( J, I ) = -TAU( I ) * V( I , J )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**T * V(i:j,i)
*
CALL DGEMV( 'Transpose', J-I, I-1, -TAU( I ),
$ V( I+1, 1 ), LDV, V( I+1, I ), 1, ONE,
$ T( 1, I ), 1 )
ELSE
* Skip any trailing zeros.
DO LASTV = N, I+1, -1
IF( V( I, LASTV ).NE.ZERO ) EXIT
END DO
DO J = 1, I-1
T( J, I ) = -TAU( I ) * V( J , I )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)**T
*
CALL DGEMV( 'No transpose', I-1, J-I, -TAU( I ),
$ V( 1, I+1 ), LDV, V( I, I+1 ), LDV, ONE,
$ T( 1, I ), 1 )
END IF
*
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
*
CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T,
$ LDT, T( 1, I ), 1 )
T( I, I ) = TAU( I )
IF( I.GT.1 ) THEN
PREVLASTV = MAX( PREVLASTV, LASTV )
ELSE
PREVLASTV = LASTV
END IF
END IF
END DO
ELSE
PREVLASTV = 1
DO I = K, 1, -1
IF( TAU( I ).EQ.ZERO ) THEN
*
* H(i) = I
*
DO J = I, K
T( J, I ) = ZERO
END DO
ELSE
*
* general case
*
IF( I.LT.K ) THEN
IF( LSAME( STOREV, 'C' ) ) THEN
* Skip any leading zeros.
DO LASTV = 1, I-1
IF( V( LASTV, I ).NE.ZERO ) EXIT
END DO
DO J = I+1, K
T( J, I ) = -TAU( I ) * V( N-K+I , J )
END DO
J = MAX( LASTV, PREVLASTV )
*
* T(i+1:k,i) = -tau(i) * V(j:n-k+i,i+1:k)**T * V(j:n-k+i,i)
*
CALL DGEMV( 'Transpose', N-K+I-J, K-I, -TAU( I ),
$ V( J, I+1 ), LDV, V( J, I ), 1, ONE,
$ T( I+1, I ), 1 )
ELSE
* Skip any leading zeros.
DO LASTV = 1, I-1
IF( V( I, LASTV ).NE.ZERO ) EXIT
END DO
DO J = I+1, K
T( J, I ) = -TAU( I ) * V( J, N-K+I )
END DO
J = MAX( LASTV, PREVLASTV )
*
* T(i+1:k,i) = -tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)**T
*
CALL DGEMV( 'No transpose', K-I, N-K+I-J,
$ -TAU( I ), V( I+1, J ), LDV, V( I, J ), LDV,
$ ONE, T( I+1, I ), 1 )
END IF
*
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
*
CALL DTRMV( 'Lower', 'No transpose', 'Non-unit', K-I,
$ T( I+1, I+1 ), LDT, T( I+1, I ), 1 )
IF( I.GT.1 ) THEN
PREVLASTV = MIN( PREVLASTV, LASTV )
ELSE
PREVLASTV = LASTV
END IF
END IF
T( I, I ) = TAU( I )
END IF
END DO
END IF
RETURN
*
* End of DLARFT
*
END

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*> \brief \b DLARTG generates a plane rotation with real cosine and real sine.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARTG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlartg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlartg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlartg.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARTG( F, G, CS, SN, R )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION CS, F, G, R, SN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARTG generate a plane rotation so that
*>
*> [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1.
*> [ -SN CS ] [ G ] [ 0 ]
*>
*> This is a slower, more accurate version of the BLAS1 routine DROTG,
*> with the following other differences:
*> F and G are unchanged on return.
*> If G=0, then CS=1 and SN=0.
*> If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any
*> floating point operations (saves work in DBDSQR when
*> there are zeros on the diagonal).
*>
*> If F exceeds G in magnitude, CS will be positive.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is DOUBLE PRECISION
*> The first component of vector to be rotated.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is DOUBLE PRECISION
*> The second component of vector to be rotated.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*> CS is DOUBLE PRECISION
*> The cosine of the rotation.
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*> SN is DOUBLE PRECISION
*> The sine of the rotation.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is DOUBLE PRECISION
*> The nonzero component of the rotated vector.
*>
*> This version has a few statements commented out for thread safety
*> (machine parameters are computed on each entry). 10 feb 03, SJH.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLARTG( F, G, CS, SN, R )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION CS, F, G, R, SN
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
* LOGICAL FIRST
INTEGER COUNT, I
DOUBLE PRECISION EPS, F1, G1, SAFMIN, SAFMN2, SAFMX2, SCALE
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, SQRT
* ..
* .. Save statement ..
* SAVE FIRST, SAFMX2, SAFMIN, SAFMN2
* ..
* .. Data statements ..
* DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
*
* IF( FIRST ) THEN
SAFMIN = DLAMCH( 'S' )
EPS = DLAMCH( 'E' )
SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
$ LOG( DLAMCH( 'B' ) ) / TWO )
SAFMX2 = ONE / SAFMN2
* FIRST = .FALSE.
* END IF
IF( G.EQ.ZERO ) THEN
CS = ONE
SN = ZERO
R = F
ELSE IF( F.EQ.ZERO ) THEN
CS = ZERO
SN = ONE
R = G
ELSE
F1 = F
G1 = G
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.GE.SAFMX2 ) THEN
COUNT = 0
10 CONTINUE
COUNT = COUNT + 1
F1 = F1*SAFMN2
G1 = G1*SAFMN2
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.GE.SAFMX2 )
$ GO TO 10
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
DO 20 I = 1, COUNT
R = R*SAFMX2
20 CONTINUE
ELSE IF( SCALE.LE.SAFMN2 ) THEN
COUNT = 0
30 CONTINUE
COUNT = COUNT + 1
F1 = F1*SAFMX2
G1 = G1*SAFMX2
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.LE.SAFMN2 )
$ GO TO 30
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
DO 40 I = 1, COUNT
R = R*SAFMN2
40 CONTINUE
ELSE
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
END IF
IF( ABS( F ).GT.ABS( G ) .AND. CS.LT.ZERO ) THEN
CS = -CS
SN = -SN
R = -R
END IF
END IF
RETURN
*
* End of DLARTG
*
END

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*> \brief \b DLAS2 computes singular values of a 2-by-2 triangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlas2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlas2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlas2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION F, G, H, SSMAX, SSMIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAS2 computes the singular values of the 2-by-2 matrix
*> [ F G ]
*> [ 0 H ].
*> On return, SSMIN is the smaller singular value and SSMAX is the
*> larger singular value.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is DOUBLE PRECISION
*> The (1,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*> SSMIN is DOUBLE PRECISION
*> The smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*> SSMAX is DOUBLE PRECISION
*> The larger singular value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Barring over/underflow, all output quantities are correct to within
*> a few units in the last place (ulps), even in the absence of a guard
*> digit in addition/subtraction.
*>
*> In IEEE arithmetic, the code works correctly if one matrix element is
*> infinite.
*>
*> Overflow will not occur unless the largest singular value itself
*> overflows, or is within a few ulps of overflow. (On machines with
*> partial overflow, like the Cray, overflow may occur if the largest
*> singular value is within a factor of 2 of overflow.)
*>
*> Underflow is harmless if underflow is gradual. Otherwise, results
*> may correspond to a matrix modified by perturbations of size near
*> the underflow threshold.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION F, G, H, SSMAX, SSMIN
* ..
*
* ====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AS, AT, AU, C, FA, FHMN, FHMX, GA, HA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
FA = ABS( F )
GA = ABS( G )
HA = ABS( H )
FHMN = MIN( FA, HA )
FHMX = MAX( FA, HA )
IF( FHMN.EQ.ZERO ) THEN
SSMIN = ZERO
IF( FHMX.EQ.ZERO ) THEN
SSMAX = GA
ELSE
SSMAX = MAX( FHMX, GA )*SQRT( ONE+
$ ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 )
END IF
ELSE
IF( GA.LT.FHMX ) THEN
AS = ONE + FHMN / FHMX
AT = ( FHMX-FHMN ) / FHMX
AU = ( GA / FHMX )**2
C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) )
SSMIN = FHMN*C
SSMAX = FHMX / C
ELSE
AU = FHMX / GA
IF( AU.EQ.ZERO ) THEN
*
* Avoid possible harmful underflow if exponent range
* asymmetric (true SSMIN may not underflow even if
* AU underflows)
*
SSMIN = ( FHMN*FHMX ) / GA
SSMAX = GA
ELSE
AS = ONE + FHMN / FHMX
AT = ( FHMX-FHMN ) / FHMX
C = ONE / ( SQRT( ONE+( AS*AU )**2 )+
$ SQRT( ONE+( AT*AU )**2 ) )
SSMIN = ( FHMN*C )*AU
SSMIN = SSMIN + SSMIN
SSMAX = GA / ( C+C )
END IF
END IF
END IF
RETURN
*
* End of DLAS2
*
END

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*> \brief \b DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASCL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlascl.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlascl.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlascl.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TYPE
* INTEGER INFO, KL, KU, LDA, M, N
* DOUBLE PRECISION CFROM, CTO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASCL multiplies the M by N real matrix A by the real scalar
*> CTO/CFROM. This is done without over/underflow as long as the final
*> result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
*> A may be full, upper triangular, lower triangular, upper Hessenberg,
*> or banded.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TYPE
*> \verbatim
*> TYPE is CHARACTER*1
*> TYPE indices the storage type of the input matrix.
*> = 'G': A is a full matrix.
*> = 'L': A is a lower triangular matrix.
*> = 'U': A is an upper triangular matrix.
*> = 'H': A is an upper Hessenberg matrix.
*> = 'B': A is a symmetric band matrix with lower bandwidth KL
*> and upper bandwidth KU and with the only the lower
*> half stored.
*> = 'Q': A is a symmetric band matrix with lower bandwidth KL
*> and upper bandwidth KU and with the only the upper
*> half stored.
*> = 'Z': A is a band matrix with lower bandwidth KL and upper
*> bandwidth KU. See DGBTRF for storage details.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The lower bandwidth of A. Referenced only if TYPE = 'B',
*> 'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The upper bandwidth of A. Referenced only if TYPE = 'B',
*> 'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] CFROM
*> \verbatim
*> CFROM is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] CTO
*> \verbatim
*> CTO is DOUBLE PRECISION
*>
*> The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
*> without over/underflow if the final result CTO*A(I,J)/CFROM
*> can be represented without over/underflow. CFROM must be
*> nonzero.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The matrix to be multiplied by CTO/CFROM. See TYPE for the
*> storage type.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> 0 - successful exit
*> <0 - if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER TYPE
INTEGER INFO, KL, KU, LDA, M, N
DOUBLE PRECISION CFROM, CTO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL DONE
INTEGER I, ITYPE, J, K1, K2, K3, K4
DOUBLE PRECISION BIGNUM, CFROM1, CFROMC, CTO1, CTOC, MUL, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
*
IF( LSAME( TYPE, 'G' ) ) THEN
ITYPE = 0
ELSE IF( LSAME( TYPE, 'L' ) ) THEN
ITYPE = 1
ELSE IF( LSAME( TYPE, 'U' ) ) THEN
ITYPE = 2
ELSE IF( LSAME( TYPE, 'H' ) ) THEN
ITYPE = 3
ELSE IF( LSAME( TYPE, 'B' ) ) THEN
ITYPE = 4
ELSE IF( LSAME( TYPE, 'Q' ) ) THEN
ITYPE = 5
ELSE IF( LSAME( TYPE, 'Z' ) ) THEN
ITYPE = 6
ELSE
ITYPE = -1
END IF
*
IF( ITYPE.EQ.-1 ) THEN
INFO = -1
ELSE IF( CFROM.EQ.ZERO .OR. DISNAN(CFROM) ) THEN
INFO = -4
ELSE IF( DISNAN(CTO) ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.4 .AND. N.NE.M ) .OR.
$ ( ITYPE.EQ.5 .AND. N.NE.M ) ) THEN
INFO = -7
ELSE IF( ITYPE.LE.3 .AND. LDA.LT.MAX( 1, M ) ) THEN
INFO = -9
ELSE IF( ITYPE.GE.4 ) THEN
IF( KL.LT.0 .OR. KL.GT.MAX( M-1, 0 ) ) THEN
INFO = -2
ELSE IF( KU.LT.0 .OR. KU.GT.MAX( N-1, 0 ) .OR.
$ ( ( ITYPE.EQ.4 .OR. ITYPE.EQ.5 ) .AND. KL.NE.KU ) )
$ THEN
INFO = -3
ELSE IF( ( ITYPE.EQ.4 .AND. LDA.LT.KL+1 ) .OR.
$ ( ITYPE.EQ.5 .AND. LDA.LT.KU+1 ) .OR.
$ ( ITYPE.EQ.6 .AND. LDA.LT.2*KL+KU+1 ) ) THEN
INFO = -9
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASCL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. M.EQ.0 )
$ RETURN
*
* Get machine parameters
*
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
*
CFROMC = CFROM
CTOC = CTO
*
10 CONTINUE
CFROM1 = CFROMC*SMLNUM
IF( CFROM1.EQ.CFROMC ) THEN
! CFROMC is an inf. Multiply by a correctly signed zero for
! finite CTOC, or a NaN if CTOC is infinite.
MUL = CTOC / CFROMC
DONE = .TRUE.
CTO1 = CTOC
ELSE
CTO1 = CTOC / BIGNUM
IF( CTO1.EQ.CTOC ) THEN
! CTOC is either 0 or an inf. In both cases, CTOC itself
! serves as the correct multiplication factor.
MUL = CTOC
DONE = .TRUE.
CFROMC = ONE
ELSE IF( ABS( CFROM1 ).GT.ABS( CTOC ) .AND. CTOC.NE.ZERO ) THEN
MUL = SMLNUM
DONE = .FALSE.
CFROMC = CFROM1
ELSE IF( ABS( CTO1 ).GT.ABS( CFROMC ) ) THEN
MUL = BIGNUM
DONE = .FALSE.
CTOC = CTO1
ELSE
MUL = CTOC / CFROMC
DONE = .TRUE.
END IF
END IF
*
IF( ITYPE.EQ.0 ) THEN
*
* Full matrix
*
DO 30 J = 1, N
DO 20 I = 1, M
A( I, J ) = A( I, J )*MUL
20 CONTINUE
30 CONTINUE
*
ELSE IF( ITYPE.EQ.1 ) THEN
*
* Lower triangular matrix
*
DO 50 J = 1, N
DO 40 I = J, M
A( I, J ) = A( I, J )*MUL
40 CONTINUE
50 CONTINUE
*
ELSE IF( ITYPE.EQ.2 ) THEN
*
* Upper triangular matrix
*
DO 70 J = 1, N
DO 60 I = 1, MIN( J, M )
A( I, J ) = A( I, J )*MUL
60 CONTINUE
70 CONTINUE
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* Upper Hessenberg matrix
*
DO 90 J = 1, N
DO 80 I = 1, MIN( J+1, M )
A( I, J ) = A( I, J )*MUL
80 CONTINUE
90 CONTINUE
*
ELSE IF( ITYPE.EQ.4 ) THEN
*
* Lower half of a symmetric band matrix
*
K3 = KL + 1
K4 = N + 1
DO 110 J = 1, N
DO 100 I = 1, MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
100 CONTINUE
110 CONTINUE
*
ELSE IF( ITYPE.EQ.5 ) THEN
*
* Upper half of a symmetric band matrix
*
K1 = KU + 2
K3 = KU + 1
DO 130 J = 1, N
DO 120 I = MAX( K1-J, 1 ), K3
A( I, J ) = A( I, J )*MUL
120 CONTINUE
130 CONTINUE
*
ELSE IF( ITYPE.EQ.6 ) THEN
*
* Band matrix
*
K1 = KL + KU + 2
K2 = KL + 1
K3 = 2*KL + KU + 1
K4 = KL + KU + 1 + M
DO 150 J = 1, N
DO 140 I = MAX( K1-J, K2 ), MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
140 CONTINUE
150 CONTINUE
*
END IF
*
IF( .NOT.DONE )
$ GO TO 10
*
RETURN
*
* End of DLASCL
*
END

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*> \brief \b DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASET + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaset.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaset.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaset.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, M, N
* DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASET initializes an m-by-n matrix A to BETA on the diagonal and
*> ALPHA on the offdiagonals.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies the part of the matrix A to be set.
*> = 'U': Upper triangular part is set; the strictly lower
*> triangular part of A is not changed.
*> = 'L': Lower triangular part is set; the strictly upper
*> triangular part of A is not changed.
*> Otherwise: All of the matrix A is set.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> The constant to which the offdiagonal elements are to be set.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> The constant to which the diagonal elements are to be set.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On exit, the leading m-by-n submatrix of A is set as follows:
*>
*> if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n,
*> if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n,
*> otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j,
*>
*> and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, M, N
DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Set the strictly upper triangular or trapezoidal part of the
* array to ALPHA.
*
DO 20 J = 2, N
DO 10 I = 1, MIN( J-1, M )
A( I, J ) = ALPHA
10 CONTINUE
20 CONTINUE
*
ELSE IF( LSAME( UPLO, 'L' ) ) THEN
*
* Set the strictly lower triangular or trapezoidal part of the
* array to ALPHA.
*
DO 40 J = 1, MIN( M, N )
DO 30 I = J + 1, M
A( I, J ) = ALPHA
30 CONTINUE
40 CONTINUE
*
ELSE
*
* Set the leading m-by-n submatrix to ALPHA.
*
DO 60 J = 1, N
DO 50 I = 1, M
A( I, J ) = ALPHA
50 CONTINUE
60 CONTINUE
END IF
*
* Set the first min(M,N) diagonal elements to BETA.
*
DO 70 I = 1, MIN( M, N )
A( I, I ) = BETA
70 CONTINUE
*
RETURN
*
* End of DLASET
*
END

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*> \brief \b DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ1 computes the singular values of a real N-by-N bidiagonal
*> matrix with diagonal D and off-diagonal E. The singular values
*> are computed to high relative accuracy, in the absence of
*> denormalization, underflow and overflow. The algorithm was first
*> presented in
*>
*> "Accurate singular values and differential qd algorithms" by K. V.
*> Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
*> 1994,
*>
*> and the present implementation is described in "An implementation of
*> the dqds Algorithm (Positive Case)", LAPACK Working Note.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, D contains the diagonal elements of the
*> bidiagonal matrix whose SVD is desired. On normal exit,
*> D contains the singular values in decreasing order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, elements E(1:N-1) contain the off-diagonal elements
*> of the bidiagonal matrix whose SVD is desired.
*> On exit, E is overwritten.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: the algorithm failed
*> = 1, a split was marked by a positive value in E
*> = 2, current block of Z not diagonalized after 100*N
*> iterations (in inner while loop) On exit D and E
*> represent a matrix with the same singular values
*> which the calling subroutine could use to finish the
*> computation, or even feed back into DLASQ1
*> = 3, termination criterion of outer while loop not met
*> (program created more than N unreduced blocks)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, IINFO
DOUBLE PRECISION EPS, SCALE, SAFMIN, SIGMN, SIGMX
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAS2, DLASCL, DLASQ2, DLASRT, XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -2
CALL XERBLA( 'DLASQ1', -INFO )
RETURN
ELSE IF( N.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
D( 1 ) = ABS( D( 1 ) )
RETURN
ELSE IF( N.EQ.2 ) THEN
CALL DLAS2( D( 1 ), E( 1 ), D( 2 ), SIGMN, SIGMX )
D( 1 ) = SIGMX
D( 2 ) = SIGMN
RETURN
END IF
*
* Estimate the largest singular value.
*
SIGMX = ZERO
DO 10 I = 1, N - 1
D( I ) = ABS( D( I ) )
SIGMX = MAX( SIGMX, ABS( E( I ) ) )
10 CONTINUE
D( N ) = ABS( D( N ) )
*
* Early return if SIGMX is zero (matrix is already diagonal).
*
IF( SIGMX.EQ.ZERO ) THEN
CALL DLASRT( 'D', N, D, IINFO )
RETURN
END IF
*
DO 20 I = 1, N
SIGMX = MAX( SIGMX, D( I ) )
20 CONTINUE
*
* Copy D and E into WORK (in the Z format) and scale (squaring the
* input data makes scaling by a power of the radix pointless).
*
EPS = DLAMCH( 'Precision' )
SAFMIN = DLAMCH( 'Safe minimum' )
SCALE = SQRT( EPS / SAFMIN )
CALL DCOPY( N, D, 1, WORK( 1 ), 2 )
CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 )
CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1,
$ IINFO )
*
* Compute the q's and e's.
*
DO 30 I = 1, 2*N - 1
WORK( I ) = WORK( I )**2
30 CONTINUE
WORK( 2*N ) = ZERO
*
CALL DLASQ2( N, WORK, INFO )
*
IF( INFO.EQ.0 ) THEN
DO 40 I = 1, N
D( I ) = SQRT( WORK( I ) )
40 CONTINUE
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
ELSE IF( INFO.EQ.2 ) THEN
*
* Maximum number of iterations exceeded. Move data from WORK
* into D and E so the calling subroutine can try to finish
*
DO I = 1, N
D( I ) = SQRT( WORK( 2*I-1 ) )
E( I ) = SQRT( WORK( 2*I ) )
END DO
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, E, N, IINFO )
END IF
*
RETURN
*
* End of DLASQ1
*
END

582
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*> \brief \b DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ2( N, Z, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ2 computes all the eigenvalues of the symmetric positive
*> definite tridiagonal matrix associated with the qd array Z to high
*> relative accuracy are computed to high relative accuracy, in the
*> absence of denormalization, underflow and overflow.
*>
*> To see the relation of Z to the tridiagonal matrix, let L be a
*> unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
*> let U be an upper bidiagonal matrix with 1's above and diagonal
*> Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
*> symmetric tridiagonal to which it is similar.
*>
*> Note : DLASQ2 defines a logical variable, IEEE, which is true
*> on machines which follow ieee-754 floating-point standard in their
*> handling of infinities and NaNs, and false otherwise. This variable
*> is passed to DLASQ3.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> On entry Z holds the qd array. On exit, entries 1 to N hold
*> the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
*> trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
*> N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
*> holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
*> shifts that failed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if the i-th argument is a scalar and had an illegal
*> value, then INFO = -i, if the i-th argument is an
*> array and the j-entry had an illegal value, then
*> INFO = -(i*100+j)
*> > 0: the algorithm failed
*> = 1, a split was marked by a positive value in E
*> = 2, current block of Z not diagonalized after 100*N
*> iterations (in inner while loop). On exit Z holds
*> a qd array with the same eigenvalues as the given Z.
*> = 3, termination criterion of outer while loop not met
*> (program created more than N unreduced blocks)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Local Variables: I0:N0 defines a current unreduced segment of Z.
*> The shifts are accumulated in SIGMA. Iteration count is in ITER.
*> Ping-pong is controlled by PP (alternates between 0 and 1).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASQ2( N, Z, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION CBIAS
PARAMETER ( CBIAS = 1.50D0 )
DOUBLE PRECISION ZERO, HALF, ONE, TWO, FOUR, HUNDRD
PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
$ TWO = 2.0D0, FOUR = 4.0D0, HUNDRD = 100.0D0 )
* ..
* .. Local Scalars ..
LOGICAL IEEE
INTEGER I0, I1, I4, IINFO, IPN4, ITER, IWHILA, IWHILB,
$ K, KMIN, N0, N1, NBIG, NDIV, NFAIL, PP, SPLT,
$ TTYPE
DOUBLE PRECISION D, DEE, DEEMIN, DESIG, DMIN, DMIN1, DMIN2, DN,
$ DN1, DN2, E, EMAX, EMIN, EPS, G, OLDEMN, QMAX,
$ QMIN, S, SAFMIN, SIGMA, T, TAU, TEMP, TOL,
$ TOL2, TRACE, ZMAX, TEMPE, TEMPQ
* ..
* .. External Subroutines ..
EXTERNAL DLASQ3, DLASRT, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments.
* (in case DLASQ2 is not called by DLASQ1)
*
INFO = 0
EPS = DLAMCH( 'Precision' )
SAFMIN = DLAMCH( 'Safe minimum' )
TOL = EPS*HUNDRD
TOL2 = TOL**2
*
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DLASQ2', 1 )
RETURN
ELSE IF( N.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
*
* 1-by-1 case.
*
IF( Z( 1 ).LT.ZERO ) THEN
INFO = -201
CALL XERBLA( 'DLASQ2', 2 )
END IF
RETURN
ELSE IF( N.EQ.2 ) THEN
*
* 2-by-2 case.
*
IF( Z( 2 ).LT.ZERO .OR. Z( 3 ).LT.ZERO ) THEN
INFO = -2
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( 3 ).GT.Z( 1 ) ) THEN
D = Z( 3 )
Z( 3 ) = Z( 1 )
Z( 1 ) = D
END IF
Z( 5 ) = Z( 1 ) + Z( 2 ) + Z( 3 )
IF( Z( 2 ).GT.Z( 3 )*TOL2 ) THEN
T = HALF*( ( Z( 1 )-Z( 3 ) )+Z( 2 ) )
S = Z( 3 )*( Z( 2 ) / T )
IF( S.LE.T ) THEN
S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) )
ELSE
S = Z( 3 )*( Z( 2 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
END IF
T = Z( 1 ) + ( S+Z( 2 ) )
Z( 3 ) = Z( 3 )*( Z( 1 ) / T )
Z( 1 ) = T
END IF
Z( 2 ) = Z( 3 )
Z( 6 ) = Z( 2 ) + Z( 1 )
RETURN
END IF
*
* Check for negative data and compute sums of q's and e's.
*
Z( 2*N ) = ZERO
EMIN = Z( 2 )
QMAX = ZERO
ZMAX = ZERO
D = ZERO
E = ZERO
*
DO 10 K = 1, 2*( N-1 ), 2
IF( Z( K ).LT.ZERO ) THEN
INFO = -( 200+K )
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( K+1 ).LT.ZERO ) THEN
INFO = -( 200+K+1 )
CALL XERBLA( 'DLASQ2', 2 )
RETURN
END IF
D = D + Z( K )
E = E + Z( K+1 )
QMAX = MAX( QMAX, Z( K ) )
EMIN = MIN( EMIN, Z( K+1 ) )
ZMAX = MAX( QMAX, ZMAX, Z( K+1 ) )
10 CONTINUE
IF( Z( 2*N-1 ).LT.ZERO ) THEN
INFO = -( 200+2*N-1 )
CALL XERBLA( 'DLASQ2', 2 )
RETURN
END IF
D = D + Z( 2*N-1 )
QMAX = MAX( QMAX, Z( 2*N-1 ) )
ZMAX = MAX( QMAX, ZMAX )
*
* Check for diagonality.
*
IF( E.EQ.ZERO ) THEN
DO 20 K = 2, N
Z( K ) = Z( 2*K-1 )
20 CONTINUE
CALL DLASRT( 'D', N, Z, IINFO )
Z( 2*N-1 ) = D
RETURN
END IF
*
TRACE = D + E
*
* Check for zero data.
*
IF( TRACE.EQ.ZERO ) THEN
Z( 2*N-1 ) = ZERO
RETURN
END IF
*
* Check whether the machine is IEEE conformable.
*
IEEE = ILAENV( 10, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 .AND.
$ ILAENV( 11, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1
*
* Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...).
*
DO 30 K = 2*N, 2, -2
Z( 2*K ) = ZERO
Z( 2*K-1 ) = Z( K )
Z( 2*K-2 ) = ZERO
Z( 2*K-3 ) = Z( K-1 )
30 CONTINUE
*
I0 = 1
N0 = N
*
* Reverse the qd-array, if warranted.
*
IF( CBIAS*Z( 4*I0-3 ).LT.Z( 4*N0-3 ) ) THEN
IPN4 = 4*( I0+N0 )
DO 40 I4 = 4*I0, 2*( I0+N0-1 ), 4
TEMP = Z( I4-3 )
Z( I4-3 ) = Z( IPN4-I4-3 )
Z( IPN4-I4-3 ) = TEMP
TEMP = Z( I4-1 )
Z( I4-1 ) = Z( IPN4-I4-5 )
Z( IPN4-I4-5 ) = TEMP
40 CONTINUE
END IF
*
* Initial split checking via dqd and Li's test.
*
PP = 0
*
DO 80 K = 1, 2
*
D = Z( 4*N0+PP-3 )
DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4
IF( Z( I4-1 ).LE.TOL2*D ) THEN
Z( I4-1 ) = -ZERO
D = Z( I4-3 )
ELSE
D = Z( I4-3 )*( D / ( D+Z( I4-1 ) ) )
END IF
50 CONTINUE
*
* dqd maps Z to ZZ plus Li's test.
*
EMIN = Z( 4*I0+PP+1 )
D = Z( 4*I0+PP-3 )
DO 60 I4 = 4*I0 + PP, 4*( N0-1 ) + PP, 4
Z( I4-2*PP-2 ) = D + Z( I4-1 )
IF( Z( I4-1 ).LE.TOL2*D ) THEN
Z( I4-1 ) = -ZERO
Z( I4-2*PP-2 ) = D
Z( I4-2*PP ) = ZERO
D = Z( I4+1 )
ELSE IF( SAFMIN*Z( I4+1 ).LT.Z( I4-2*PP-2 ) .AND.
$ SAFMIN*Z( I4-2*PP-2 ).LT.Z( I4+1 ) ) THEN
TEMP = Z( I4+1 ) / Z( I4-2*PP-2 )
Z( I4-2*PP ) = Z( I4-1 )*TEMP
D = D*TEMP
ELSE
Z( I4-2*PP ) = Z( I4+1 )*( Z( I4-1 ) / Z( I4-2*PP-2 ) )
D = Z( I4+1 )*( D / Z( I4-2*PP-2 ) )
END IF
EMIN = MIN( EMIN, Z( I4-2*PP ) )
60 CONTINUE
Z( 4*N0-PP-2 ) = D
*
* Now find qmax.
*
QMAX = Z( 4*I0-PP-2 )
DO 70 I4 = 4*I0 - PP + 2, 4*N0 - PP - 2, 4
QMAX = MAX( QMAX, Z( I4 ) )
70 CONTINUE
*
* Prepare for the next iteration on K.
*
PP = 1 - PP
80 CONTINUE
*
* Initialise variables to pass to DLASQ3.
*
TTYPE = 0
DMIN1 = ZERO
DMIN2 = ZERO
DN = ZERO
DN1 = ZERO
DN2 = ZERO
G = ZERO
TAU = ZERO
*
ITER = 2
NFAIL = 0
NDIV = 2*( N0-I0 )
*
DO 160 IWHILA = 1, N + 1
IF( N0.LT.1 )
$ GO TO 170
*
* While array unfinished do
*
* E(N0) holds the value of SIGMA when submatrix in I0:N0
* splits from the rest of the array, but is negated.
*
DESIG = ZERO
IF( N0.EQ.N ) THEN
SIGMA = ZERO
ELSE
SIGMA = -Z( 4*N0-1 )
END IF
IF( SIGMA.LT.ZERO ) THEN
INFO = 1
RETURN
END IF
*
* Find last unreduced submatrix's top index I0, find QMAX and
* EMIN. Find Gershgorin-type bound if Q's much greater than E's.
*
EMAX = ZERO
IF( N0.GT.I0 ) THEN
EMIN = ABS( Z( 4*N0-5 ) )
ELSE
EMIN = ZERO
END IF
QMIN = Z( 4*N0-3 )
QMAX = QMIN
DO 90 I4 = 4*N0, 8, -4
IF( Z( I4-5 ).LE.ZERO )
$ GO TO 100
IF( QMIN.GE.FOUR*EMAX ) THEN
QMIN = MIN( QMIN, Z( I4-3 ) )
EMAX = MAX( EMAX, Z( I4-5 ) )
END IF
QMAX = MAX( QMAX, Z( I4-7 )+Z( I4-5 ) )
EMIN = MIN( EMIN, Z( I4-5 ) )
90 CONTINUE
I4 = 4
*
100 CONTINUE
I0 = I4 / 4
PP = 0
*
IF( N0-I0.GT.1 ) THEN
DEE = Z( 4*I0-3 )
DEEMIN = DEE
KMIN = I0
DO 110 I4 = 4*I0+1, 4*N0-3, 4
DEE = Z( I4 )*( DEE /( DEE+Z( I4-2 ) ) )
IF( DEE.LE.DEEMIN ) THEN
DEEMIN = DEE
KMIN = ( I4+3 )/4
END IF
110 CONTINUE
IF( (KMIN-I0)*2.LT.N0-KMIN .AND.
$ DEEMIN.LE.HALF*Z(4*N0-3) ) THEN
IPN4 = 4*( I0+N0 )
PP = 2
DO 120 I4 = 4*I0, 2*( I0+N0-1 ), 4
TEMP = Z( I4-3 )
Z( I4-3 ) = Z( IPN4-I4-3 )
Z( IPN4-I4-3 ) = TEMP
TEMP = Z( I4-2 )
Z( I4-2 ) = Z( IPN4-I4-2 )
Z( IPN4-I4-2 ) = TEMP
TEMP = Z( I4-1 )
Z( I4-1 ) = Z( IPN4-I4-5 )
Z( IPN4-I4-5 ) = TEMP
TEMP = Z( I4 )
Z( I4 ) = Z( IPN4-I4-4 )
Z( IPN4-I4-4 ) = TEMP
120 CONTINUE
END IF
END IF
*
* Put -(initial shift) into DMIN.
*
DMIN = -MAX( ZERO, QMIN-TWO*SQRT( QMIN )*SQRT( EMAX ) )
*
* Now I0:N0 is unreduced.
* PP = 0 for ping, PP = 1 for pong.
* PP = 2 indicates that flipping was applied to the Z array and
* and that the tests for deflation upon entry in DLASQ3
* should not be performed.
*
NBIG = 100*( N0-I0+1 )
DO 140 IWHILB = 1, NBIG
IF( I0.GT.N0 )
$ GO TO 150
*
* While submatrix unfinished take a good dqds step.
*
CALL DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
$ ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
$ DN2, G, TAU )
*
PP = 1 - PP
*
* When EMIN is very small check for splits.
*
IF( PP.EQ.0 .AND. N0-I0.GE.3 ) THEN
IF( Z( 4*N0 ).LE.TOL2*QMAX .OR.
$ Z( 4*N0-1 ).LE.TOL2*SIGMA ) THEN
SPLT = I0 - 1
QMAX = Z( 4*I0-3 )
EMIN = Z( 4*I0-1 )
OLDEMN = Z( 4*I0 )
DO 130 I4 = 4*I0, 4*( N0-3 ), 4
IF( Z( I4 ).LE.TOL2*Z( I4-3 ) .OR.
$ Z( I4-1 ).LE.TOL2*SIGMA ) THEN
Z( I4-1 ) = -SIGMA
SPLT = I4 / 4
QMAX = ZERO
EMIN = Z( I4+3 )
OLDEMN = Z( I4+4 )
ELSE
QMAX = MAX( QMAX, Z( I4+1 ) )
EMIN = MIN( EMIN, Z( I4-1 ) )
OLDEMN = MIN( OLDEMN, Z( I4 ) )
END IF
130 CONTINUE
Z( 4*N0-1 ) = EMIN
Z( 4*N0 ) = OLDEMN
I0 = SPLT + 1
END IF
END IF
*
140 CONTINUE
*
INFO = 2
*
* Maximum number of iterations exceeded, restore the shift
* SIGMA and place the new d's and e's in a qd array.
* This might need to be done for several blocks
*
I1 = I0
N1 = N0
145 CONTINUE
TEMPQ = Z( 4*I0-3 )
Z( 4*I0-3 ) = Z( 4*I0-3 ) + SIGMA
DO K = I0+1, N0
TEMPE = Z( 4*K-5 )
Z( 4*K-5 ) = Z( 4*K-5 ) * (TEMPQ / Z( 4*K-7 ))
TEMPQ = Z( 4*K-3 )
Z( 4*K-3 ) = Z( 4*K-3 ) + SIGMA + TEMPE - Z( 4*K-5 )
END DO
*
* Prepare to do this on the previous block if there is one
*
IF( I1.GT.1 ) THEN
N1 = I1-1
DO WHILE( ( I1.GE.2 ) .AND. ( Z(4*I1-5).GE.ZERO ) )
I1 = I1 - 1
END DO
SIGMA = -Z(4*N1-1)
GO TO 145
END IF
DO K = 1, N
Z( 2*K-1 ) = Z( 4*K-3 )
*
* Only the block 1..N0 is unfinished. The rest of the e's
* must be essentially zero, although sometimes other data
* has been stored in them.
*
IF( K.LT.N0 ) THEN
Z( 2*K ) = Z( 4*K-1 )
ELSE
Z( 2*K ) = 0
END IF
END DO
RETURN
*
* end IWHILB
*
150 CONTINUE
*
160 CONTINUE
*
INFO = 3
RETURN
*
* end IWHILA
*
170 CONTINUE
*
* Move q's to the front.
*
DO 180 K = 2, N
Z( K ) = Z( 4*K-3 )
180 CONTINUE
*
* Sort and compute sum of eigenvalues.
*
CALL DLASRT( 'D', N, Z, IINFO )
*
E = ZERO
DO 190 K = N, 1, -1
E = E + Z( K )
190 CONTINUE
*
* Store trace, sum(eigenvalues) and information on performance.
*
Z( 2*N+1 ) = TRACE
Z( 2*N+2 ) = E
Z( 2*N+3 ) = DBLE( ITER )
Z( 2*N+4 ) = DBLE( NDIV ) / DBLE( N**2 )
Z( 2*N+5 ) = HUNDRD*NFAIL / DBLE( ITER )
RETURN
*
* End of DLASQ2
*
END

421
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*> \brief \b DLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
* ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
* DN2, G, TAU )
*
* .. Scalar Arguments ..
* LOGICAL IEEE
* INTEGER I0, ITER, N0, NDIV, NFAIL, PP
* DOUBLE PRECISION DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G,
* $ QMAX, SIGMA, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
*> In case of failure it changes shifts, and tries again until output
*> is positive.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in,out] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> Z holds the qd array.
*> \endverbatim
*>
*> \param[in,out] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> PP=2 indicates that flipping was applied to the Z array
*> and that the initial tests for deflation should not be
*> performed.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> Sum of shifts used in current segment.
*> \endverbatim
*>
*> \param[in,out] DESIG
*> \verbatim
*> DESIG is DOUBLE PRECISION
*> Lower order part of SIGMA
*> \endverbatim
*>
*> \param[in] QMAX
*> \verbatim
*> QMAX is DOUBLE PRECISION
*> Maximum value of q.
*> \endverbatim
*>
*> \param[out] NFAIL
*> \verbatim
*> NFAIL is INTEGER
*> Number of times shift was too big.
*> \endverbatim
*>
*> \param[out] ITER
*> \verbatim
*> ITER is INTEGER
*> Number of iterations.
*> \endverbatim
*>
*> \param[out] NDIV
*> \verbatim
*> NDIV is INTEGER
*> Number of divisions.
*> \endverbatim
*>
*> \param[in] IEEE
*> \verbatim
*> IEEE is LOGICAL
*> Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).
*> \endverbatim
*>
*> \param[in,out] TTYPE
*> \verbatim
*> TTYPE is INTEGER
*> Shift type.
*> \endverbatim
*>
*> \param[in,out] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN1
*> \verbatim
*> DN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN2
*> \verbatim
*> DN2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] G
*> \verbatim
*> G is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*>
*> These are passed as arguments in order to save their values
*> between calls to DLASQ3.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
$ ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
$ DN2, G, TAU )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
LOGICAL IEEE
INTEGER I0, ITER, N0, NDIV, NFAIL, PP
DOUBLE PRECISION DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G,
$ QMAX, SIGMA, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION CBIAS
PARAMETER ( CBIAS = 1.50D0 )
DOUBLE PRECISION ZERO, QURTR, HALF, ONE, TWO, HUNDRD
PARAMETER ( ZERO = 0.0D0, QURTR = 0.250D0, HALF = 0.5D0,
$ ONE = 1.0D0, TWO = 2.0D0, HUNDRD = 100.0D0 )
* ..
* .. Local Scalars ..
INTEGER IPN4, J4, N0IN, NN, TTYPE
DOUBLE PRECISION EPS, S, T, TEMP, TOL, TOL2
* ..
* .. External Subroutines ..
EXTERNAL DLASQ4, DLASQ5, DLASQ6
* ..
* .. External Function ..
DOUBLE PRECISION DLAMCH
LOGICAL DISNAN
EXTERNAL DISNAN, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
N0IN = N0
EPS = DLAMCH( 'Precision' )
TOL = EPS*HUNDRD
TOL2 = TOL**2
*
* Check for deflation.
*
10 CONTINUE
*
IF( N0.LT.I0 )
$ RETURN
IF( N0.EQ.I0 )
$ GO TO 20
NN = 4*N0 + PP
IF( N0.EQ.( I0+1 ) )
$ GO TO 40
*
* Check whether E(N0-1) is negligible, 1 eigenvalue.
*
IF( Z( NN-5 ).GT.TOL2*( SIGMA+Z( NN-3 ) ) .AND.
$ Z( NN-2*PP-4 ).GT.TOL2*Z( NN-7 ) )
$ GO TO 30
*
20 CONTINUE
*
Z( 4*N0-3 ) = Z( 4*N0+PP-3 ) + SIGMA
N0 = N0 - 1
GO TO 10
*
* Check whether E(N0-2) is negligible, 2 eigenvalues.
*
30 CONTINUE
*
IF( Z( NN-9 ).GT.TOL2*SIGMA .AND.
$ Z( NN-2*PP-8 ).GT.TOL2*Z( NN-11 ) )
$ GO TO 50
*
40 CONTINUE
*
IF( Z( NN-3 ).GT.Z( NN-7 ) ) THEN
S = Z( NN-3 )
Z( NN-3 ) = Z( NN-7 )
Z( NN-7 ) = S
END IF
T = HALF*( ( Z( NN-7 )-Z( NN-3 ) )+Z( NN-5 ) )
IF( Z( NN-5 ).GT.Z( NN-3 )*TOL2.AND.T.NE.ZERO ) THEN
S = Z( NN-3 )*( Z( NN-5 ) / T )
IF( S.LE.T ) THEN
S = Z( NN-3 )*( Z( NN-5 ) /
$ ( T*( ONE+SQRT( ONE+S / T ) ) ) )
ELSE
S = Z( NN-3 )*( Z( NN-5 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
END IF
T = Z( NN-7 ) + ( S+Z( NN-5 ) )
Z( NN-3 ) = Z( NN-3 )*( Z( NN-7 ) / T )
Z( NN-7 ) = T
END IF
Z( 4*N0-7 ) = Z( NN-7 ) + SIGMA
Z( 4*N0-3 ) = Z( NN-3 ) + SIGMA
N0 = N0 - 2
GO TO 10
*
50 CONTINUE
IF( PP.EQ.2 )
$ PP = 0
*
* Reverse the qd-array, if warranted.
*
IF( DMIN.LE.ZERO .OR. N0.LT.N0IN ) THEN
IF( CBIAS*Z( 4*I0+PP-3 ).LT.Z( 4*N0+PP-3 ) ) THEN
IPN4 = 4*( I0+N0 )
DO 60 J4 = 4*I0, 2*( I0+N0-1 ), 4
TEMP = Z( J4-3 )
Z( J4-3 ) = Z( IPN4-J4-3 )
Z( IPN4-J4-3 ) = TEMP
TEMP = Z( J4-2 )
Z( J4-2 ) = Z( IPN4-J4-2 )
Z( IPN4-J4-2 ) = TEMP
TEMP = Z( J4-1 )
Z( J4-1 ) = Z( IPN4-J4-5 )
Z( IPN4-J4-5 ) = TEMP
TEMP = Z( J4 )
Z( J4 ) = Z( IPN4-J4-4 )
Z( IPN4-J4-4 ) = TEMP
60 CONTINUE
IF( N0-I0.LE.4 ) THEN
Z( 4*N0+PP-1 ) = Z( 4*I0+PP-1 )
Z( 4*N0-PP ) = Z( 4*I0-PP )
END IF
DMIN2 = MIN( DMIN2, Z( 4*N0+PP-1 ) )
Z( 4*N0+PP-1 ) = MIN( Z( 4*N0+PP-1 ), Z( 4*I0+PP-1 ),
$ Z( 4*I0+PP+3 ) )
Z( 4*N0-PP ) = MIN( Z( 4*N0-PP ), Z( 4*I0-PP ),
$ Z( 4*I0-PP+4 ) )
QMAX = MAX( QMAX, Z( 4*I0+PP-3 ), Z( 4*I0+PP+1 ) )
DMIN = -ZERO
END IF
END IF
*
* Choose a shift.
*
CALL DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, DN1,
$ DN2, TAU, TTYPE, G )
*
* Call dqds until DMIN > 0.
*
70 CONTINUE
*
CALL DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN,
$ DN1, DN2, IEEE, EPS )
*
NDIV = NDIV + ( N0-I0+2 )
ITER = ITER + 1
*
* Check status.
*
IF( DMIN.GE.ZERO .AND. DMIN1.GE.ZERO ) THEN
*
* Success.
*
GO TO 90
*
ELSE IF( DMIN.LT.ZERO .AND. DMIN1.GT.ZERO .AND.
$ Z( 4*( N0-1 )-PP ).LT.TOL*( SIGMA+DN1 ) .AND.
$ ABS( DN ).LT.TOL*SIGMA ) THEN
*
* Convergence hidden by negative DN.
*
Z( 4*( N0-1 )-PP+2 ) = ZERO
DMIN = ZERO
GO TO 90
ELSE IF( DMIN.LT.ZERO ) THEN
*
* TAU too big. Select new TAU and try again.
*
NFAIL = NFAIL + 1
IF( TTYPE.LT.-22 ) THEN
*
* Failed twice. Play it safe.
*
TAU = ZERO
ELSE IF( DMIN1.GT.ZERO ) THEN
*
* Late failure. Gives excellent shift.
*
TAU = ( TAU+DMIN )*( ONE-TWO*EPS )
TTYPE = TTYPE - 11
ELSE
*
* Early failure. Divide by 4.
*
TAU = QURTR*TAU
TTYPE = TTYPE - 12
END IF
GO TO 70
ELSE IF( DISNAN( DMIN ) ) THEN
*
* NaN.
*
IF( TAU.EQ.ZERO ) THEN
GO TO 80
ELSE
TAU = ZERO
GO TO 70
END IF
ELSE
*
* Possible underflow. Play it safe.
*
GO TO 80
END IF
*
* Risk of underflow.
*
80 CONTINUE
CALL DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, DN1, DN2 )
NDIV = NDIV + ( N0-I0+2 )
ITER = ITER + 1
TAU = ZERO
*
90 CONTINUE
IF( TAU.LT.SIGMA ) THEN
DESIG = DESIG + TAU
T = SIGMA + DESIG
DESIG = DESIG - ( T-SIGMA )
ELSE
T = SIGMA + TAU
DESIG = SIGMA - ( T-TAU ) + DESIG
END IF
SIGMA = T
*
RETURN
*
* End of DLASQ3
*
END

425
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@ -0,0 +1,425 @@
*> \brief \b DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq4.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
* DN1, DN2, TAU, TTYPE, G )
*
* .. Scalar Arguments ..
* INTEGER I0, N0, N0IN, PP, TTYPE
* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ4 computes an approximation TAU to the smallest eigenvalue
*> using values of d from the previous transform.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> Z holds the qd array.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[in] N0IN
*> \verbatim
*> N0IN is INTEGER
*> The value of N0 at start of EIGTEST.
*> \endverbatim
*>
*> \param[in] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[in] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[in] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[in] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> d(N)
*> \endverbatim
*>
*> \param[in] DN1
*> \verbatim
*> DN1 is DOUBLE PRECISION
*> d(N-1)
*> \endverbatim
*>
*> \param[in] DN2
*> \verbatim
*> DN2 is DOUBLE PRECISION
*> d(N-2)
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> This is the shift.
*> \endverbatim
*>
*> \param[out] TTYPE
*> \verbatim
*> TTYPE is INTEGER
*> Shift type.
*> \endverbatim
*>
*> \param[in,out] G
*> \verbatim
*> G is REAL
*> G is passed as an argument in order to save its value between
*> calls to DLASQ4.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> CNST1 = 9/16
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
$ DN1, DN2, TAU, TTYPE, G )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER I0, N0, N0IN, PP, TTYPE
DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION CNST1, CNST2, CNST3
PARAMETER ( CNST1 = 0.5630D0, CNST2 = 1.010D0,
$ CNST3 = 1.050D0 )
DOUBLE PRECISION QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
PARAMETER ( QURTR = 0.250D0, THIRD = 0.3330D0,
$ HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, HUNDRD = 100.0D0 )
* ..
* .. Local Scalars ..
INTEGER I4, NN, NP
DOUBLE PRECISION A2, B1, B2, GAM, GAP1, GAP2, S
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* A negative DMIN forces the shift to take that absolute value
* TTYPE records the type of shift.
*
IF( DMIN.LE.ZERO ) THEN
TAU = -DMIN
TTYPE = -1
RETURN
END IF
*
NN = 4*N0 + PP
IF( N0IN.EQ.N0 ) THEN
*
* No eigenvalues deflated.
*
IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
*
B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
A2 = Z( NN-7 ) + Z( NN-5 )
*
* Cases 2 and 3.
*
IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
GAP2 = DMIN2 - A2 - DMIN2*QURTR
IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
GAP1 = A2 - DN - ( B2 / GAP2 )*B2
ELSE
GAP1 = A2 - DN - ( B1+B2 )
END IF
IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
TTYPE = -2
ELSE
S = ZERO
IF( DN.GT.B1 )
$ S = DN - B1
IF( A2.GT.( B1+B2 ) )
$ S = MIN( S, A2-( B1+B2 ) )
S = MAX( S, THIRD*DMIN )
TTYPE = -3
END IF
ELSE
*
* Case 4.
*
TTYPE = -4
S = QURTR*DMIN
IF( DMIN.EQ.DN ) THEN
GAM = DN
A2 = ZERO
IF( Z( NN-5 ) .GT. Z( NN-7 ) )
$ RETURN
B2 = Z( NN-5 ) / Z( NN-7 )
NP = NN - 9
ELSE
NP = NN - 2*PP
B2 = Z( NP-2 )
GAM = DN1
IF( Z( NP-4 ) .GT. Z( NP-2 ) )
$ RETURN
A2 = Z( NP-4 ) / Z( NP-2 )
IF( Z( NN-9 ) .GT. Z( NN-11 ) )
$ RETURN
B2 = Z( NN-9 ) / Z( NN-11 )
NP = NN - 13
END IF
*
* Approximate contribution to norm squared from I < NN-1.
*
A2 = A2 + B2
DO 10 I4 = NP, 4*I0 - 1 + PP, -4
IF( B2.EQ.ZERO )
$ GO TO 20
B1 = B2
IF( Z( I4 ) .GT. Z( I4-2 ) )
$ RETURN
B2 = B2*( Z( I4 ) / Z( I4-2 ) )
A2 = A2 + B2
IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
$ GO TO 20
10 CONTINUE
20 CONTINUE
A2 = CNST3*A2
*
* Rayleigh quotient residual bound.
*
IF( A2.LT.CNST1 )
$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
END IF
ELSE IF( DMIN.EQ.DN2 ) THEN
*
* Case 5.
*
TTYPE = -5
S = QURTR*DMIN
*
* Compute contribution to norm squared from I > NN-2.
*
NP = NN - 2*PP
B1 = Z( NP-2 )
B2 = Z( NP-6 )
GAM = DN2
IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
$ RETURN
A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
*
* Approximate contribution to norm squared from I < NN-2.
*
IF( N0-I0.GT.2 ) THEN
B2 = Z( NN-13 ) / Z( NN-15 )
A2 = A2 + B2
DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
IF( B2.EQ.ZERO )
$ GO TO 40
B1 = B2
IF( Z( I4 ) .GT. Z( I4-2 ) )
$ RETURN
B2 = B2*( Z( I4 ) / Z( I4-2 ) )
A2 = A2 + B2
IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
$ GO TO 40
30 CONTINUE
40 CONTINUE
A2 = CNST3*A2
END IF
*
IF( A2.LT.CNST1 )
$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
ELSE
*
* Case 6, no information to guide us.
*
IF( TTYPE.EQ.-6 ) THEN
G = G + THIRD*( ONE-G )
ELSE IF( TTYPE.EQ.-18 ) THEN
G = QURTR*THIRD
ELSE
G = QURTR
END IF
S = G*DMIN
TTYPE = -6
END IF
*
ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
*
* One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
*
IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN
*
* Cases 7 and 8.
*
TTYPE = -7
S = THIRD*DMIN1
IF( Z( NN-5 ).GT.Z( NN-7 ) )
$ RETURN
B1 = Z( NN-5 ) / Z( NN-7 )
B2 = B1
IF( B2.EQ.ZERO )
$ GO TO 60
DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
A2 = B1
IF( Z( I4 ).GT.Z( I4-2 ) )
$ RETURN
B1 = B1*( Z( I4 ) / Z( I4-2 ) )
B2 = B2 + B1
IF( HUNDRD*MAX( B1, A2 ).LT.B2 )
$ GO TO 60
50 CONTINUE
60 CONTINUE
B2 = SQRT( CNST3*B2 )
A2 = DMIN1 / ( ONE+B2**2 )
GAP2 = HALF*DMIN2 - A2
IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
ELSE
S = MAX( S, A2*( ONE-CNST2*B2 ) )
TTYPE = -8
END IF
ELSE
*
* Case 9.
*
S = QURTR*DMIN1
IF( DMIN1.EQ.DN1 )
$ S = HALF*DMIN1
TTYPE = -9
END IF
*
ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
*
* Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
*
* Cases 10 and 11.
*
IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN
TTYPE = -10
S = THIRD*DMIN2
IF( Z( NN-5 ).GT.Z( NN-7 ) )
$ RETURN
B1 = Z( NN-5 ) / Z( NN-7 )
B2 = B1
IF( B2.EQ.ZERO )
$ GO TO 80
DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
IF( Z( I4 ).GT.Z( I4-2 ) )
$ RETURN
B1 = B1*( Z( I4 ) / Z( I4-2 ) )
B2 = B2 + B1
IF( HUNDRD*B1.LT.B2 )
$ GO TO 80
70 CONTINUE
80 CONTINUE
B2 = SQRT( CNST3*B2 )
A2 = DMIN2 / ( ONE+B2**2 )
GAP2 = Z( NN-7 ) + Z( NN-9 ) -
$ SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
ELSE
S = MAX( S, A2*( ONE-CNST2*B2 ) )
END IF
ELSE
S = QURTR*DMIN2
TTYPE = -11
END IF
ELSE IF( N0IN.GT.( N0+2 ) ) THEN
*
* Case 12, more than two eigenvalues deflated. No information.
*
S = ZERO
TTYPE = -12
END IF
*
TAU = S
RETURN
*
* End of DLASQ4
*
END

410
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*> \brief \b DLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq5.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN,
* DNM1, DNM2, IEEE, EPS )
*
* .. Scalar Arguments ..
* LOGICAL IEEE
* INTEGER I0, N0, PP
* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU, SIGMA, EPS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ5 computes one dqds transform in ping-pong form, one
*> version for IEEE machines another for non IEEE machines.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*> an extra argument.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> This is the shift.
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> This is the accumulated shift up to this step.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[out] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[out] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[out] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> d(N0), the last value of d.
*> \endverbatim
*>
*> \param[out] DNM1
*> \verbatim
*> DNM1 is DOUBLE PRECISION
*> d(N0-1).
*> \endverbatim
*>
*> \param[out] DNM2
*> \verbatim
*> DNM2 is DOUBLE PRECISION
*> d(N0-2).
*> \endverbatim
*>
*> \param[in] IEEE
*> \verbatim
*> IEEE is LOGICAL
*> Flag for IEEE or non IEEE arithmetic.
*> \endverbatim
*
*> \param[in] EPS
*> \verbatim
*> EPS is DOUBLE PRECISION
*> This is the value of epsilon used.
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2,
$ DN, DNM1, DNM2, IEEE, EPS )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
LOGICAL IEEE
INTEGER I0, N0, PP
DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU,
$ SIGMA, EPS
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameter ..
DOUBLE PRECISION ZERO, HALF
PARAMETER ( ZERO = 0.0D0, HALF = 0.5 )
* ..
* .. Local Scalars ..
INTEGER J4, J4P2
DOUBLE PRECISION D, EMIN, TEMP, DTHRESH
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( ( N0-I0-1 ).LE.0 )
$ RETURN
*
DTHRESH = EPS*(SIGMA+TAU)
IF( TAU.LT.DTHRESH*HALF ) TAU = ZERO
IF( TAU.NE.ZERO ) THEN
J4 = 4*I0 + PP - 3
EMIN = Z( J4+4 )
D = Z( J4 ) - TAU
DMIN = D
DMIN1 = -Z( J4 )
*
IF( IEEE ) THEN
*
* Code for IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 10 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
TEMP = Z( J4+1 ) / Z( J4-2 )
D = D*TEMP - TAU
DMIN = MIN( DMIN, D )
Z( J4 ) = Z( J4-1 )*TEMP
EMIN = MIN( Z( J4 ), EMIN )
10 CONTINUE
ELSE
DO 20 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
TEMP = Z( J4+2 ) / Z( J4-3 )
D = D*TEMP - TAU
DMIN = MIN( DMIN, D )
Z( J4-1 ) = Z( J4 )*TEMP
EMIN = MIN( Z( J4-1 ), EMIN )
20 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DN )
*
ELSE
*
* Code for non IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 30 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
D = Z( J4+1 )*( D / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4 ) )
30 CONTINUE
ELSE
DO 40 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
D = Z( J4+2 )*( D / Z( J4-3 ) ) - TAU
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4-1 ) )
40 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
IF( DNM2.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
IF( DNM1.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DN )
*
END IF
ELSE
* This is the version that sets d's to zero if they are small enough
J4 = 4*I0 + PP - 3
EMIN = Z( J4+4 )
D = Z( J4 ) - TAU
DMIN = D
DMIN1 = -Z( J4 )
IF( IEEE ) THEN
*
* Code for IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 50 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
TEMP = Z( J4+1 ) / Z( J4-2 )
D = D*TEMP - TAU
IF( D.LT.DTHRESH ) D = ZERO
DMIN = MIN( DMIN, D )
Z( J4 ) = Z( J4-1 )*TEMP
EMIN = MIN( Z( J4 ), EMIN )
50 CONTINUE
ELSE
DO 60 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
TEMP = Z( J4+2 ) / Z( J4-3 )
D = D*TEMP - TAU
IF( D.LT.DTHRESH ) D = ZERO
DMIN = MIN( DMIN, D )
Z( J4-1 ) = Z( J4 )*TEMP
EMIN = MIN( Z( J4-1 ), EMIN )
60 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DN )
*
ELSE
*
* Code for non IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 70 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
D = Z( J4+1 )*( D / Z( J4-2 ) ) - TAU
END IF
IF( D.LT.DTHRESH) D = ZERO
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4 ) )
70 CONTINUE
ELSE
DO 80 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
D = Z( J4+2 )*( D / Z( J4-3 ) ) - TAU
END IF
IF( D.LT.DTHRESH) D = ZERO
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4-1 ) )
80 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
IF( DNM2.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
IF( DNM1.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DN )
*
END IF
END IF
*
Z( J4+2 ) = DN
Z( 4*N0-PP ) = EMIN
RETURN
*
* End of DLASQ5
*
END

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*> \brief \b DLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq6.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN,
* DNM1, DNM2 )
*
* .. Scalar Arguments ..
* INTEGER I0, N0, PP
* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ6 computes one dqd (shift equal to zero) transform in
*> ping-pong form, with protection against underflow and overflow.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*> an extra argument.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[out] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[out] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[out] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> d(N0), the last value of d.
*> \endverbatim
*>
*> \param[out] DNM1
*> \verbatim
*> DNM1 is DOUBLE PRECISION
*> d(N0-1).
*> \endverbatim
*>
*> \param[out] DNM2
*> \verbatim
*> DNM2 is DOUBLE PRECISION
*> d(N0-2).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN,
$ DNM1, DNM2 )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER I0, N0, PP
DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameter ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER J4, J4P2
DOUBLE PRECISION D, EMIN, SAFMIN, TEMP
* ..
* .. External Function ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( ( N0-I0-1 ).LE.0 )
$ RETURN
*
SAFMIN = DLAMCH( 'Safe minimum' )
J4 = 4*I0 + PP - 3
EMIN = Z( J4+4 )
D = Z( J4 )
DMIN = D
*
IF( PP.EQ.0 ) THEN
DO 10 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
IF( Z( J4-2 ).EQ.ZERO ) THEN
Z( J4 ) = ZERO
D = Z( J4+1 )
DMIN = D
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4+1 ).LT.Z( J4-2 ) .AND.
$ SAFMIN*Z( J4-2 ).LT.Z( J4+1 ) ) THEN
TEMP = Z( J4+1 ) / Z( J4-2 )
Z( J4 ) = Z( J4-1 )*TEMP
D = D*TEMP
ELSE
Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
D = Z( J4+1 )*( D / Z( J4-2 ) )
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4 ) )
10 CONTINUE
ELSE
DO 20 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
IF( Z( J4-3 ).EQ.ZERO ) THEN
Z( J4-1 ) = ZERO
D = Z( J4+2 )
DMIN = D
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4+2 ).LT.Z( J4-3 ) .AND.
$ SAFMIN*Z( J4-3 ).LT.Z( J4+2 ) ) THEN
TEMP = Z( J4+2 ) / Z( J4-3 )
Z( J4-1 ) = Z( J4 )*TEMP
D = D*TEMP
ELSE
Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
D = Z( J4+2 )*( D / Z( J4-3 ) )
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4-1 ) )
20 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
IF( Z( J4-2 ).EQ.ZERO ) THEN
Z( J4 ) = ZERO
DNM1 = Z( J4P2+2 )
DMIN = DNM1
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND.
$ SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN
TEMP = Z( J4P2+2 ) / Z( J4-2 )
Z( J4 ) = Z( J4P2 )*TEMP
DNM1 = DNM2*TEMP
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) )
END IF
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
IF( Z( J4-2 ).EQ.ZERO ) THEN
Z( J4 ) = ZERO
DN = Z( J4P2+2 )
DMIN = DN
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND.
$ SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN
TEMP = Z( J4P2+2 ) / Z( J4-2 )
Z( J4 ) = Z( J4P2 )*TEMP
DN = DNM1*TEMP
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) )
END IF
DMIN = MIN( DMIN, DN )
*
Z( J4+2 ) = DN
Z( 4*N0-PP ) = EMIN
RETURN
*
* End of DLASQ6
*
END

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*> \brief \b DLASR applies a sequence of plane rotations to a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, PIVOT, SIDE
* INTEGER LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASR applies a sequence of plane rotations to a real matrix A,
*> from either the left or the right.
*>
*> When SIDE = 'L', the transformation takes the form
*>
*> A := P*A
*>
*> and when SIDE = 'R', the transformation takes the form
*>
*> A := A*P**T
*>
*> where P is an orthogonal matrix consisting of a sequence of z plane
*> rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
*> and P**T is the transpose of P.
*>
*> When DIRECT = 'F' (Forward sequence), then
*>
*> P = P(z-1) * ... * P(2) * P(1)
*>
*> and when DIRECT = 'B' (Backward sequence), then
*>
*> P = P(1) * P(2) * ... * P(z-1)
*>
*> where P(k) is a plane rotation matrix defined by the 2-by-2 rotation
*>
*> R(k) = ( c(k) s(k) )
*> = ( -s(k) c(k) ).
*>
*> When PIVOT = 'V' (Variable pivot), the rotation is performed
*> for the plane (k,k+1), i.e., P(k) has the form
*>
*> P(k) = ( 1 )
*> ( ... )
*> ( 1 )
*> ( c(k) s(k) )
*> ( -s(k) c(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*>
*> where R(k) appears as a rank-2 modification to the identity matrix in
*> rows and columns k and k+1.
*>
*> When PIVOT = 'T' (Top pivot), the rotation is performed for the
*> plane (1,k+1), so P(k) has the form
*>
*> P(k) = ( c(k) s(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*> ( -s(k) c(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*>
*> where R(k) appears in rows and columns 1 and k+1.
*>
*> Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
*> performed for the plane (k,z), giving P(k) the form
*>
*> P(k) = ( 1 )
*> ( ... )
*> ( 1 )
*> ( c(k) s(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*> ( -s(k) c(k) )
*>
*> where R(k) appears in rows and columns k and z. The rotations are
*> performed without ever forming P(k) explicitly.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> Specifies whether the plane rotation matrix P is applied to
*> A on the left or the right.
*> = 'L': Left, compute A := P*A
*> = 'R': Right, compute A:= A*P**T
*> \endverbatim
*>
*> \param[in] PIVOT
*> \verbatim
*> PIVOT is CHARACTER*1
*> Specifies the plane for which P(k) is a plane rotation
*> matrix.
*> = 'V': Variable pivot, the plane (k,k+1)
*> = 'T': Top pivot, the plane (1,k+1)
*> = 'B': Bottom pivot, the plane (k,z)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Specifies whether P is a forward or backward sequence of
*> plane rotations.
*> = 'F': Forward, P = P(z-1)*...*P(2)*P(1)
*> = 'B': Backward, P = P(1)*P(2)*...*P(z-1)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. If m <= 1, an immediate
*> return is effected.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. If n <= 1, an
*> immediate return is effected.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> The cosines c(k) of the plane rotations.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> The sines s(k) of the plane rotations. The 2-by-2 plane
*> rotation part of the matrix P(k), R(k), has the form
*> R(k) = ( c(k) s(k) )
*> ( -s(k) c(k) ).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The M-by-N matrix A. On exit, A is overwritten by P*A if
*> SIDE = 'R' or by A*P**T if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER DIRECT, PIVOT, SIDE
INTEGER LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J
DOUBLE PRECISION CTEMP, STEMP, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( .NOT.( LSAME( SIDE, 'L' ) .OR. LSAME( SIDE, 'R' ) ) ) THEN
INFO = 1
ELSE IF( .NOT.( LSAME( PIVOT, 'V' ) .OR. LSAME( PIVOT,
$ 'T' ) .OR. LSAME( PIVOT, 'B' ) ) ) THEN
INFO = 2
ELSE IF( .NOT.( LSAME( DIRECT, 'F' ) .OR. LSAME( DIRECT, 'B' ) ) )
$ THEN
INFO = 3
ELSE IF( M.LT.0 ) THEN
INFO = 4
ELSE IF( N.LT.0 ) THEN
INFO = 5
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = 9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASR ', INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )
$ RETURN
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form P * A
*
IF( LSAME( PIVOT, 'V' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 20 J = 1, M - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 10 I = 1, N
TEMP = A( J+1, I )
A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
10 CONTINUE
END IF
20 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 40 J = M - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 30 I = 1, N
TEMP = A( J+1, I )
A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
30 CONTINUE
END IF
40 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 60 J = 2, M
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 50 I = 1, N
TEMP = A( J, I )
A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
50 CONTINUE
END IF
60 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 80 J = M, 2, -1
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 70 I = 1, N
TEMP = A( J, I )
A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
70 CONTINUE
END IF
80 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 100 J = 1, M - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 90 I = 1, N
TEMP = A( J, I )
A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
90 CONTINUE
END IF
100 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 120 J = M - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 110 I = 1, N
TEMP = A( J, I )
A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
110 CONTINUE
END IF
120 CONTINUE
END IF
END IF
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form A * P**T
*
IF( LSAME( PIVOT, 'V' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 140 J = 1, N - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 130 I = 1, M
TEMP = A( I, J+1 )
A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
130 CONTINUE
END IF
140 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 160 J = N - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 150 I = 1, M
TEMP = A( I, J+1 )
A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
150 CONTINUE
END IF
160 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 180 J = 2, N
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 170 I = 1, M
TEMP = A( I, J )
A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
170 CONTINUE
END IF
180 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 200 J = N, 2, -1
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 190 I = 1, M
TEMP = A( I, J )
A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
190 CONTINUE
END IF
200 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 220 J = 1, N - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 210 I = 1, M
TEMP = A( I, J )
A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
210 CONTINUE
END IF
220 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 240 J = N - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 230 I = 1, M
TEMP = A( I, J )
A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
230 CONTINUE
END IF
240 CONTINUE
END IF
END IF
END IF
*
RETURN
*
* End of DLASR
*
END

303
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*> \brief \b DLASRT sorts numbers in increasing or decreasing order.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASRT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasrt.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasrt.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasrt.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASRT( ID, N, D, INFO )
*
* .. Scalar Arguments ..
* CHARACTER ID
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Sort the numbers in D in increasing order (if ID = 'I') or
*> in decreasing order (if ID = 'D' ).
*>
*> Use Quick Sort, reverting to Insertion sort on arrays of
*> size <= 20. Dimension of STACK limits N to about 2**32.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ID
*> \verbatim
*> ID is CHARACTER*1
*> = 'I': sort D in increasing order;
*> = 'D': sort D in decreasing order.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The length of the array D.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the array to be sorted.
*> On exit, D has been sorted into increasing order
*> (D(1) <= ... <= D(N) ) or into decreasing order
*> (D(1) >= ... >= D(N) ), depending on ID.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DLASRT( ID, N, D, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER ID
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER SELECT
PARAMETER ( SELECT = 20 )
* ..
* .. Local Scalars ..
INTEGER DIR, ENDD, I, J, START, STKPNT
DOUBLE PRECISION D1, D2, D3, DMNMX, TMP
* ..
* .. Local Arrays ..
INTEGER STACK( 2, 32 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input paramters.
*
INFO = 0
DIR = -1
IF( LSAME( ID, 'D' ) ) THEN
DIR = 0
ELSE IF( LSAME( ID, 'I' ) ) THEN
DIR = 1
END IF
IF( DIR.EQ.-1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASRT', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
STKPNT = 1
STACK( 1, 1 ) = 1
STACK( 2, 1 ) = N
10 CONTINUE
START = STACK( 1, STKPNT )
ENDD = STACK( 2, STKPNT )
STKPNT = STKPNT - 1
IF( ENDD-START.LE.SELECT .AND. ENDD-START.GT.0 ) THEN
*
* Do Insertion sort on D( START:ENDD )
*
IF( DIR.EQ.0 ) THEN
*
* Sort into decreasing order
*
DO 30 I = START + 1, ENDD
DO 20 J = I, START + 1, -1
IF( D( J ).GT.D( J-1 ) ) THEN
DMNMX = D( J )
D( J ) = D( J-1 )
D( J-1 ) = DMNMX
ELSE
GO TO 30
END IF
20 CONTINUE
30 CONTINUE
*
ELSE
*
* Sort into increasing order
*
DO 50 I = START + 1, ENDD
DO 40 J = I, START + 1, -1
IF( D( J ).LT.D( J-1 ) ) THEN
DMNMX = D( J )
D( J ) = D( J-1 )
D( J-1 ) = DMNMX
ELSE
GO TO 50
END IF
40 CONTINUE
50 CONTINUE
*
END IF
*
ELSE IF( ENDD-START.GT.SELECT ) THEN
*
* Partition D( START:ENDD ) and stack parts, largest one first
*
* Choose partition entry as median of 3
*
D1 = D( START )
D2 = D( ENDD )
I = ( START+ENDD ) / 2
D3 = D( I )
IF( D1.LT.D2 ) THEN
IF( D3.LT.D1 ) THEN
DMNMX = D1
ELSE IF( D3.LT.D2 ) THEN
DMNMX = D3
ELSE
DMNMX = D2
END IF
ELSE
IF( D3.LT.D2 ) THEN
DMNMX = D2
ELSE IF( D3.LT.D1 ) THEN
DMNMX = D3
ELSE
DMNMX = D1
END IF
END IF
*
IF( DIR.EQ.0 ) THEN
*
* Sort into decreasing order
*
I = START - 1
J = ENDD + 1
60 CONTINUE
70 CONTINUE
J = J - 1
IF( D( J ).LT.DMNMX )
$ GO TO 70
80 CONTINUE
I = I + 1
IF( D( I ).GT.DMNMX )
$ GO TO 80
IF( I.LT.J ) THEN
TMP = D( I )
D( I ) = D( J )
D( J ) = TMP
GO TO 60
END IF
IF( J-START.GT.ENDD-J-1 ) THEN
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
ELSE
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
END IF
ELSE
*
* Sort into increasing order
*
I = START - 1
J = ENDD + 1
90 CONTINUE
100 CONTINUE
J = J - 1
IF( D( J ).GT.DMNMX )
$ GO TO 100
110 CONTINUE
I = I + 1
IF( D( I ).LT.DMNMX )
$ GO TO 110
IF( I.LT.J ) THEN
TMP = D( I )
D( I ) = D( J )
D( J ) = TMP
GO TO 90
END IF
IF( J-START.GT.ENDD-J-1 ) THEN
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
ELSE
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
END IF
END IF
END IF
IF( STKPNT.GT.0 )
$ GO TO 10
RETURN
*
* End of DLASRT
*
END

325
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*> \brief \b DLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASV2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasv2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasv2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasv2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASV2 computes the singular value decomposition of a 2-by-2
*> triangular matrix
*> [ F G ]
*> [ 0 H ].
*> On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the
*> smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and
*> right singular vectors for abs(SSMAX), giving the decomposition
*>
*> [ CSL SNL ] [ F G ] [ CSR -SNR ] = [ SSMAX 0 ]
*> [-SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ].
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is DOUBLE PRECISION
*> The (1,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*> SSMIN is DOUBLE PRECISION
*> abs(SSMIN) is the smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*> SSMAX is DOUBLE PRECISION
*> abs(SSMAX) is the larger singular value.
*> \endverbatim
*>
*> \param[out] SNL
*> \verbatim
*> SNL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] CSL
*> \verbatim
*> CSL is DOUBLE PRECISION
*> The vector (CSL, SNL) is a unit left singular vector for the
*> singular value abs(SSMAX).
*> \endverbatim
*>
*> \param[out] SNR
*> \verbatim
*> SNR is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] CSR
*> \verbatim
*> CSR is DOUBLE PRECISION
*> The vector (CSR, SNR) is a unit right singular vector for the
*> singular value abs(SSMAX).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Any input parameter may be aliased with any output parameter.
*>
*> Barring over/underflow and assuming a guard digit in subtraction, all
*> output quantities are correct to within a few units in the last
*> place (ulps).
*>
*> In IEEE arithmetic, the code works correctly if one matrix element is
*> infinite.
*>
*> Overflow will not occur unless the largest singular value itself
*> overflows or is within a few ulps of overflow. (On machines with
*> partial overflow, like the Cray, overflow may occur if the largest
*> singular value is within a factor of 2 of overflow.)
*>
*> Underflow is harmless if underflow is gradual. Otherwise, results
*> may correspond to a matrix modified by perturbations of size near
*> the underflow threshold.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
DOUBLE PRECISION CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
DOUBLE PRECISION FOUR
PARAMETER ( FOUR = 4.0D0 )
* ..
* .. Local Scalars ..
LOGICAL GASMAL, SWAP
INTEGER PMAX
DOUBLE PRECISION A, CLT, CRT, D, FA, FT, GA, GT, HA, HT, L, M,
$ MM, R, S, SLT, SRT, T, TEMP, TSIGN, TT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Executable Statements ..
*
FT = F
FA = ABS( FT )
HT = H
HA = ABS( H )
*
* PMAX points to the maximum absolute element of matrix
* PMAX = 1 if F largest in absolute values
* PMAX = 2 if G largest in absolute values
* PMAX = 3 if H largest in absolute values
*
PMAX = 1
SWAP = ( HA.GT.FA )
IF( SWAP ) THEN
PMAX = 3
TEMP = FT
FT = HT
HT = TEMP
TEMP = FA
FA = HA
HA = TEMP
*
* Now FA .ge. HA
*
END IF
GT = G
GA = ABS( GT )
IF( GA.EQ.ZERO ) THEN
*
* Diagonal matrix
*
SSMIN = HA
SSMAX = FA
CLT = ONE
CRT = ONE
SLT = ZERO
SRT = ZERO
ELSE
GASMAL = .TRUE.
IF( GA.GT.FA ) THEN
PMAX = 2
IF( ( FA / GA ).LT.DLAMCH( 'EPS' ) ) THEN
*
* Case of very large GA
*
GASMAL = .FALSE.
SSMAX = GA
IF( HA.GT.ONE ) THEN
SSMIN = FA / ( GA / HA )
ELSE
SSMIN = ( FA / GA )*HA
END IF
CLT = ONE
SLT = HT / GT
SRT = ONE
CRT = FT / GT
END IF
END IF
IF( GASMAL ) THEN
*
* Normal case
*
D = FA - HA
IF( D.EQ.FA ) THEN
*
* Copes with infinite F or H
*
L = ONE
ELSE
L = D / FA
END IF
*
* Note that 0 .le. L .le. 1
*
M = GT / FT
*
* Note that abs(M) .le. 1/macheps
*
T = TWO - L
*
* Note that T .ge. 1
*
MM = M*M
TT = T*T
S = SQRT( TT+MM )
*
* Note that 1 .le. S .le. 1 + 1/macheps
*
IF( L.EQ.ZERO ) THEN
R = ABS( M )
ELSE
R = SQRT( L*L+MM )
END IF
*
* Note that 0 .le. R .le. 1 + 1/macheps
*
A = HALF*( S+R )
*
* Note that 1 .le. A .le. 1 + abs(M)
*
SSMIN = HA / A
SSMAX = FA*A
IF( MM.EQ.ZERO ) THEN
*
* Note that M is very tiny
*
IF( L.EQ.ZERO ) THEN
T = SIGN( TWO, FT )*SIGN( ONE, GT )
ELSE
T = GT / SIGN( D, FT ) + M / T
END IF
ELSE
T = ( M / ( S+T )+M / ( R+L ) )*( ONE+A )
END IF
L = SQRT( T*T+FOUR )
CRT = TWO / L
SRT = T / L
CLT = ( CRT+SRT*M ) / A
SLT = ( HT / FT )*SRT / A
END IF
END IF
IF( SWAP ) THEN
CSL = SRT
SNL = CRT
CSR = SLT
SNR = CLT
ELSE
CSL = CLT
SNL = SLT
CSR = CRT
SNR = SRT
END IF
*
* Correct signs of SSMAX and SSMIN
*
IF( PMAX.EQ.1 )
$ TSIGN = SIGN( ONE, CSR )*SIGN( ONE, CSL )*SIGN( ONE, F )
IF( PMAX.EQ.2 )
$ TSIGN = SIGN( ONE, SNR )*SIGN( ONE, CSL )*SIGN( ONE, G )
IF( PMAX.EQ.3 )
$ TSIGN = SIGN( ONE, SNR )*SIGN( ONE, SNL )*SIGN( ONE, H )
SSMAX = SIGN( SSMAX, TSIGN )
SSMIN = SIGN( SSMIN, TSIGN*SIGN( ONE, F )*SIGN( ONE, H ) )
RETURN
*
* End of DLASV2
*
END

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*> \brief \b DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDW, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATRD reduces NB rows and columns of a real symmetric matrix A to
*> symmetric tridiagonal form by an orthogonal similarity
*> transformation Q**T * A * Q, and returns the matrices V and W which are
*> needed to apply the transformation to the unreduced part of A.
*>
*> If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
*> matrix, of which the upper triangle is supplied;
*> if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
*> matrix, of which the lower triangle is supplied.
*>
*> This is an auxiliary routine called by DSYTRD.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of rows and columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n-by-n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n-by-n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> On exit:
*> if UPLO = 'U', the last NB columns have been reduced to
*> tridiagonal form, with the diagonal elements overwriting
*> the diagonal elements of A; the elements above the diagonal
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors;
*> if UPLO = 'L', the first NB columns have been reduced to
*> tridiagonal form, with the diagonal elements overwriting
*> the diagonal elements of A; the elements below the diagonal
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= (1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
*> elements of the last NB columns of the reduced matrix;
*> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
*> the first NB columns of the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors, stored in
*> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
*> See Further Details.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (LDW,NB)
*> The n-by-nb matrix W required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDW
*> \verbatim
*> LDW is INTEGER
*> The leading dimension of the array W. LDW >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n) H(n-1) . . . H(n-nb+1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
*> and tau in TAU(i-1).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(nb).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
*> and tau in TAU(i).
*>
*> The elements of the vectors v together form the n-by-nb matrix V
*> which is needed, with W, to apply the transformation to the unreduced
*> part of the matrix, using a symmetric rank-2k update of the form:
*> A := A - V*W**T - W*V**T.
*>
*> The contents of A on exit are illustrated by the following examples
*> with n = 5 and nb = 2:
*>
*> if UPLO = 'U': if UPLO = 'L':
*>
*> ( a a a v4 v5 ) ( d )
*> ( a a v4 v5 ) ( 1 d )
*> ( a 1 v5 ) ( v1 1 a )
*> ( d 1 ) ( v1 v2 a a )
*> ( d ) ( v1 v2 a a a )
*>
*> where d denotes a diagonal element of the reduced matrix, a denotes
*> an element of the original matrix that is unchanged, and vi denotes
*> an element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDW, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, HALF
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IW
DOUBLE PRECISION ALPHA
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DGEMV, DLARFG, DSCAL, DSYMV
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Reduce last NB columns of upper triangle
*
DO 10 I = N, N - NB + 1, -1
IW = I - N + NB
IF( I.LT.N ) THEN
*
* Update A(1:i,i)
*
CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
$ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
$ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
END IF
IF( I.GT.1 ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(1:i-2,i)
*
CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) )
E( I-1 ) = A( I-1, I )
A( I-1, I ) = ONE
*
* Compute W(1:i-1,i)
*
CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
$ ZERO, W( 1, IW ), 1 )
IF( I.LT.N ) THEN
CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ),
$ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
$ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
$ W( 1, IW ), 1 )
CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ),
$ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
$ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
$ W( 1, IW ), 1 )
END IF
CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1,
$ A( 1, I ), 1 )
CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
END IF
*
10 CONTINUE
ELSE
*
* Reduce first NB columns of lower triangle
*
DO 20 I = 1, NB
*
* Update A(i:n,i)
*
CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
$ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
$ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
IF( I.LT.N ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(i+2:n,i)
*
CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
$ TAU( I ) )
E( I ) = A( I+1, I )
A( I+1, I ) = ONE
*
* Compute W(i+1:n,i)
*
CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
$ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW,
$ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
$ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA,
$ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
$ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1,
$ A( I+1, I ), 1 )
CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
END IF
*
20 CONTINUE
END IF
*
RETURN
*
* End of DLATRD
*
END

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*> \brief \b DNRM2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DNRM2(N,X,INCX)
*
* .. Scalar Arguments ..
* INTEGER INCX,N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION X(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DNRM2 returns the euclidean norm of a vector via the function
*> name, so that
*>
*> DNRM2 := sqrt( x'*x )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_blas_level1
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> -- This version written on 25-October-1982.
*> Modified on 14-October-1993 to inline the call to DLASSQ.
*> Sven Hammarling, Nag Ltd.
*> \endverbatim
*>
* =====================================================================
DOUBLE PRECISION FUNCTION DNRM2(N,X,INCX)
*
* -- Reference BLAS level1 routine (version 3.4.0) --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INCX,N
* ..
* .. Array Arguments ..
DOUBLE PRECISION X(*)
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE,ZERO
PARAMETER (ONE=1.0D+0,ZERO=0.0D+0)
* ..
* .. Local Scalars ..
DOUBLE PRECISION ABSXI,NORM,SCALE,SSQ
INTEGER IX
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS,SQRT
* ..
IF (N.LT.1 .OR. INCX.LT.1) THEN
NORM = ZERO
ELSE IF (N.EQ.1) THEN
NORM = ABS(X(1))
ELSE
SCALE = ZERO
SSQ = ONE
* The following loop is equivalent to this call to the LAPACK
* auxiliary routine:
* CALL DLASSQ( N, X, INCX, SCALE, SSQ )
*
DO 10 IX = 1,1 + (N-1)*INCX,INCX
IF (X(IX).NE.ZERO) THEN
ABSXI = ABS(X(IX))
IF (SCALE.LT.ABSXI) THEN
SSQ = ONE + SSQ* (SCALE/ABSXI)**2
SCALE = ABSXI
ELSE
SSQ = SSQ + (ABSXI/SCALE)**2
END IF
END IF
10 CONTINUE
NORM = SCALE*SQRT(SSQ)
END IF
*
DNRM2 = NORM
RETURN
*
* End of DNRM2.
*
END

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*> \brief \b DORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORG2R + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorg2r.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorg2r.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorg2r.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORG2R generates an m by n real matrix Q with orthonormal columns,
*> which is defined as the first n columns of a product of k elementary
*> reflectors of order m
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th column must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGEQRF in the first k columns of its array
*> argument A.
*> On exit, the m-by-n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, L
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORG2R', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
* Initialise columns k+1:n to columns of the unit matrix
*
DO 20 J = K + 1, N
DO 10 L = 1, M
A( L, J ) = ZERO
10 CONTINUE
A( J, J ) = ONE
20 CONTINUE
*
DO 40 I = K, 1, -1
*
* Apply H(i) to A(i:m,i:n) from the left
*
IF( I.LT.N ) THEN
A( I, I ) = ONE
CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
$ A( I, I+1 ), LDA, WORK )
END IF
IF( I.LT.M )
$ CALL DSCAL( M-I, -TAU( I ), A( I+1, I ), 1 )
A( I, I ) = ONE - TAU( I )
*
* Set A(1:i-1,i) to zero
*
DO 30 L = 1, I - 1
A( L, I ) = ZERO
30 CONTINUE
40 CONTINUE
RETURN
*
* End of DORG2R
*
END

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*> \brief \b DORGBR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGBR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgbr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgbr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgbr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGBR( VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER VECT
* INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGBR generates one of the real orthogonal matrices Q or P**T
*> determined by DGEBRD when reducing a real matrix A to bidiagonal
*> form: A = Q * B * P**T. Q and P**T are defined as products of
*> elementary reflectors H(i) or G(i) respectively.
*>
*> If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
*> is of order M:
*> if m >= k, Q = H(1) H(2) . . . H(k) and DORGBR returns the first n
*> columns of Q, where m >= n >= k;
*> if m < k, Q = H(1) H(2) . . . H(m-1) and DORGBR returns Q as an
*> M-by-M matrix.
*>
*> If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T
*> is of order N:
*> if k < n, P**T = G(k) . . . G(2) G(1) and DORGBR returns the first m
*> rows of P**T, where n >= m >= k;
*> if k >= n, P**T = G(n-1) . . . G(2) G(1) and DORGBR returns P**T as
*> an N-by-N matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> Specifies whether the matrix Q or the matrix P**T is
*> required, as defined in the transformation applied by DGEBRD:
*> = 'Q': generate Q;
*> = 'P': generate P**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q or P**T to be returned.
*> M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q or P**T to be returned.
*> N >= 0.
*> If VECT = 'Q', M >= N >= min(M,K);
*> if VECT = 'P', N >= M >= min(N,K).
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> If VECT = 'Q', the number of columns in the original M-by-K
*> matrix reduced by DGEBRD.
*> If VECT = 'P', the number of rows in the original K-by-N
*> matrix reduced by DGEBRD.
*> K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the vectors which define the elementary reflectors,
*> as returned by DGEBRD.
*> On exit, the M-by-N matrix Q or P**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension
*> (min(M,K)) if VECT = 'Q'
*> (min(N,K)) if VECT = 'P'
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i) or G(i), which determines Q or P**T, as
*> returned by DGEBRD in its array argument TAUQ or TAUP.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,min(M,N)).
*> For optimum performance LWORK >= min(M,N)*NB, where NB
*> is the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date April 2012
*
*> \ingroup doubleGBcomputational
*
* =====================================================================
SUBROUTINE DORGBR( VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* April 2012
*
* .. Scalar Arguments ..
CHARACTER VECT
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WANTQ
INTEGER I, IINFO, J, LWKOPT, MN
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DORGLQ, DORGQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
WANTQ = LSAME( VECT, 'Q' )
MN = MIN( M, N )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.WANTQ .AND. .NOT.LSAME( VECT, 'P' ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 .OR. ( WANTQ .AND. ( N.GT.M .OR. N.LT.MIN( M,
$ K ) ) ) .OR. ( .NOT.WANTQ .AND. ( M.GT.N .OR. M.LT.
$ MIN( N, K ) ) ) ) THEN
INFO = -3
ELSE IF( K.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LWORK.LT.MAX( 1, MN ) .AND. .NOT.LQUERY ) THEN
INFO = -9
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = 1
IF( WANTQ ) THEN
IF( M.GE.K ) THEN
CALL DORGQR( M, N, K, A, LDA, TAU, WORK, -1, IINFO )
ELSE
IF( M.GT.1 ) THEN
CALL DORGQR( M-1, M-1, M-1, A( 2, 2 ), LDA, TAU, WORK,
$ -1, IINFO )
END IF
END IF
ELSE
IF( K.LT.N ) THEN
CALL DORGLQ( M, N, K, A, LDA, TAU, WORK, -1, IINFO )
ELSE
IF( N.GT.1 ) THEN
CALL DORGLQ( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK,
$ -1, IINFO )
END IF
END IF
END IF
LWKOPT = WORK( 1 )
LWKOPT = MAX (LWKOPT, MN)
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGBR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
WORK( 1 ) = LWKOPT
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( WANTQ ) THEN
*
* Form Q, determined by a call to DGEBRD to reduce an m-by-k
* matrix
*
IF( M.GE.K ) THEN
*
* If m >= k, assume m >= n >= k
*
CALL DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, IINFO )
*
ELSE
*
* If m < k, assume m = n
*
* Shift the vectors which define the elementary reflectors one
* column to the right, and set the first row and column of Q
* to those of the unit matrix
*
DO 20 J = M, 2, -1
A( 1, J ) = ZERO
DO 10 I = J + 1, M
A( I, J ) = A( I, J-1 )
10 CONTINUE
20 CONTINUE
A( 1, 1 ) = ONE
DO 30 I = 2, M
A( I, 1 ) = ZERO
30 CONTINUE
IF( M.GT.1 ) THEN
*
* Form Q(2:m,2:m)
*
CALL DORGQR( M-1, M-1, M-1, A( 2, 2 ), LDA, TAU, WORK,
$ LWORK, IINFO )
END IF
END IF
ELSE
*
* Form P**T, determined by a call to DGEBRD to reduce a k-by-n
* matrix
*
IF( K.LT.N ) THEN
*
* If k < n, assume k <= m <= n
*
CALL DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, IINFO )
*
ELSE
*
* If k >= n, assume m = n
*
* Shift the vectors which define the elementary reflectors one
* row downward, and set the first row and column of P**T to
* those of the unit matrix
*
A( 1, 1 ) = ONE
DO 40 I = 2, N
A( I, 1 ) = ZERO
40 CONTINUE
DO 60 J = 2, N
DO 50 I = J - 1, 2, -1
A( I, J ) = A( I-1, J )
50 CONTINUE
A( 1, J ) = ZERO
60 CONTINUE
IF( N.GT.1 ) THEN
*
* Form P**T(2:n,2:n)
*
CALL DORGLQ( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK,
$ LWORK, IINFO )
END IF
END IF
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORGBR
*
END

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*> \brief \b DORGL2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGL2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgl2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgl2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgl2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGL2( M, N, K, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGL2 generates an m by n real matrix Q with orthonormal rows,
*> which is defined as the first m rows of a product of k elementary
*> reflectors of order n
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th row must contain the vector which defines
*> the elementary reflector H(i), for i = 1,2,...,k, as returned
*> by DGELQF in the first k rows of its array argument A.
*> On exit, the m-by-n matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORGL2( M, N, K, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, L
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGL2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.LE.0 )
$ RETURN
*
IF( K.LT.M ) THEN
*
* Initialise rows k+1:m to rows of the unit matrix
*
DO 20 J = 1, N
DO 10 L = K + 1, M
A( L, J ) = ZERO
10 CONTINUE
IF( J.GT.K .AND. J.LE.M )
$ A( J, J ) = ONE
20 CONTINUE
END IF
*
DO 40 I = K, 1, -1
*
* Apply H(i) to A(i:m,i:n) from the right
*
IF( I.LT.N ) THEN
IF( I.LT.M ) THEN
A( I, I ) = ONE
CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
$ TAU( I ), A( I+1, I ), LDA, WORK )
END IF
CALL DSCAL( N-I, -TAU( I ), A( I, I+1 ), LDA )
END IF
A( I, I ) = ONE - TAU( I )
*
* Set A(i,1:i-1) to zero
*
DO 30 L = 1, I - 1
A( I, L ) = ZERO
30 CONTINUE
40 CONTINUE
RETURN
*
* End of DORGL2
*
END

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*> \brief \b DORGLQ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGLQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorglq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorglq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorglq.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGLQ generates an M-by-N real matrix Q with orthonormal rows,
*> which is defined as the first M rows of a product of K elementary
*> reflectors of order N
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th row must contain the vector which defines
*> the elementary reflector H(i), for i = 1,2,...,k, as returned
*> by DGELQF in the first k rows of its array argument A.
*> On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,M).
*> For optimum performance LWORK >= M*NB, where NB is
*> the optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, J, KI, KK, L, LDWORK,
$ LWKOPT, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORGL2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NB = ILAENV( 1, 'DORGLQ', ' ', M, N, K, -1 )
LWKOPT = MAX( 1, M )*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGLQ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.LE.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = M
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DORGLQ', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = M
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORGLQ', ' ', M, N, K, -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the last block.
* The first kk rows are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
* Set A(kk+1:m,1:kk) to zero.
*
DO 20 J = 1, KK
DO 10 I = KK + 1, M
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the last or only block.
*
IF( KK.LT.M )
$ CALL DORGL2( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA,
$ TAU( KK+1 ), WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF( I+IB.LE.M ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I, I ),
$ LDA, TAU( I ), WORK, LDWORK )
*
* Apply H**T to A(i+ib:m,i:n) from the right
*
CALL DLARFB( 'Right', 'Transpose', 'Forward', 'Rowwise',
$ M-I-IB+1, N-I+1, IB, A( I, I ), LDA, WORK,
$ LDWORK, A( I+IB, I ), LDA, WORK( IB+1 ),
$ LDWORK )
END IF
*
* Apply H**T to columns i:n of current block
*
CALL DORGL2( IB, N-I+1, IB, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
*
* Set columns 1:i-1 of current block to zero
*
DO 40 J = 1, I - 1
DO 30 L = I, I + IB - 1
A( L, J ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DORGLQ
*
END

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*> \brief \b DORGQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORGQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorgqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorgqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorgqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORGQR generates an M-by-N real matrix Q with orthonormal columns,
*> which is defined as the first N columns of a product of K elementary
*> reflectors of order M
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Q. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines the
*> matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the i-th column must contain the vector which
*> defines the elementary reflector H(i), for i = 1,2,...,k, as
*> returned by DGEQRF in the first k columns of its array
*> argument A.
*> On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*> For optimum performance LWORK >= N*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument has an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, J, KI, KK, L, LDWORK,
$ LWKOPT, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORG2R, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
NB = ILAENV( 1, 'DORGQR', ' ', M, N, K, -1 )
LWKOPT = MAX( 1, N )*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGQR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DORGQR', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORGQR', ' ', M, N, K, -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the last block.
* The first kk columns are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
* Set A(1:kk,kk+1:n) to zero.
*
DO 20 J = KK + 1, N
DO 10 I = 1, KK
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the last or only block.
*
IF( KK.LT.N )
$ CALL DORG2R( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA,
$ TAU( KK+1 ), WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF( I+IB.LE.N ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
$ A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(i:m,i+ib:n) from the left
*
CALL DLARFB( 'Left', 'No transpose', 'Forward',
$ 'Columnwise', M-I+1, N-I-IB+1, IB,
$ A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
$ LDA, WORK( IB+1 ), LDWORK )
END IF
*
* Apply H to rows i:m of current block
*
CALL DORG2R( M-I+1, IB, IB, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
*
* Set rows 1:i-1 of current block to zero
*
DO 40 J = I, I + IB - 1
DO 30 L = 1, I - 1
A( L, J ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DORGQR
*
END

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*> \brief \b DORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORM2L + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorm2l.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorm2l.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorm2l.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORM2L overwrites the general real m by n matrix C with
*>
*> Q * C if SIDE = 'L' and TRANS = 'N', or
*>
*> Q**T * C if SIDE = 'L' and TRANS = 'T', or
*>
*> C * Q if SIDE = 'R' and TRANS = 'N', or
*>
*> C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left
*> = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply Q (No transpose)
*> = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQLF in the last k columns of its array argument A.
*> A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L',
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, NOTRAN
INTEGER I, I1, I2, I3, MI, NI, NQ
DOUBLE PRECISION AII
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORM2L', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
$ RETURN
*
IF( ( LEFT .AND. NOTRAN ) .OR. ( .NOT.LEFT .AND. .NOT.NOTRAN ) )
$ THEN
I1 = 1
I2 = K
I3 = 1
ELSE
I1 = K
I2 = 1
I3 = -1
END IF
*
IF( LEFT ) THEN
NI = N
ELSE
MI = M
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) is applied to C(1:m-k+i,1:n)
*
MI = M - K + I
ELSE
*
* H(i) is applied to C(1:m,1:n-k+i)
*
NI = N - K + I
END IF
*
* Apply H(i)
*
AII = A( NQ-K+I, I )
A( NQ-K+I, I ) = ONE
CALL DLARF( SIDE, MI, NI, A( 1, I ), 1, TAU( I ), C, LDC,
$ WORK )
A( NQ-K+I, I ) = AII
10 CONTINUE
RETURN
*
* End of DORM2L
*
END

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*> \brief \b DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORM2R + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorm2r.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorm2r.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorm2r.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORM2R overwrites the general real m by n matrix C with
*>
*> Q * C if SIDE = 'L' and TRANS = 'N', or
*>
*> Q**T* C if SIDE = 'L' and TRANS = 'T', or
*>
*> C * Q if SIDE = 'R' and TRANS = 'N', or
*>
*> C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left
*> = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply Q (No transpose)
*> = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQRF in the first k columns of its array argument A.
*> A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L',
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, NOTRAN
INTEGER I, I1, I2, I3, IC, JC, MI, NI, NQ
DOUBLE PRECISION AII
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORM2R', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
$ RETURN
*
IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. ( .NOT.LEFT .AND. NOTRAN ) )
$ THEN
I1 = 1
I2 = K
I3 = 1
ELSE
I1 = K
I2 = 1
I3 = -1
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H(i) is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H(i)
*
AII = A( I, I )
A( I, I ) = ONE
CALL DLARF( SIDE, MI, NI, A( I, I ), 1, TAU( I ), C( IC, JC ),
$ LDC, WORK )
A( I, I ) = AII
10 CONTINUE
RETURN
*
* End of DORM2R
*
END

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*> \brief \b DORMBR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMBR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormbr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormbr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormbr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMBR( VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C,
* LDC, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS, VECT
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> If VECT = 'Q', DORMBR overwrites the general real M-by-N matrix C
*> with
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> If VECT = 'P', DORMBR overwrites the general real M-by-N matrix C
*> with
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': P * C C * P
*> TRANS = 'T': P**T * C C * P**T
*>
*> Here Q and P**T are the orthogonal matrices determined by DGEBRD when
*> reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
*> P**T are defined as products of elementary reflectors H(i) and G(i)
*> respectively.
*>
*> Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
*> order of the orthogonal matrix Q or P**T that is applied.
*>
*> If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
*> if nq >= k, Q = H(1) H(2) . . . H(k);
*> if nq < k, Q = H(1) H(2) . . . H(nq-1).
*>
*> If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
*> if k < nq, P = G(1) G(2) . . . G(k);
*> if k >= nq, P = G(1) G(2) . . . G(nq-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> = 'Q': apply Q or Q**T;
*> = 'P': apply P or P**T.
*> \endverbatim
*>
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q, Q**T, P or P**T from the Left;
*> = 'R': apply Q, Q**T, P or P**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q or P;
*> = 'T': Transpose, apply Q**T or P**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> If VECT = 'Q', the number of columns in the original
*> matrix reduced by DGEBRD.
*> If VECT = 'P', the number of rows in the original
*> matrix reduced by DGEBRD.
*> K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,min(nq,K)) if VECT = 'Q'
*> (LDA,nq) if VECT = 'P'
*> The vectors which define the elementary reflectors H(i) and
*> G(i), whose products determine the matrices Q and P, as
*> returned by DGEBRD.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If VECT = 'Q', LDA >= max(1,nq);
*> if VECT = 'P', LDA >= max(1,min(nq,K)).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (min(nq,K))
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i) or G(i) which determines Q or P, as returned
*> by DGEBRD in the array argument TAUQ or TAUP.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
*> or P*C or P**T*C or C*P or C*P**T.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For optimum performance LWORK >= N*NB if SIDE = 'L', and
*> LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*> blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORMBR( VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C,
$ LDC, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS, VECT
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL APPLYQ, LEFT, LQUERY, NOTRAN
CHARACTER TRANST
INTEGER I1, I2, IINFO, LWKOPT, MI, NB, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DORMLQ, DORMQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
APPLYQ = LSAME( VECT, 'Q' )
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q or P and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = N
ELSE
NQ = N
NW = M
END IF
IF( .NOT.APPLYQ .AND. .NOT.LSAME( VECT, 'P' ) ) THEN
INFO = -1
ELSE IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( K.LT.0 ) THEN
INFO = -6
ELSE IF( ( APPLYQ .AND. LDA.LT.MAX( 1, NQ ) ) .OR.
$ ( .NOT.APPLYQ .AND. LDA.LT.MAX( 1, MIN( NQ, K ) ) ) )
$ THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -11
ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
*
IF( INFO.EQ.0 ) THEN
IF( APPLYQ ) THEN
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M-1, N, M-1,
$ -1 )
ELSE
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N-1, N-1,
$ -1 )
END IF
ELSE
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMLQ', SIDE // TRANS, M-1, N, M-1,
$ -1 )
ELSE
NB = ILAENV( 1, 'DORMLQ', SIDE // TRANS, M, N-1, N-1,
$ -1 )
END IF
END IF
LWKOPT = MAX( 1, NW )*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMBR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
WORK( 1 ) = 1
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
IF( APPLYQ ) THEN
*
* Apply Q
*
IF( NQ.GE.K ) THEN
*
* Q was determined by a call to DGEBRD with nq >= k
*
CALL DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, IINFO )
ELSE IF( NQ.GT.1 ) THEN
*
* Q was determined by a call to DGEBRD with nq < k
*
IF( LEFT ) THEN
MI = M - 1
NI = N
I1 = 2
I2 = 1
ELSE
MI = M
NI = N - 1
I1 = 1
I2 = 2
END IF
CALL DORMQR( SIDE, TRANS, MI, NI, NQ-1, A( 2, 1 ), LDA, TAU,
$ C( I1, I2 ), LDC, WORK, LWORK, IINFO )
END IF
ELSE
*
* Apply P
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
IF( NQ.GT.K ) THEN
*
* P was determined by a call to DGEBRD with nq > k
*
CALL DORMLQ( SIDE, TRANST, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, IINFO )
ELSE IF( NQ.GT.1 ) THEN
*
* P was determined by a call to DGEBRD with nq <= k
*
IF( LEFT ) THEN
MI = M - 1
NI = N
I1 = 2
I2 = 1
ELSE
MI = M
NI = N - 1
I1 = 1
I2 = 2
END IF
CALL DORMLQ( SIDE, TRANST, MI, NI, NQ-1, A( 1, 2 ), LDA,
$ TAU, C( I1, I2 ), LDC, WORK, LWORK, IINFO )
END IF
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMBR
*
END

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*> \brief \b DORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORML2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorml2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorml2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorml2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORML2 overwrites the general real m by n matrix C with
*>
*> Q * C if SIDE = 'L' and TRANS = 'N', or
*>
*> Q**T* C if SIDE = 'L' and TRANS = 'T', or
*>
*> C * Q if SIDE = 'R' and TRANS = 'N', or
*>
*> C * Q**T if SIDE = 'R' and TRANS = 'T',
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF. Q is of order m if SIDE = 'L' and of order n
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left
*> = 'R': apply Q or Q**T from the Right
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': apply Q (No transpose)
*> = 'T': apply Q**T (Transpose)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L',
*> (LDA,N) if SIDE = 'R'
*> The i-th row must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGELQF in the first k rows of its array argument A.
*> A is modified by the routine but restored on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the m by n matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension
*> (N) if SIDE = 'L',
*> (M) if SIDE = 'R'
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, NOTRAN
INTEGER I, I1, I2, I3, IC, JC, MI, NI, NQ
DOUBLE PRECISION AII
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLARF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
*
* NQ is the order of Q
*
IF( LEFT ) THEN
NQ = M
ELSE
NQ = N
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORML2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 )
$ RETURN
*
IF( ( LEFT .AND. NOTRAN ) .OR. ( .NOT.LEFT .AND. .NOT.NOTRAN ) )
$ THEN
I1 = 1
I2 = K
I3 = 1
ELSE
I1 = K
I2 = 1
I3 = -1
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
DO 10 I = I1, I2, I3
IF( LEFT ) THEN
*
* H(i) is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H(i) is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H(i)
*
AII = A( I, I )
A( I, I ) = ONE
CALL DLARF( SIDE, MI, NI, A( I, I ), LDA, TAU( I ),
$ C( IC, JC ), LDC, WORK )
A( I, I ) = AII
10 CONTINUE
RETURN
*
* End of DORML2
*
END

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*> \brief \b DORMLQ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMLQ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormlq.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormlq.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormlq.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMLQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMLQ overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGELQF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L',
*> (LDA,N) if SIDE = 'R'
*> The i-th row must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGELQF in the first k rows of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,K).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGELQF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For optimum performance LWORK >= N*NB if SIDE = 'L', and
*> LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*> blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORMLQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT
PARAMETER ( NBMAX = 64, LDT = NBMAX+1 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
CHARACTER TRANST
INTEGER I, I1, I2, I3, IB, IC, IINFO, IWS, JC, LDWORK,
$ LWKOPT, MI, NB, NBMIN, NI, NQ, NW
* ..
* .. Local Arrays ..
DOUBLE PRECISION T( LDT, NBMAX )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORML2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = N
ELSE
NQ = N
NW = M
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Determine the block size. NB may be at most NBMAX, where NBMAX
* is used to define the local array T.
*
NB = MIN( NBMAX, ILAENV( 1, 'DORMLQ', SIDE // TRANS, M, N, K,
$ -1 ) )
LWKOPT = MAX( 1, NW )*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMLQ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IWS = NW*NB
IF( LWORK.LT.IWS ) THEN
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORMLQ', SIDE // TRANS, M, N, K,
$ -1 ) )
END IF
ELSE
IWS = NW
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK,
$ IINFO )
ELSE
*
* Use blocked code
*
IF( ( LEFT .AND. NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Rowwise', NQ-I+1, IB, A( I, I ),
$ LDA, TAU( I ), T, LDT )
IF( LEFT ) THEN
*
* H or H**T is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H or H**T is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H or H**T
*
CALL DLARFB( SIDE, TRANST, 'Forward', 'Rowwise', MI, NI, IB,
$ A( I, I ), LDA, T, LDT, C( IC, JC ), LDC, WORK,
$ LDWORK )
10 CONTINUE
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMLQ
*
END

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*> \brief \b DORMQL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMQL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormql.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormql.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormql.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMQL( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMQL overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(k) . . . H(2) H(1)
*>
*> as returned by DGEQLF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQLF in the last k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQLF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For optimum performance LWORK >= N*NB if SIDE = 'L', and
*> LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*> blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORMQL( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT
PARAMETER ( NBMAX = 64, LDT = NBMAX+1 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
INTEGER I, I1, I2, I3, IB, IINFO, IWS, LDWORK, LWKOPT,
$ MI, NB, NBMIN, NI, NQ, NW
* ..
* .. Local Arrays ..
DOUBLE PRECISION T( LDT, NBMAX )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORM2L, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
*
IF( INFO.EQ.0 ) THEN
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
LWKOPT = 1
ELSE
*
* Determine the block size. NB may be at most NBMAX, where
* NBMAX is used to define the local array T.
*
NB = MIN( NBMAX, ILAENV( 1, 'DORMQL', SIDE // TRANS, M, N,
$ K, -1 ) )
LWKOPT = NW*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMQL', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IWS = NW*NB
IF( LWORK.LT.IWS ) THEN
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORMQL', SIDE // TRANS, M, N, K,
$ -1 ) )
END IF
ELSE
IWS = NW
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK,
$ IINFO )
ELSE
*
* Use blocked code
*
IF( ( LEFT .AND. NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
ELSE
MI = M
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARFT( 'Backward', 'Columnwise', NQ-K+I+IB-1, IB,
$ A( 1, I ), LDA, TAU( I ), T, LDT )
IF( LEFT ) THEN
*
* H or H**T is applied to C(1:m-k+i+ib-1,1:n)
*
MI = M - K + I + IB - 1
ELSE
*
* H or H**T is applied to C(1:m,1:n-k+i+ib-1)
*
NI = N - K + I + IB - 1
END IF
*
* Apply H or H**T
*
CALL DLARFB( SIDE, TRANS, 'Backward', 'Columnwise', MI, NI,
$ IB, A( 1, I ), LDA, T, LDT, C, LDC, WORK,
$ LDWORK )
10 CONTINUE
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMQL
*
END

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*> \brief \b DORMQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS
* INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMQR overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix defined as the product of k
*> elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k)
*>
*> as returned by DGEQRF. Q is of order M if SIDE = 'L' and of order N
*> if SIDE = 'R'.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of elementary reflectors whose product defines
*> the matrix Q.
*> If SIDE = 'L', M >= K >= 0;
*> if SIDE = 'R', N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,K)
*> The i-th column must contain the vector which defines the
*> elementary reflector H(i), for i = 1,2,...,k, as returned by
*> DGEQRF in the first k columns of its array argument A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If SIDE = 'L', LDA >= max(1,M);
*> if SIDE = 'R', LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DGEQRF.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For optimum performance LWORK >= N*NB if SIDE = 'L', and
*> LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*> blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT
PARAMETER ( NBMAX = 64, LDT = NBMAX+1 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
INTEGER I, I1, I2, I3, IB, IC, IINFO, IWS, JC, LDWORK,
$ LWKOPT, MI, NB, NBMIN, NI, NQ, NW
* ..
* .. Local Arrays ..
DOUBLE PRECISION T( LDT, NBMAX )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORM2R, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = N
ELSE
NQ = N
NW = M
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Determine the block size. NB may be at most NBMAX, where NBMAX
* is used to define the local array T.
*
NB = MIN( NBMAX, ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N, K,
$ -1 ) )
LWKOPT = MAX( 1, NW )*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMQR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IWS = NW*NB
IF( LWORK.LT.IWS ) THEN
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORMQR', SIDE // TRANS, M, N, K,
$ -1 ) )
END IF
ELSE
IWS = NW
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK,
$ IINFO )
ELSE
*
* Use blocked code
*
IF( ( LEFT .AND. .NOT.NOTRAN ) .OR.
$ ( .NOT.LEFT .AND. NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
JC = 1
ELSE
MI = M
IC = 1
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL DLARFT( 'Forward', 'Columnwise', NQ-I+1, IB, A( I, I ),
$ LDA, TAU( I ), T, LDT )
IF( LEFT ) THEN
*
* H or H**T is applied to C(i:m,1:n)
*
MI = M - I + 1
IC = I
ELSE
*
* H or H**T is applied to C(1:m,i:n)
*
NI = N - I + 1
JC = I
END IF
*
* Apply H or H**T
*
CALL DLARFB( SIDE, TRANS, 'Forward', 'Columnwise', MI, NI,
$ IB, A( I, I ), LDA, T, LDT, C( IC, JC ), LDC,
$ WORK, LDWORK )
10 CONTINUE
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMQR
*
END

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*> \brief \b DORMTR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DORMTR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dormtr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dormtr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dormtr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DORMTR( SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER SIDE, TRANS, UPLO
* INTEGER INFO, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORMTR overwrites the general real M-by-N matrix C with
*>
*> SIDE = 'L' SIDE = 'R'
*> TRANS = 'N': Q * C C * Q
*> TRANS = 'T': Q**T * C C * Q**T
*>
*> where Q is a real orthogonal matrix of order nq, with nq = m if
*> SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
*> nq-1 elementary reflectors, as returned by DSYTRD:
*>
*> if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
*>
*> if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'L': apply Q or Q**T from the Left;
*> = 'R': apply Q or Q**T from the Right.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A contains elementary reflectors
*> from DSYTRD;
*> = 'L': Lower triangle of A contains elementary reflectors
*> from DSYTRD.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': No transpose, apply Q;
*> = 'T': Transpose, apply Q**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix C. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix C. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension
*> (LDA,M) if SIDE = 'L'
*> (LDA,N) if SIDE = 'R'
*> The vectors which define the elementary reflectors, as
*> returned by DSYTRD.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i), as returned by DSYTRD.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension (LDC,N)
*> On entry, the M-by-N matrix C.
*> On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If SIDE = 'L', LWORK >= max(1,N);
*> if SIDE = 'R', LWORK >= max(1,M).
*> For optimum performance LWORK >= N*NB if SIDE = 'L', and
*> LWORK >= M*NB if SIDE = 'R', where NB is the optimal
*> blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DORMTR( SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC,
$ WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, UPPER
INTEGER I1, I2, IINFO, LWKOPT, MI, NB, NI, NQ, NW
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DORMQL, DORMQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = N
ELSE
NQ = N
NW = M
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.LSAME( TRANS, 'T' ) )
$ THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
*
IF( INFO.EQ.0 ) THEN
IF( UPPER ) THEN
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMQL', SIDE // TRANS, M-1, N, M-1,
$ -1 )
ELSE
NB = ILAENV( 1, 'DORMQL', SIDE // TRANS, M, N-1, N-1,
$ -1 )
END IF
ELSE
IF( LEFT ) THEN
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M-1, N, M-1,
$ -1 )
ELSE
NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N-1, N-1,
$ -1 )
END IF
END IF
LWKOPT = MAX( 1, NW )*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORMTR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. NQ.EQ.1 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( LEFT ) THEN
MI = M - 1
NI = N
ELSE
MI = M
NI = N - 1
END IF
*
IF( UPPER ) THEN
*
* Q was determined by a call to DSYTRD with UPLO = 'U'
*
CALL DORMQL( SIDE, TRANS, MI, NI, NQ-1, A( 1, 2 ), LDA, TAU, C,
$ LDC, WORK, LWORK, IINFO )
ELSE
*
* Q was determined by a call to DSYTRD with UPLO = 'L'
*
IF( LEFT ) THEN
I1 = 2
I2 = 1
ELSE
I1 = 1
I2 = 2
END IF
CALL DORMQR( SIDE, TRANS, MI, NI, NQ-1, A( 2, 1 ), LDA, TAU,
$ C( I1, I2 ), LDC, WORK, LWORK, IINFO )
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of DORMTR
*
END

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*> \brief \b DPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotf2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOTF2 computes the Cholesky factorization of a real symmetric
*> positive definite matrix A.
*>
*> The factorization has the form
*> A = U**T * U , if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n by n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n by n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization A = U**T *U or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, the leading minor of order k is not
*> positive definite, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doublePOcomputational
*
* =====================================================================
SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U**T *U.
*
DO 10 J = 1, N
*
* Compute U(J,J) and test for non-positive-definiteness.
*
AJJ = A( J, J ) - DDOT( J-1, A( 1, J ), 1, A( 1, J ), 1 )
IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of row J.
*
IF( J.LT.N ) THEN
CALL DGEMV( 'Transpose', J-1, N-J, -ONE, A( 1, J+1 ),
$ LDA, A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
END IF
10 CONTINUE
ELSE
*
* Compute the Cholesky factorization A = L*L**T.
*
DO 20 J = 1, N
*
* Compute L(J,J) and test for non-positive-definiteness.
*
AJJ = A( J, J ) - DDOT( J-1, A( J, 1 ), LDA, A( J, 1 ),
$ LDA )
IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of column J.
*
IF( J.LT.N ) THEN
CALL DGEMV( 'No transpose', N-J, J-1, -ONE, A( J+1, 1 ),
$ LDA, A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
END IF
20 CONTINUE
END IF
GO TO 40
*
30 CONTINUE
INFO = J
*
40 CONTINUE
RETURN
*
* End of DPOTF2
*
END

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*> \brief \b DPOTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPOTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpotrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpotrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpotrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOTRF computes the Cholesky factorization of a real symmetric
*> positive definite matrix A.
*>
*> The factorization has the form
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the block version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the factor U or L from the Cholesky
*> factorization A = U**T*U or A = L*L**T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading minor of order i is not
*> positive definite, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doublePOcomputational
*
* =====================================================================
SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, JB, NB
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DPOTF2, DSYRK, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'DPOTRF', UPLO, N, -1, -1, -1 )
IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
* Use unblocked code.
*
CALL DPOTF2( UPLO, N, A, LDA, INFO )
ELSE
*
* Use blocked code.
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U**T*U.
*
DO 10 J = 1, N, NB
*
* Update and factorize the current diagonal block and test
* for non-positive-definiteness.
*
JB = MIN( NB, N-J+1 )
CALL DSYRK( 'Upper', 'Transpose', JB, J-1, -ONE,
$ A( 1, J ), LDA, ONE, A( J, J ), LDA )
CALL DPOTF2( 'Upper', JB, A( J, J ), LDA, INFO )
IF( INFO.NE.0 )
$ GO TO 30
IF( J+JB.LE.N ) THEN
*
* Compute the current block row.
*
CALL DGEMM( 'Transpose', 'No transpose', JB, N-J-JB+1,
$ J-1, -ONE, A( 1, J ), LDA, A( 1, J+JB ),
$ LDA, ONE, A( J, J+JB ), LDA )
CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit',
$ JB, N-J-JB+1, ONE, A( J, J ), LDA,
$ A( J, J+JB ), LDA )
END IF
10 CONTINUE
*
ELSE
*
* Compute the Cholesky factorization A = L*L**T.
*
DO 20 J = 1, N, NB
*
* Update and factorize the current diagonal block and test
* for non-positive-definiteness.
*
JB = MIN( NB, N-J+1 )
CALL DSYRK( 'Lower', 'No transpose', JB, J-1, -ONE,
$ A( J, 1 ), LDA, ONE, A( J, J ), LDA )
CALL DPOTF2( 'Lower', JB, A( J, J ), LDA, INFO )
IF( INFO.NE.0 )
$ GO TO 30
IF( J+JB.LE.N ) THEN
*
* Compute the current block column.
*
CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
$ J-1, -ONE, A( J+JB, 1 ), LDA, A( J, 1 ),
$ LDA, ONE, A( J+JB, J ), LDA )
CALL DTRSM( 'Right', 'Lower', 'Transpose', 'Non-unit',
$ N-J-JB+1, JB, ONE, A( J, J ), LDA,
$ A( J+JB, J ), LDA )
END IF
20 CONTINUE
END IF
END IF
GO TO 40
*
30 CONTINUE
INFO = INFO + J - 1
*
40 CONTINUE
RETURN
*
* End of DPOTRF
*
END

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*> \brief \b DROT
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DROT(N,DX,INCX,DY,INCY,C,S)
*
* .. Scalar Arguments ..
* DOUBLE PRECISION C,S
* INTEGER INCX,INCY,N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION DX(*),DY(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DROT applies a plane rotation.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_blas_level1
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> jack dongarra, linpack, 3/11/78.
*> modified 12/3/93, array(1) declarations changed to array(*)
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DROT(N,DX,INCX,DY,INCY,C,S)
*
* -- Reference BLAS level1 routine (version 3.4.0) --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
DOUBLE PRECISION C,S
INTEGER INCX,INCY,N
* ..
* .. Array Arguments ..
DOUBLE PRECISION DX(*),DY(*)
* ..
*
* =====================================================================
*
* .. Local Scalars ..
DOUBLE PRECISION DTEMP
INTEGER I,IX,IY
* ..
IF (N.LE.0) RETURN
IF (INCX.EQ.1 .AND. INCY.EQ.1) THEN
*
* code for both increments equal to 1
*
DO I = 1,N
DTEMP = C*DX(I) + S*DY(I)
DY(I) = C*DY(I) - S*DX(I)
DX(I) = DTEMP
END DO
ELSE
*
* code for unequal increments or equal increments not equal
* to 1
*
IX = 1
IY = 1
IF (INCX.LT.0) IX = (-N+1)*INCX + 1
IF (INCY.LT.0) IY = (-N+1)*INCY + 1
DO I = 1,N
DTEMP = C*DX(IX) + S*DY(IY)
DY(IY) = C*DY(IY) - S*DX(IX)
DX(IX) = DTEMP
IX = IX + INCX
IY = IY + INCY
END DO
END IF
RETURN
END

489
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*> \brief \b DSTEBZ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEDC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstedc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstedc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstedc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
* LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPZ
* INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEDC computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric tridiagonal matrix using the divide and conquer method.
*> The eigenvectors of a full or band real symmetric matrix can also be
*> found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this
*> matrix to tridiagonal form.
*>
*> This code makes very mild assumptions about floating point
*> arithmetic. It will work on machines with a guard digit in
*> add/subtract, or on those binary machines without guard digits
*> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
*> It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none. See DLAED3 for details.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Compute eigenvalues only.
*> = 'I': Compute eigenvectors of tridiagonal matrix also.
*> = 'V': Compute eigenvectors of original dense symmetric
*> matrix also. On entry, Z contains the orthogonal
*> matrix used to reduce the original matrix to
*> tridiagonal form.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the diagonal elements of the tridiagonal matrix.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the subdiagonal elements of the tridiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> On entry, if COMPZ = 'V', then Z contains the orthogonal
*> matrix used in the reduction to tridiagonal form.
*> On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
*> orthonormal eigenvectors of the original symmetric matrix,
*> and if COMPZ = 'I', Z contains the orthonormal eigenvectors
*> of the symmetric tridiagonal matrix.
*> If COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1.
*> If eigenvectors are desired, then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
*> If COMPZ = 'V' and N > 1 then LWORK must be at least
*> ( 1 + 3*N + 2*N*lg N + 4*N**2 ),
*> where lg( N ) = smallest integer k such
*> that 2**k >= N.
*> If COMPZ = 'I' and N > 1 then LWORK must be at least
*> ( 1 + 4*N + N**2 ).
*> Note that for COMPZ = 'I' or 'V', then if N is less than or
*> equal to the minimum divide size, usually 25, then LWORK need
*> only be max(1,2*(N-1)).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
*> If COMPZ = 'V' and N > 1 then LIWORK must be at least
*> ( 6 + 6*N + 5*N*lg N ).
*> If COMPZ = 'I' and N > 1 then LIWORK must be at least
*> ( 3 + 5*N ).
*> Note that for COMPZ = 'I' or 'V', then if N is less than or
*> equal to the minimum divide size, usually 25, then LIWORK
*> need only be 1.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the IWORK array,
*> returns this value as the first entry of the IWORK array, and
*> no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: The algorithm failed to compute an eigenvalue while
*> working on the submatrix lying in rows and columns
*> INFO/(N+1) through mod(INFO,N+1).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK,
$ LIWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN,
$ LWMIN, M, SMLSIZ, START, STOREZ, STRTRW
DOUBLE PRECISION EPS, ORGNRM, P, TINY
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL LSAME, ILAENV, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLAED0, DLASCL, DLASET, DLASRT,
$ DSTEQR, DSTERF, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG, MAX, MOD, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ICOMPZ = 0
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ICOMPZ = 2
ELSE
ICOMPZ = -1
END IF
IF( ICOMPZ.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ( LDZ.LT.1 ) .OR.
$ ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
INFO = -6
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Compute the workspace requirements
*
SMLSIZ = ILAENV( 9, 'DSTEDC', ' ', 0, 0, 0, 0 )
IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE IF( N.LE.SMLSIZ ) THEN
LIWMIN = 1
LWMIN = 2*( N - 1 )
ELSE
LGN = INT( LOG( DBLE( N ) )/LOG( TWO ) )
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( ICOMPZ.EQ.1 ) THEN
LWMIN = 1 + 3*N + 2*N*LGN + 4*N**2
LIWMIN = 6 + 6*N + 5*N*LGN
ELSE IF( ICOMPZ.EQ.2 ) THEN
LWMIN = 1 + 4*N + N**2
LIWMIN = 3 + 5*N
END IF
END IF
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT. LQUERY ) THEN
INFO = -8
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT. LQUERY ) THEN
INFO = -10
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEDC', -INFO )
RETURN
ELSE IF (LQUERY) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( N.EQ.1 ) THEN
IF( ICOMPZ.NE.0 )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* If the following conditional clause is removed, then the routine
* will use the Divide and Conquer routine to compute only the
* eigenvalues, which requires (3N + 3N**2) real workspace and
* (2 + 5N + 2N lg(N)) integer workspace.
* Since on many architectures DSTERF is much faster than any other
* algorithm for finding eigenvalues only, it is used here
* as the default. If the conditional clause is removed, then
* information on the size of workspace needs to be changed.
*
* If COMPZ = 'N', use DSTERF to compute the eigenvalues.
*
IF( ICOMPZ.EQ.0 ) THEN
CALL DSTERF( N, D, E, INFO )
GO TO 50
END IF
*
* If N is smaller than the minimum divide size (SMLSIZ+1), then
* solve the problem with another solver.
*
IF( N.LE.SMLSIZ ) THEN
*
CALL DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
ELSE
*
* If COMPZ = 'V', the Z matrix must be stored elsewhere for later
* use.
*
IF( ICOMPZ.EQ.1 ) THEN
STOREZ = 1 + N*N
ELSE
STOREZ = 1
END IF
*
IF( ICOMPZ.EQ.2 ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
END IF
*
* Scale.
*
ORGNRM = DLANST( 'M', N, D, E )
IF( ORGNRM.EQ.ZERO )
$ GO TO 50
*
EPS = DLAMCH( 'Epsilon' )
*
START = 1
*
* while ( START <= N )
*
10 CONTINUE
IF( START.LE.N ) THEN
*
* Let FINISH be the position of the next subdiagonal entry
* such that E( FINISH ) <= TINY or FINISH = N if no such
* subdiagonal exists. The matrix identified by the elements
* between START and FINISH constitutes an independent
* sub-problem.
*
FINISH = START
20 CONTINUE
IF( FINISH.LT.N ) THEN
TINY = EPS*SQRT( ABS( D( FINISH ) ) )*
$ SQRT( ABS( D( FINISH+1 ) ) )
IF( ABS( E( FINISH ) ).GT.TINY ) THEN
FINISH = FINISH + 1
GO TO 20
END IF
END IF
*
* (Sub) Problem determined. Compute its size and solve it.
*
M = FINISH - START + 1
IF( M.EQ.1 ) THEN
START = FINISH + 1
GO TO 10
END IF
IF( M.GT.SMLSIZ ) THEN
*
* Scale.
*
ORGNRM = DLANST( 'M', M, D( START ), E( START ) )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ), M,
$ INFO )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1, E( START ),
$ M-1, INFO )
*
IF( ICOMPZ.EQ.1 ) THEN
STRTRW = 1
ELSE
STRTRW = START
END IF
CALL DLAED0( ICOMPZ, N, M, D( START ), E( START ),
$ Z( STRTRW, START ), LDZ, WORK( 1 ), N,
$ WORK( STOREZ ), IWORK, INFO )
IF( INFO.NE.0 ) THEN
INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) +
$ MOD( INFO, ( M+1 ) ) + START - 1
GO TO 50
END IF
*
* Scale back.
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ), M,
$ INFO )
*
ELSE
IF( ICOMPZ.EQ.1 ) THEN
*
* Since QR won't update a Z matrix which is larger than
* the length of D, we must solve the sub-problem in a
* workspace and then multiply back into Z.
*
CALL DSTEQR( 'I', M, D( START ), E( START ), WORK, M,
$ WORK( M*M+1 ), INFO )
CALL DLACPY( 'A', N, M, Z( 1, START ), LDZ,
$ WORK( STOREZ ), N )
CALL DGEMM( 'N', 'N', N, M, M, ONE,
$ WORK( STOREZ ), N, WORK, M, ZERO,
$ Z( 1, START ), LDZ )
ELSE IF( ICOMPZ.EQ.2 ) THEN
CALL DSTEQR( 'I', M, D( START ), E( START ),
$ Z( START, START ), LDZ, WORK, INFO )
ELSE
CALL DSTERF( M, D( START ), E( START ), INFO )
END IF
IF( INFO.NE.0 ) THEN
INFO = START*( N+1 ) + FINISH
GO TO 50
END IF
END IF
*
START = FINISH + 1
GO TO 10
END IF
*
* endwhile
*
* If the problem split any number of times, then the eigenvalues
* will not be properly ordered. Here we permute the eigenvalues
* (and the associated eigenvectors) into ascending order.
*
IF( M.NE.N ) THEN
IF( ICOMPZ.EQ.0 ) THEN
*
* Use Quick Sort
*
CALL DLASRT( 'I', N, D, INFO )
*
ELSE
*
* Use Selection Sort to minimize swaps of eigenvectors
*
DO 40 II = 2, N
I = II - 1
K = I
P = D( I )
DO 30 J = II, N
IF( D( J ).LT.P ) THEN
K = J
P = D( J )
END IF
30 CONTINUE
IF( K.NE.I ) THEN
D( K ) = D( I )
D( I ) = P
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
END IF
40 CONTINUE
END IF
END IF
END IF
*
50 CONTINUE
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DSTEDC
*
END

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*> \brief \b DSTEQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsteqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsteqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsteqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPZ
* INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEQR computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric tridiagonal matrix using the implicit QL or QR method.
*> The eigenvectors of a full or band symmetric matrix can also be found
*> if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to
*> tridiagonal form.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Compute eigenvalues only.
*> = 'V': Compute eigenvalues and eigenvectors of the original
*> symmetric matrix. On entry, Z must contain the
*> orthogonal matrix used to reduce the original matrix
*> to tridiagonal form.
*> = 'I': Compute eigenvalues and eigenvectors of the
*> tridiagonal matrix. Z is initialized to the identity
*> matrix.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the diagonal elements of the tridiagonal matrix.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, N)
*> On entry, if COMPZ = 'V', then Z contains the orthogonal
*> matrix used in the reduction to tridiagonal form.
*> On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
*> orthonormal eigenvectors of the original symmetric matrix,
*> and if COMPZ = 'I', Z contains the orthonormal eigenvectors
*> of the symmetric tridiagonal matrix.
*> If COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> eigenvectors are desired, then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
*> If COMPZ = 'N', then WORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: the algorithm has failed to find all the eigenvalues in
*> a total of 30*N iterations; if INFO = i, then i
*> elements of E have not converged to zero; on exit, D
*> and E contain the elements of a symmetric tridiagonal
*> matrix which is orthogonally similar to the original
*> matrix.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, THREE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0 )
INTEGER MAXIT
PARAMETER ( MAXIT = 30 )
* ..
* .. Local Scalars ..
INTEGER I, ICOMPZ, II, ISCALE, J, JTOT, K, L, L1, LEND,
$ LENDM1, LENDP1, LENDSV, LM1, LSV, M, MM, MM1,
$ NM1, NMAXIT
DOUBLE PRECISION ANORM, B, C, EPS, EPS2, F, G, P, R, RT1, RT2,
$ S, SAFMAX, SAFMIN, SSFMAX, SSFMIN, TST
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANST, DLAPY2
EXTERNAL LSAME, DLAMCH, DLANST, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DLAE2, DLAEV2, DLARTG, DLASCL, DLASET, DLASR,
$ DLASRT, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ICOMPZ = 0
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ICOMPZ = 2
ELSE
ICOMPZ = -1
END IF
IF( ICOMPZ.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
$ N ) ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEQR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ICOMPZ.EQ.2 )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Determine the unit roundoff and over/underflow thresholds.
*
EPS = DLAMCH( 'E' )
EPS2 = EPS**2
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SSFMAX = SQRT( SAFMAX ) / THREE
SSFMIN = SQRT( SAFMIN ) / EPS2
*
* Compute the eigenvalues and eigenvectors of the tridiagonal
* matrix.
*
IF( ICOMPZ.EQ.2 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
NMAXIT = N*MAXIT
JTOT = 0
*
* Determine where the matrix splits and choose QL or QR iteration
* for each block, according to whether top or bottom diagonal
* element is smaller.
*
L1 = 1
NM1 = N - 1
*
10 CONTINUE
IF( L1.GT.N )
$ GO TO 160
IF( L1.GT.1 )
$ E( L1-1 ) = ZERO
IF( L1.LE.NM1 ) THEN
DO 20 M = L1, NM1
TST = ABS( E( M ) )
IF( TST.EQ.ZERO )
$ GO TO 30
IF( TST.LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
$ 1 ) ) ) )*EPS ) THEN
E( M ) = ZERO
GO TO 30
END IF
20 CONTINUE
END IF
M = N
*
30 CONTINUE
L = L1
LSV = L
LEND = M
LENDSV = LEND
L1 = M + 1
IF( LEND.EQ.L )
$ GO TO 10
*
* Scale submatrix in rows and columns L to LEND
*
ANORM = DLANST( 'M', LEND-L+1, D( L ), E( L ) )
ISCALE = 0
IF( ANORM.EQ.ZERO )
$ GO TO 10
IF( ANORM.GT.SSFMAX ) THEN
ISCALE = 1
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
$ INFO )
ELSE IF( ANORM.LT.SSFMIN ) THEN
ISCALE = 2
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
$ INFO )
END IF
*
* Choose between QL and QR iteration
*
IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
LEND = LSV
L = LENDSV
END IF
*
IF( LEND.GT.L ) THEN
*
* QL Iteration
*
* Look for small subdiagonal element.
*
40 CONTINUE
IF( L.NE.LEND ) THEN
LENDM1 = LEND - 1
DO 50 M = L, LENDM1
TST = ABS( E( M ) )**2
IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M+1 ) )+
$ SAFMIN )GO TO 60
50 CONTINUE
END IF
*
M = LEND
*
60 CONTINUE
IF( M.LT.LEND )
$ E( M ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 80
*
* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
* to compute its eigensystem.
*
IF( M.EQ.L+1 ) THEN
IF( ICOMPZ.GT.0 ) THEN
CALL DLAEV2( D( L ), E( L ), D( L+1 ), RT1, RT2, C, S )
WORK( L ) = C
WORK( N-1+L ) = S
CALL DLASR( 'R', 'V', 'B', N, 2, WORK( L ),
$ WORK( N-1+L ), Z( 1, L ), LDZ )
ELSE
CALL DLAE2( D( L ), E( L ), D( L+1 ), RT1, RT2 )
END IF
D( L ) = RT1
D( L+1 ) = RT2
E( L ) = ZERO
L = L + 2
IF( L.LE.LEND )
$ GO TO 40
GO TO 140
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 140
JTOT = JTOT + 1
*
* Form shift.
*
G = ( D( L+1 )-P ) / ( TWO*E( L ) )
R = DLAPY2( G, ONE )
G = D( M ) - P + ( E( L ) / ( G+SIGN( R, G ) ) )
*
S = ONE
C = ONE
P = ZERO
*
* Inner loop
*
MM1 = M - 1
DO 70 I = MM1, L, -1
F = S*E( I )
B = C*E( I )
CALL DLARTG( G, F, C, S, R )
IF( I.NE.M-1 )
$ E( I+1 ) = R
G = D( I+1 ) - P
R = ( D( I )-G )*S + TWO*C*B
P = S*R
D( I+1 ) = G + P
G = C*R - B
*
* If eigenvectors are desired, then save rotations.
*
IF( ICOMPZ.GT.0 ) THEN
WORK( I ) = C
WORK( N-1+I ) = -S
END IF
*
70 CONTINUE
*
* If eigenvectors are desired, then apply saved rotations.
*
IF( ICOMPZ.GT.0 ) THEN
MM = M - L + 1
CALL DLASR( 'R', 'V', 'B', N, MM, WORK( L ), WORK( N-1+L ),
$ Z( 1, L ), LDZ )
END IF
*
D( L ) = D( L ) - P
E( L ) = G
GO TO 40
*
* Eigenvalue found.
*
80 CONTINUE
D( L ) = P
*
L = L + 1
IF( L.LE.LEND )
$ GO TO 40
GO TO 140
*
ELSE
*
* QR Iteration
*
* Look for small superdiagonal element.
*
90 CONTINUE
IF( L.NE.LEND ) THEN
LENDP1 = LEND + 1
DO 100 M = L, LENDP1, -1
TST = ABS( E( M-1 ) )**2
IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M-1 ) )+
$ SAFMIN )GO TO 110
100 CONTINUE
END IF
*
M = LEND
*
110 CONTINUE
IF( M.GT.LEND )
$ E( M-1 ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 130
*
* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
* to compute its eigensystem.
*
IF( M.EQ.L-1 ) THEN
IF( ICOMPZ.GT.0 ) THEN
CALL DLAEV2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2, C, S )
WORK( M ) = C
WORK( N-1+M ) = S
CALL DLASR( 'R', 'V', 'F', N, 2, WORK( M ),
$ WORK( N-1+M ), Z( 1, L-1 ), LDZ )
ELSE
CALL DLAE2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2 )
END IF
D( L-1 ) = RT1
D( L ) = RT2
E( L-1 ) = ZERO
L = L - 2
IF( L.GE.LEND )
$ GO TO 90
GO TO 140
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 140
JTOT = JTOT + 1
*
* Form shift.
*
G = ( D( L-1 )-P ) / ( TWO*E( L-1 ) )
R = DLAPY2( G, ONE )
G = D( M ) - P + ( E( L-1 ) / ( G+SIGN( R, G ) ) )
*
S = ONE
C = ONE
P = ZERO
*
* Inner loop
*
LM1 = L - 1
DO 120 I = M, LM1
F = S*E( I )
B = C*E( I )
CALL DLARTG( G, F, C, S, R )
IF( I.NE.M )
$ E( I-1 ) = R
G = D( I ) - P
R = ( D( I+1 )-G )*S + TWO*C*B
P = S*R
D( I ) = G + P
G = C*R - B
*
* If eigenvectors are desired, then save rotations.
*
IF( ICOMPZ.GT.0 ) THEN
WORK( I ) = C
WORK( N-1+I ) = S
END IF
*
120 CONTINUE
*
* If eigenvectors are desired, then apply saved rotations.
*
IF( ICOMPZ.GT.0 ) THEN
MM = L - M + 1
CALL DLASR( 'R', 'V', 'F', N, MM, WORK( M ), WORK( N-1+M ),
$ Z( 1, M ), LDZ )
END IF
*
D( L ) = D( L ) - P
E( LM1 ) = G
GO TO 90
*
* Eigenvalue found.
*
130 CONTINUE
D( L ) = P
*
L = L - 1
IF( L.GE.LEND )
$ GO TO 90
GO TO 140
*
END IF
*
* Undo scaling if necessary
*
140 CONTINUE
IF( ISCALE.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV, 1, E( LSV ),
$ N, INFO )
ELSE IF( ISCALE.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV, 1, E( LSV ),
$ N, INFO )
END IF
*
* Check for no convergence to an eigenvalue after a total
* of N*MAXIT iterations.
*
IF( JTOT.LT.NMAXIT )
$ GO TO 10
DO 150 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
150 CONTINUE
GO TO 190
*
* Order eigenvalues and eigenvectors.
*
160 CONTINUE
IF( ICOMPZ.EQ.0 ) THEN
*
* Use Quick Sort
*
CALL DLASRT( 'I', N, D, INFO )
*
ELSE
*
* Use Selection Sort to minimize swaps of eigenvectors
*
DO 180 II = 2, N
I = II - 1
K = I
P = D( I )
DO 170 J = II, N
IF( D( J ).LT.P ) THEN
K = J
P = D( J )
END IF
170 CONTINUE
IF( K.NE.I ) THEN
D( K ) = D( I )
D( I ) = P
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
END IF
180 CONTINUE
END IF
*
190 CONTINUE
RETURN
*
* End of DSTEQR
*
END

426
lib/linalg/dsterf.f Normal file
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@ -0,0 +1,426 @@
*> \brief \b DSTERF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTERF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsterf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsterf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsterf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTERF( N, D, E, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTERF computes all eigenvalues of a symmetric tridiagonal matrix
*> using the Pal-Walker-Kahan variant of the QL or QR algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix.
*> On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: the algorithm failed to find all of the eigenvalues in
*> a total of 30*N iterations; if INFO = i, then i
*> elements of E have not converged to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DSTERF( N, D, E, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, THREE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0 )
INTEGER MAXIT
PARAMETER ( MAXIT = 30 )
* ..
* .. Local Scalars ..
INTEGER I, ISCALE, JTOT, L, L1, LEND, LENDSV, LSV, M,
$ NMAXIT
DOUBLE PRECISION ALPHA, ANORM, BB, C, EPS, EPS2, GAMMA, OLDC,
$ OLDGAM, P, R, RT1, RT2, RTE, S, SAFMAX, SAFMIN,
$ SIGMA, SSFMAX, SSFMIN, RMAX
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANST, DLAPY2
EXTERNAL DLAMCH, DLANST, DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL DLAE2, DLASCL, DLASRT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
* Quick return if possible
*
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DSTERF', -INFO )
RETURN
END IF
IF( N.LE.1 )
$ RETURN
*
* Determine the unit roundoff for this environment.
*
EPS = DLAMCH( 'E' )
EPS2 = EPS**2
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SSFMAX = SQRT( SAFMAX ) / THREE
SSFMIN = SQRT( SAFMIN ) / EPS2
RMAX = DLAMCH( 'O' )
*
* Compute the eigenvalues of the tridiagonal matrix.
*
NMAXIT = N*MAXIT
SIGMA = ZERO
JTOT = 0
*
* Determine where the matrix splits and choose QL or QR iteration
* for each block, according to whether top or bottom diagonal
* element is smaller.
*
L1 = 1
*
10 CONTINUE
IF( L1.GT.N )
$ GO TO 170
IF( L1.GT.1 )
$ E( L1-1 ) = ZERO
DO 20 M = L1, N - 1
IF( ABS( E( M ) ).LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
$ 1 ) ) ) )*EPS ) THEN
E( M ) = ZERO
GO TO 30
END IF
20 CONTINUE
M = N
*
30 CONTINUE
L = L1
LSV = L
LEND = M
LENDSV = LEND
L1 = M + 1
IF( LEND.EQ.L )
$ GO TO 10
*
* Scale submatrix in rows and columns L to LEND
*
ANORM = DLANST( 'M', LEND-L+1, D( L ), E( L ) )
ISCALE = 0
IF( ANORM.EQ.ZERO )
$ GO TO 10
IF( (ANORM.GT.SSFMAX) ) THEN
ISCALE = 1
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
$ INFO )
ELSE IF( ANORM.LT.SSFMIN ) THEN
ISCALE = 2
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
$ INFO )
END IF
*
DO 40 I = L, LEND - 1
E( I ) = E( I )**2
40 CONTINUE
*
* Choose between QL and QR iteration
*
IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
LEND = LSV
L = LENDSV
END IF
*
IF( LEND.GE.L ) THEN
*
* QL Iteration
*
* Look for small subdiagonal element.
*
50 CONTINUE
IF( L.NE.LEND ) THEN
DO 60 M = L, LEND - 1
IF( ABS( E( M ) ).LE.EPS2*ABS( D( M )*D( M+1 ) ) )
$ GO TO 70
60 CONTINUE
END IF
M = LEND
*
70 CONTINUE
IF( M.LT.LEND )
$ E( M ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 90
*
* If remaining matrix is 2 by 2, use DLAE2 to compute its
* eigenvalues.
*
IF( M.EQ.L+1 ) THEN
RTE = SQRT( E( L ) )
CALL DLAE2( D( L ), RTE, D( L+1 ), RT1, RT2 )
D( L ) = RT1
D( L+1 ) = RT2
E( L ) = ZERO
L = L + 2
IF( L.LE.LEND )
$ GO TO 50
GO TO 150
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 150
JTOT = JTOT + 1
*
* Form shift.
*
RTE = SQRT( E( L ) )
SIGMA = ( D( L+1 )-P ) / ( TWO*RTE )
R = DLAPY2( SIGMA, ONE )
SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
C = ONE
S = ZERO
GAMMA = D( M ) - SIGMA
P = GAMMA*GAMMA
*
* Inner loop
*
DO 80 I = M - 1, L, -1
BB = E( I )
R = P + BB
IF( I.NE.M-1 )
$ E( I+1 ) = S*R
OLDC = C
C = P / R
S = BB / R
OLDGAM = GAMMA
ALPHA = D( I )
GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
D( I+1 ) = OLDGAM + ( ALPHA-GAMMA )
IF( C.NE.ZERO ) THEN
P = ( GAMMA*GAMMA ) / C
ELSE
P = OLDC*BB
END IF
80 CONTINUE
*
E( L ) = S*P
D( L ) = SIGMA + GAMMA
GO TO 50
*
* Eigenvalue found.
*
90 CONTINUE
D( L ) = P
*
L = L + 1
IF( L.LE.LEND )
$ GO TO 50
GO TO 150
*
ELSE
*
* QR Iteration
*
* Look for small superdiagonal element.
*
100 CONTINUE
DO 110 M = L, LEND + 1, -1
IF( ABS( E( M-1 ) ).LE.EPS2*ABS( D( M )*D( M-1 ) ) )
$ GO TO 120
110 CONTINUE
M = LEND
*
120 CONTINUE
IF( M.GT.LEND )
$ E( M-1 ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 140
*
* If remaining matrix is 2 by 2, use DLAE2 to compute its
* eigenvalues.
*
IF( M.EQ.L-1 ) THEN
RTE = SQRT( E( L-1 ) )
CALL DLAE2( D( L ), RTE, D( L-1 ), RT1, RT2 )
D( L ) = RT1
D( L-1 ) = RT2
E( L-1 ) = ZERO
L = L - 2
IF( L.GE.LEND )
$ GO TO 100
GO TO 150
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 150
JTOT = JTOT + 1
*
* Form shift.
*
RTE = SQRT( E( L-1 ) )
SIGMA = ( D( L-1 )-P ) / ( TWO*RTE )
R = DLAPY2( SIGMA, ONE )
SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
C = ONE
S = ZERO
GAMMA = D( M ) - SIGMA
P = GAMMA*GAMMA
*
* Inner loop
*
DO 130 I = M, L - 1
BB = E( I )
R = P + BB
IF( I.NE.M )
$ E( I-1 ) = S*R
OLDC = C
C = P / R
S = BB / R
OLDGAM = GAMMA
ALPHA = D( I+1 )
GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
D( I ) = OLDGAM + ( ALPHA-GAMMA )
IF( C.NE.ZERO ) THEN
P = ( GAMMA*GAMMA ) / C
ELSE
P = OLDC*BB
END IF
130 CONTINUE
*
E( L-1 ) = S*P
D( L ) = SIGMA + GAMMA
GO TO 100
*
* Eigenvalue found.
*
140 CONTINUE
D( L ) = P
*
L = L - 1
IF( L.GE.LEND )
$ GO TO 100
GO TO 150
*
END IF
*
* Undo scaling if necessary
*
150 CONTINUE
IF( ISCALE.EQ.1 )
$ CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
IF( ISCALE.EQ.2 )
$ CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
*
* Check for no convergence to an eigenvalue after a total
* of N*MAXIT iterations.
*
IF( JTOT.LT.NMAXIT )
$ GO TO 10
DO 160 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
160 CONTINUE
GO TO 180
*
* Sort eigenvalues in increasing order.
*
170 CONTINUE
CALL DLASRT( 'I', N, D, INFO )
*
180 CONTINUE
RETURN
*
* End of DSTERF
*
END

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*> \brief <b> DSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK,
* LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, LDA, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYEVD computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric matrix A. If eigenvectors are desired, it uses a
*> divide and conquer algorithm.
*>
*> The divide and conquer algorithm makes very mild assumptions about
*> floating point arithmetic. It will work on machines with a guard
*> digit in add/subtract, or on those binary machines without guard
*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*>
*> Because of large use of BLAS of level 3, DSYEVD needs N**2 more
*> workspace than DSYEVX.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of A contains the
*> upper triangular part of the matrix A. If UPLO = 'L',
*> the leading N-by-N lower triangular part of A contains
*> the lower triangular part of the matrix A.
*> On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*> orthonormal eigenvectors of the matrix A.
*> If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
*> or the upper triangle (if UPLO='U') of A, including the
*> diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N <= 1, LWORK must be at least 1.
*> If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1.
*> If JOBZ = 'V' and N > 1, LWORK must be at least
*> 1 + 6*N + 2*N**2.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If N <= 1, LIWORK must be at least 1.
*> If JOBZ = 'N' and N > 1, LIWORK must be at least 1.
*> If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i and JOBZ = 'N', then the algorithm failed
*> to converge; i off-diagonal elements of an intermediate
*> tridiagonal form did not converge to zero;
*> if INFO = i and JOBZ = 'V', then the algorithm failed
*> to compute an eigenvalue while working on the submatrix
*> lying in rows and columns INFO/(N+1) through
*> mod(INFO,N+1).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleSYeigen
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee \n
*> Modified description of INFO. Sven, 16 Feb 05. \n
*>
* =====================================================================
SUBROUTINE DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK,
$ LIWORK, INFO )
*
* -- LAPACK driver routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, LDA, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
*
LOGICAL LOWER, LQUERY, WANTZ
INTEGER IINFO, INDE, INDTAU, INDWK2, INDWRK, ISCALE,
$ LIOPT, LIWMIN, LLWORK, LLWRK2, LOPT, LWMIN
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, DLAMCH, DLANSY, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLASCL, DORMTR, DSCAL, DSTEDC, DSTERF,
$ DSYTRD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.LE.1 ) THEN
LIWMIN = 1
LWMIN = 1
LOPT = LWMIN
LIOPT = LIWMIN
ELSE
IF( WANTZ ) THEN
LIWMIN = 3 + 5*N
LWMIN = 1 + 6*N + 2*N**2
ELSE
LIWMIN = 1
LWMIN = 2*N + 1
END IF
LOPT = MAX( LWMIN, 2*N +
$ ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
LIOPT = LIWMIN
END IF
WORK( 1 ) = LOPT
IWORK( 1 ) = LIOPT
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -8
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYEVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
W( 1 ) = A( 1, 1 )
IF( WANTZ )
$ A( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 )
$ CALL DLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
*
* Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
INDE = 1
INDTAU = INDE + N
INDWRK = INDTAU + N
LLWORK = LWORK - INDWRK + 1
INDWK2 = INDWRK + N*N
LLWRK2 = LWORK - INDWK2 + 1
*
CALL DSYTRD( UPLO, N, A, LDA, W, WORK( INDE ), WORK( INDTAU ),
$ WORK( INDWRK ), LLWORK, IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, first call
* DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
* tridiagonal matrix, then call DORMTR to multiply it by the
* Householder transformations stored in A.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
$ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
CALL DORMTR( 'L', UPLO, 'N', N, N, A, LDA, WORK( INDTAU ),
$ WORK( INDWRK ), N, WORK( INDWK2 ), LLWRK2, IINFO )
CALL DLACPY( 'A', N, N, WORK( INDWRK ), N, A, LDA )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 )
$ CALL DSCAL( N, ONE / SIGMA, W, 1 )
*
WORK( 1 ) = LOPT
IWORK( 1 ) = LIOPT
*
RETURN
*
* End of DSYEVD
*
END

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*> \brief \b DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygs2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygs2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygs2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYGS2 reduces a real symmetric-definite generalized eigenproblem
*> to standard form.
*>
*> If ITYPE = 1, the problem is A*x = lambda*B*x,
*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*>
*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T *A*L.
*>
*> B must have been previously factorized as U**T *U or L*L**T by DPOTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*> = 2 or 3: compute U*A*U**T or L**T *A*L.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored, and how B has been factorized.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n by n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n by n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the transformed matrix, stored in the
*> same format as A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> The triangular factor from the Cholesky factorization of B,
*> as returned by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleSYcomputational
*
* =====================================================================
SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, HALF
PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER K
DOUBLE PRECISION AKK, BKK, CT
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DSCAL, DSYR2, DTRMV, DTRSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYGS2', -INFO )
RETURN
END IF
*
IF( ITYPE.EQ.1 ) THEN
IF( UPPER ) THEN
*
* Compute inv(U**T)*A*inv(U)
*
DO 10 K = 1, N
*
* Update the upper triangle of A(k:n,k:n)
*
AKK = A( K, K )
BKK = B( K, K )
AKK = AKK / BKK**2
A( K, K ) = AKK
IF( K.LT.N ) THEN
CALL DSCAL( N-K, ONE / BKK, A( K, K+1 ), LDA )
CT = -HALF*AKK
CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
$ LDA )
CALL DSYR2( UPLO, N-K, -ONE, A( K, K+1 ), LDA,
$ B( K, K+1 ), LDB, A( K+1, K+1 ), LDA )
CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
$ LDA )
CALL DTRSV( UPLO, 'Transpose', 'Non-unit', N-K,
$ B( K+1, K+1 ), LDB, A( K, K+1 ), LDA )
END IF
10 CONTINUE
ELSE
*
* Compute inv(L)*A*inv(L**T)
*
DO 20 K = 1, N
*
* Update the lower triangle of A(k:n,k:n)
*
AKK = A( K, K )
BKK = B( K, K )
AKK = AKK / BKK**2
A( K, K ) = AKK
IF( K.LT.N ) THEN
CALL DSCAL( N-K, ONE / BKK, A( K+1, K ), 1 )
CT = -HALF*AKK
CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 )
CALL DSYR2( UPLO, N-K, -ONE, A( K+1, K ), 1,
$ B( K+1, K ), 1, A( K+1, K+1 ), LDA )
CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 )
CALL DTRSV( UPLO, 'No transpose', 'Non-unit', N-K,
$ B( K+1, K+1 ), LDB, A( K+1, K ), 1 )
END IF
20 CONTINUE
END IF
ELSE
IF( UPPER ) THEN
*
* Compute U*A*U**T
*
DO 30 K = 1, N
*
* Update the upper triangle of A(1:k,1:k)
*
AKK = A( K, K )
BKK = B( K, K )
CALL DTRMV( UPLO, 'No transpose', 'Non-unit', K-1, B,
$ LDB, A( 1, K ), 1 )
CT = HALF*AKK
CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
CALL DSYR2( UPLO, K-1, ONE, A( 1, K ), 1, B( 1, K ), 1,
$ A, LDA )
CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
CALL DSCAL( K-1, BKK, A( 1, K ), 1 )
A( K, K ) = AKK*BKK**2
30 CONTINUE
ELSE
*
* Compute L**T *A*L
*
DO 40 K = 1, N
*
* Update the lower triangle of A(1:k,1:k)
*
AKK = A( K, K )
BKK = B( K, K )
CALL DTRMV( UPLO, 'Transpose', 'Non-unit', K-1, B, LDB,
$ A( K, 1 ), LDA )
CT = HALF*AKK
CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
CALL DSYR2( UPLO, K-1, ONE, A( K, 1 ), LDA, B( K, 1 ),
$ LDB, A, LDA )
CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
CALL DSCAL( K-1, BKK, A( K, 1 ), LDA )
A( K, K ) = AKK*BKK**2
40 CONTINUE
END IF
END IF
RETURN
*
* End of DSYGS2
*
END

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*> \brief \b DSYGST
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygst.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYGST reduces a real symmetric-definite generalized eigenproblem
*> to standard form.
*>
*> If ITYPE = 1, the problem is A*x = lambda*B*x,
*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*>
*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
*>
*> B must have been previously factorized as U**T*U or L*L**T by DPOTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*> = 2 or 3: compute U*A*U**T or L**T*A*L.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored and B is factored as
*> U**T*U;
*> = 'L': Lower triangle of A is stored and B is factored as
*> L*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the transformed matrix, stored in the
*> same format as A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> The triangular factor from the Cholesky factorization of B,
*> as returned by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleSYcomputational
*
* =====================================================================
SUBROUTINE DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, HALF
PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER K, KB, NB
* ..
* .. External Subroutines ..
EXTERNAL DSYGS2, DSYMM, DSYR2K, DTRMM, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYGST', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'DSYGST', UPLO, N, -1, -1, -1 )
*
IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
* Use unblocked code
*
CALL DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
ELSE
*
* Use blocked code
*
IF( ITYPE.EQ.1 ) THEN
IF( UPPER ) THEN
*
* Compute inv(U**T)*A*inv(U)
*
DO 10 K = 1, N, NB
KB = MIN( N-K+1, NB )
*
* Update the upper triangle of A(k:n,k:n)
*
CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
$ B( K, K ), LDB, INFO )
IF( K+KB.LE.N ) THEN
CALL DTRSM( 'Left', UPLO, 'Transpose', 'Non-unit',
$ KB, N-K-KB+1, ONE, B( K, K ), LDB,
$ A( K, K+KB ), LDA )
CALL DSYMM( 'Left', UPLO, KB, N-K-KB+1, -HALF,
$ A( K, K ), LDA, B( K, K+KB ), LDB, ONE,
$ A( K, K+KB ), LDA )
CALL DSYR2K( UPLO, 'Transpose', N-K-KB+1, KB, -ONE,
$ A( K, K+KB ), LDA, B( K, K+KB ), LDB,
$ ONE, A( K+KB, K+KB ), LDA )
CALL DSYMM( 'Left', UPLO, KB, N-K-KB+1, -HALF,
$ A( K, K ), LDA, B( K, K+KB ), LDB, ONE,
$ A( K, K+KB ), LDA )
CALL DTRSM( 'Right', UPLO, 'No transpose',
$ 'Non-unit', KB, N-K-KB+1, ONE,
$ B( K+KB, K+KB ), LDB, A( K, K+KB ),
$ LDA )
END IF
10 CONTINUE
ELSE
*
* Compute inv(L)*A*inv(L**T)
*
DO 20 K = 1, N, NB
KB = MIN( N-K+1, NB )
*
* Update the lower triangle of A(k:n,k:n)
*
CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
$ B( K, K ), LDB, INFO )
IF( K+KB.LE.N ) THEN
CALL DTRSM( 'Right', UPLO, 'Transpose', 'Non-unit',
$ N-K-KB+1, KB, ONE, B( K, K ), LDB,
$ A( K+KB, K ), LDA )
CALL DSYMM( 'Right', UPLO, N-K-KB+1, KB, -HALF,
$ A( K, K ), LDA, B( K+KB, K ), LDB, ONE,
$ A( K+KB, K ), LDA )
CALL DSYR2K( UPLO, 'No transpose', N-K-KB+1, KB,
$ -ONE, A( K+KB, K ), LDA, B( K+KB, K ),
$ LDB, ONE, A( K+KB, K+KB ), LDA )
CALL DSYMM( 'Right', UPLO, N-K-KB+1, KB, -HALF,
$ A( K, K ), LDA, B( K+KB, K ), LDB, ONE,
$ A( K+KB, K ), LDA )
CALL DTRSM( 'Left', UPLO, 'No transpose',
$ 'Non-unit', N-K-KB+1, KB, ONE,
$ B( K+KB, K+KB ), LDB, A( K+KB, K ),
$ LDA )
END IF
20 CONTINUE
END IF
ELSE
IF( UPPER ) THEN
*
* Compute U*A*U**T
*
DO 30 K = 1, N, NB
KB = MIN( N-K+1, NB )
*
* Update the upper triangle of A(1:k+kb-1,1:k+kb-1)
*
CALL DTRMM( 'Left', UPLO, 'No transpose', 'Non-unit',
$ K-1, KB, ONE, B, LDB, A( 1, K ), LDA )
CALL DSYMM( 'Right', UPLO, K-1, KB, HALF, A( K, K ),
$ LDA, B( 1, K ), LDB, ONE, A( 1, K ), LDA )
CALL DSYR2K( UPLO, 'No transpose', K-1, KB, ONE,
$ A( 1, K ), LDA, B( 1, K ), LDB, ONE, A,
$ LDA )
CALL DSYMM( 'Right', UPLO, K-1, KB, HALF, A( K, K ),
$ LDA, B( 1, K ), LDB, ONE, A( 1, K ), LDA )
CALL DTRMM( 'Right', UPLO, 'Transpose', 'Non-unit',
$ K-1, KB, ONE, B( K, K ), LDB, A( 1, K ),
$ LDA )
CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
$ B( K, K ), LDB, INFO )
30 CONTINUE
ELSE
*
* Compute L**T*A*L
*
DO 40 K = 1, N, NB
KB = MIN( N-K+1, NB )
*
* Update the lower triangle of A(1:k+kb-1,1:k+kb-1)
*
CALL DTRMM( 'Right', UPLO, 'No transpose', 'Non-unit',
$ KB, K-1, ONE, B, LDB, A( K, 1 ), LDA )
CALL DSYMM( 'Left', UPLO, KB, K-1, HALF, A( K, K ),
$ LDA, B( K, 1 ), LDB, ONE, A( K, 1 ), LDA )
CALL DSYR2K( UPLO, 'Transpose', K-1, KB, ONE,
$ A( K, 1 ), LDA, B( K, 1 ), LDB, ONE, A,
$ LDA )
CALL DSYMM( 'Left', UPLO, KB, K-1, HALF, A( K, K ),
$ LDA, B( K, 1 ), LDB, ONE, A( K, 1 ), LDA )
CALL DTRMM( 'Left', UPLO, 'Transpose', 'Non-unit', KB,
$ K-1, ONE, B( K, K ), LDB, A( K, 1 ), LDA )
CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA,
$ B( K, K ), LDB, INFO )
40 CONTINUE
END IF
END IF
END IF
RETURN
*
* End of DSYGST
*
END

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*> \brief \b DSYGST
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
* LWORK, IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYGVD computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
*> B are assumed to be symmetric and B is also positive definite.
*> If eigenvectors are desired, it uses a divide and conquer algorithm.
*>
*> The divide and conquer algorithm makes very mild assumptions about
*> floating point arithmetic. It will work on machines with a guard
*> digit in add/subtract, or on those binary machines without guard
*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> Specifies the problem type to be solved:
*> = 1: A*x = (lambda)*B*x
*> = 2: A*B*x = (lambda)*x
*> = 3: B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangles of A and B are stored;
*> = 'L': Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the
*> leading N-by-N upper triangular part of A contains the
*> upper triangular part of the matrix A. If UPLO = 'L',
*> the leading N-by-N lower triangular part of A contains
*> the lower triangular part of the matrix A.
*>
*> On exit, if JOBZ = 'V', then if INFO = 0, A contains the
*> matrix Z of eigenvectors. The eigenvectors are normalized
*> as follows:
*> if ITYPE = 1 or 2, Z**T*B*Z = I;
*> if ITYPE = 3, Z**T*inv(B)*Z = I.
*> If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
*> or the lower triangle (if UPLO='L') of A, including the
*> diagonal, is destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the symmetric matrix B. If UPLO = 'U', the
*> leading N-by-N upper triangular part of B contains the
*> upper triangular part of the matrix B. If UPLO = 'L',
*> the leading N-by-N lower triangular part of B contains
*> the lower triangular part of the matrix B.
*>
*> On exit, if INFO <= N, the part of B containing the matrix is
*> overwritten by the triangular factor U or L from the Cholesky
*> factorization B = U**T*U or B = L*L**T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> If N <= 1, LWORK >= 1.
*> If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
*> If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal sizes of the WORK and IWORK
*> arrays, returns these values as the first entries of the WORK
*> and IWORK arrays, and no error message related to LWORK or
*> LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> If N <= 1, LIWORK >= 1.
*> If JOBZ = 'N' and N > 1, LIWORK >= 1.
*> If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK and IWORK arrays, and no error message related to
*> LWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: DPOTRF or DSYEVD returned an error code:
*> <= N: if INFO = i and JOBZ = 'N', then the algorithm
*> failed to converge; i off-diagonal elements of an
*> intermediate tridiagonal form did not converge to
*> zero;
*> if INFO = i and JOBZ = 'V', then the algorithm
*> failed to compute an eigenvalue while working on
*> the submatrix lying in rows and columns INFO/(N+1)
*> through mod(INFO,N+1);
*> > N: if INFO = N + i, for 1 <= i <= N, then the leading
*> minor of order i of B is not positive definite.
*> The factorization of B could not be completed and
*> no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleSYeigen
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Modified so that no backsubstitution is performed if DSYEVD fails to
*> converge (NEIG in old code could be greater than N causing out of
*> bounds reference to A - reported by Ralf Meyer). Also corrected the
*> description of INFO and the test on ITYPE. Sven, 16 Feb 05.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*>
* =====================================================================
SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
$ LWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER, WANTZ
CHARACTER TRANS
INTEGER LIOPT, LIWMIN, LOPT, LWMIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPOTRF, DSYEVD, DSYGST, DTRMM, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( N.LE.1 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE IF( WANTZ ) THEN
LIWMIN = 3 + 5*N
LWMIN = 1 + 6*N + 2*N**2
ELSE
LIWMIN = 1
LWMIN = 2*N + 1
END IF
LOPT = LWMIN
LIOPT = LIWMIN
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LOPT
IWORK( 1 ) = LIOPT
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -11
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYGVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Form a Cholesky factorization of B.
*
CALL DPOTRF( UPLO, N, B, LDB, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem and solve.
*
CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
CALL DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK,
$ INFO )
LOPT = MAX( DBLE( LOPT ), DBLE( WORK( 1 ) ) )
LIOPT = MAX( DBLE( LIOPT ), DBLE( IWORK( 1 ) ) )
*
IF( WANTZ .AND. INFO.EQ.0 ) THEN
*
* Backtransform eigenvectors to the original problem.
*
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
* backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
IF( UPPER ) THEN
TRANS = 'N'
ELSE
TRANS = 'T'
END IF
*
CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, N, ONE,
$ B, LDB, A, LDA )
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* For B*A*x=(lambda)*x;
* backtransform eigenvectors: x = L*y or U**T*y
*
IF( UPPER ) THEN
TRANS = 'T'
ELSE
TRANS = 'N'
END IF
*
CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, N, ONE,
$ B, LDB, A, LDA )
END IF
END IF
*
WORK( 1 ) = LOPT
IWORK( 1 ) = LIOPT
*
RETURN
*
* End of DSYGVD
*
END

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*> \brief \b DSYMM
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DSYMM(SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
*
* .. Scalar Arguments ..
* DOUBLE PRECISION ALPHA,BETA
* INTEGER LDA,LDB,LDC,M,N
* CHARACTER SIDE,UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A(LDA,*),B(LDB,*),C(LDC,*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYMM performs one of the matrix-matrix operations
*>
*> C := alpha*A*B + beta*C,
*>
*> or
*>
*> C := alpha*B*A + beta*C,
*>
*> where alpha and beta are scalars, A is a symmetric matrix and B and
*> C are m by n matrices.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> On entry, SIDE specifies whether the symmetric matrix A
*> appears on the left or right in the operation as follows:
*>
*> SIDE = 'L' or 'l' C := alpha*A*B + beta*C,
*>
*> SIDE = 'R' or 'r' C := alpha*B*A + beta*C,
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the upper or lower
*> triangular part of the symmetric matrix A is to be
*> referenced as follows:
*>
*> UPLO = 'U' or 'u' Only the upper triangular part of the
*> symmetric matrix is to be referenced.
*>
*> UPLO = 'L' or 'l' Only the lower triangular part of the
*> symmetric matrix is to be referenced.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> On entry, M specifies the number of rows of the matrix C.
*> M must be at least zero.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the number of columns of the matrix C.
*> N must be at least zero.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION.
*> On entry, ALPHA specifies the scalar alpha.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is
*> m when SIDE = 'L' or 'l' and is n otherwise.
*> Before entry with SIDE = 'L' or 'l', the m by m part of
*> the array A must contain the symmetric matrix, such that
*> when UPLO = 'U' or 'u', the leading m by m upper triangular
*> part of the array A must contain the upper triangular part
*> of the symmetric matrix and the strictly lower triangular
*> part of A is not referenced, and when UPLO = 'L' or 'l',
*> the leading m by m lower triangular part of the array A
*> must contain the lower triangular part of the symmetric
*> matrix and the strictly upper triangular part of A is not
*> referenced.
*> Before entry with SIDE = 'R' or 'r', the n by n part of
*> the array A must contain the symmetric matrix, such that
*> when UPLO = 'U' or 'u', the leading n by n upper triangular
*> part of the array A must contain the upper triangular part
*> of the symmetric matrix and the strictly lower triangular
*> part of A is not referenced, and when UPLO = 'L' or 'l',
*> the leading n by n lower triangular part of the array A
*> must contain the lower triangular part of the symmetric
*> matrix and the strictly upper triangular part of A is not
*> referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> On entry, LDA specifies the first dimension of A as declared
*> in the calling (sub) program. When SIDE = 'L' or 'l' then
*> LDA must be at least max( 1, m ), otherwise LDA must be at
*> least max( 1, n ).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array of DIMENSION ( LDB, n ).
*> Before entry, the leading m by n part of the array B must
*> contain the matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> On entry, LDB specifies the first dimension of B as declared
*> in the calling (sub) program. LDB must be at least
*> max( 1, m ).
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION.
*> On entry, BETA specifies the scalar beta. When BETA is
*> supplied as zero then C need not be set on input.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array of DIMENSION ( LDC, n ).
*> Before entry, the leading m by n part of the array C must
*> contain the matrix C, except when beta is zero, in which
*> case C need not be set on entry.
*> On exit, the array C is overwritten by the m by n updated
*> matrix.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> On entry, LDC specifies the first dimension of C as declared
*> in the calling (sub) program. LDC must be at least
*> max( 1, m ).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_blas_level3
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Level 3 Blas routine.
*>
*> -- Written on 8-February-1989.
*> Jack Dongarra, Argonne National Laboratory.
*> Iain Duff, AERE Harwell.
*> Jeremy Du Croz, Numerical Algorithms Group Ltd.
*> Sven Hammarling, Numerical Algorithms Group Ltd.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYMM(SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
*
* -- Reference BLAS level3 routine (version 3.4.0) --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA,BETA
INTEGER LDA,LDB,LDC,M,N
CHARACTER SIDE,UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),B(LDB,*),C(LDC,*)
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP1,TEMP2
INTEGER I,INFO,J,K,NROWA
LOGICAL UPPER
* ..
* .. Parameters ..
DOUBLE PRECISION ONE,ZERO
PARAMETER (ONE=1.0D+0,ZERO=0.0D+0)
* ..
*
* Set NROWA as the number of rows of A.
*
IF (LSAME(SIDE,'L')) THEN
NROWA = M
ELSE
NROWA = N
END IF
UPPER = LSAME(UPLO,'U')
*
* Test the input parameters.
*
INFO = 0
IF ((.NOT.LSAME(SIDE,'L')) .AND. (.NOT.LSAME(SIDE,'R'))) THEN
INFO = 1
ELSE IF ((.NOT.UPPER) .AND. (.NOT.LSAME(UPLO,'L'))) THEN
INFO = 2
ELSE IF (M.LT.0) THEN
INFO = 3
ELSE IF (N.LT.0) THEN
INFO = 4
ELSE IF (LDA.LT.MAX(1,NROWA)) THEN
INFO = 7
ELSE IF (LDB.LT.MAX(1,M)) THEN
INFO = 9
ELSE IF (LDC.LT.MAX(1,M)) THEN
INFO = 12
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DSYMM ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((M.EQ.0) .OR. (N.EQ.0) .OR.
+ ((ALPHA.EQ.ZERO).AND. (BETA.EQ.ONE))) RETURN
*
* And when alpha.eq.zero.
*
IF (ALPHA.EQ.ZERO) THEN
IF (BETA.EQ.ZERO) THEN
DO 20 J = 1,N
DO 10 I = 1,M
C(I,J) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1,N
DO 30 I = 1,M
C(I,J) = BETA*C(I,J)
30 CONTINUE
40 CONTINUE
END IF
RETURN
END IF
*
* Start the operations.
*
IF (LSAME(SIDE,'L')) THEN
*
* Form C := alpha*A*B + beta*C.
*
IF (UPPER) THEN
DO 70 J = 1,N
DO 60 I = 1,M
TEMP1 = ALPHA*B(I,J)
TEMP2 = ZERO
DO 50 K = 1,I - 1
C(K,J) = C(K,J) + TEMP1*A(K,I)
TEMP2 = TEMP2 + B(K,J)*A(K,I)
50 CONTINUE
IF (BETA.EQ.ZERO) THEN
C(I,J) = TEMP1*A(I,I) + ALPHA*TEMP2
ELSE
C(I,J) = BETA*C(I,J) + TEMP1*A(I,I) +
+ ALPHA*TEMP2
END IF
60 CONTINUE
70 CONTINUE
ELSE
DO 100 J = 1,N
DO 90 I = M,1,-1
TEMP1 = ALPHA*B(I,J)
TEMP2 = ZERO
DO 80 K = I + 1,M
C(K,J) = C(K,J) + TEMP1*A(K,I)
TEMP2 = TEMP2 + B(K,J)*A(K,I)
80 CONTINUE
IF (BETA.EQ.ZERO) THEN
C(I,J) = TEMP1*A(I,I) + ALPHA*TEMP2
ELSE
C(I,J) = BETA*C(I,J) + TEMP1*A(I,I) +
+ ALPHA*TEMP2
END IF
90 CONTINUE
100 CONTINUE
END IF
ELSE
*
* Form C := alpha*B*A + beta*C.
*
DO 170 J = 1,N
TEMP1 = ALPHA*A(J,J)
IF (BETA.EQ.ZERO) THEN
DO 110 I = 1,M
C(I,J) = TEMP1*B(I,J)
110 CONTINUE
ELSE
DO 120 I = 1,M
C(I,J) = BETA*C(I,J) + TEMP1*B(I,J)
120 CONTINUE
END IF
DO 140 K = 1,J - 1
IF (UPPER) THEN
TEMP1 = ALPHA*A(K,J)
ELSE
TEMP1 = ALPHA*A(J,K)
END IF
DO 130 I = 1,M
C(I,J) = C(I,J) + TEMP1*B(I,K)
130 CONTINUE
140 CONTINUE
DO 160 K = J + 1,N
IF (UPPER) THEN
TEMP1 = ALPHA*A(J,K)
ELSE
TEMP1 = ALPHA*A(K,J)
END IF
DO 150 I = 1,M
C(I,J) = C(I,J) + TEMP1*B(I,K)
150 CONTINUE
160 CONTINUE
170 CONTINUE
END IF
*
RETURN
*
* End of DSYMM .
*
END

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*> \brief \b DSYMV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DSYMV(UPLO,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY)
*
* .. Scalar Arguments ..
* DOUBLE PRECISION ALPHA,BETA
* INTEGER INCX,INCY,LDA,N
* CHARACTER UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A(LDA,*),X(*),Y(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYMV performs the matrix-vector operation
*>
*> y := alpha*A*x + beta*y,
*>
*> where alpha and beta are scalars, x and y are n element vectors and
*> A is an n by n symmetric matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the upper or lower
*> triangular part of the array A is to be referenced as
*> follows:
*>
*> UPLO = 'U' or 'u' Only the upper triangular part of A
*> is to be referenced.
*>
*> UPLO = 'L' or 'l' Only the lower triangular part of A
*> is to be referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of the matrix A.
*> N must be at least zero.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION.
*> On entry, ALPHA specifies the scalar alpha.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array of DIMENSION ( LDA, n ).
*> Before entry with UPLO = 'U' or 'u', the leading n by n
*> upper triangular part of the array A must contain the upper
*> triangular part of the symmetric matrix and the strictly
*> lower triangular part of A is not referenced.
*> Before entry with UPLO = 'L' or 'l', the leading n by n
*> lower triangular part of the array A must contain the lower
*> triangular part of the symmetric matrix and the strictly
*> upper triangular part of A is not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> On entry, LDA specifies the first dimension of A as declared
*> in the calling (sub) program. LDA must be at least
*> max( 1, n ).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array of dimension at least
*> ( 1 + ( n - 1 )*abs( INCX ) ).
*> Before entry, the incremented array X must contain the n
*> element vector x.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> On entry, INCX specifies the increment for the elements of
*> X. INCX must not be zero.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION.
*> On entry, BETA specifies the scalar beta. When BETA is
*> supplied as zero then Y need not be set on input.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array of dimension at least
*> ( 1 + ( n - 1 )*abs( INCY ) ).
*> Before entry, the incremented array Y must contain the n
*> element vector y. On exit, Y is overwritten by the updated
*> vector y.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*> INCY is INTEGER
*> On entry, INCY specifies the increment for the elements of
*> Y. INCY must not be zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_blas_level2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Level 2 Blas routine.
*> The vector and matrix arguments are not referenced when N = 0, or M = 0
*>
*> -- Written on 22-October-1986.
*> Jack Dongarra, Argonne National Lab.
*> Jeremy Du Croz, Nag Central Office.
*> Sven Hammarling, Nag Central Office.
*> Richard Hanson, Sandia National Labs.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYMV(UPLO,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY)
*
* -- Reference BLAS level2 routine (version 3.4.0) --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA,BETA
INTEGER INCX,INCY,LDA,N
CHARACTER UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),X(*),Y(*)
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE,ZERO
PARAMETER (ONE=1.0D+0,ZERO=0.0D+0)
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP1,TEMP2
INTEGER I,INFO,IX,IY,J,JX,JY,KX,KY
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
*
* Test the input parameters.
*
INFO = 0
IF (.NOT.LSAME(UPLO,'U') .AND. .NOT.LSAME(UPLO,'L')) THEN
INFO = 1
ELSE IF (N.LT.0) THEN
INFO = 2
ELSE IF (LDA.LT.MAX(1,N)) THEN
INFO = 5
ELSE IF (INCX.EQ.0) THEN
INFO = 7
ELSE IF (INCY.EQ.0) THEN
INFO = 10
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DSYMV ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((N.EQ.0) .OR. ((ALPHA.EQ.ZERO).AND. (BETA.EQ.ONE))) RETURN
*
* Set up the start points in X and Y.
*
IF (INCX.GT.0) THEN
KX = 1
ELSE
KX = 1 - (N-1)*INCX
END IF
IF (INCY.GT.0) THEN
KY = 1
ELSE
KY = 1 - (N-1)*INCY
END IF
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through the triangular part
* of A.
*
* First form y := beta*y.
*
IF (BETA.NE.ONE) THEN
IF (INCY.EQ.1) THEN
IF (BETA.EQ.ZERO) THEN
DO 10 I = 1,N
Y(I) = ZERO
10 CONTINUE
ELSE
DO 20 I = 1,N
Y(I) = BETA*Y(I)
20 CONTINUE
END IF
ELSE
IY = KY
IF (BETA.EQ.ZERO) THEN
DO 30 I = 1,N
Y(IY) = ZERO
IY = IY + INCY
30 CONTINUE
ELSE
DO 40 I = 1,N
Y(IY) = BETA*Y(IY)
IY = IY + INCY
40 CONTINUE
END IF
END IF
END IF
IF (ALPHA.EQ.ZERO) RETURN
IF (LSAME(UPLO,'U')) THEN
*
* Form y when A is stored in upper triangle.
*
IF ((INCX.EQ.1) .AND. (INCY.EQ.1)) THEN
DO 60 J = 1,N
TEMP1 = ALPHA*X(J)
TEMP2 = ZERO
DO 50 I = 1,J - 1
Y(I) = Y(I) + TEMP1*A(I,J)
TEMP2 = TEMP2 + A(I,J)*X(I)
50 CONTINUE
Y(J) = Y(J) + TEMP1*A(J,J) + ALPHA*TEMP2
60 CONTINUE
ELSE
JX = KX
JY = KY
DO 80 J = 1,N
TEMP1 = ALPHA*X(JX)
TEMP2 = ZERO
IX = KX
IY = KY
DO 70 I = 1,J - 1
Y(IY) = Y(IY) + TEMP1*A(I,J)
TEMP2 = TEMP2 + A(I,J)*X(IX)
IX = IX + INCX
IY = IY + INCY
70 CONTINUE
Y(JY) = Y(JY) + TEMP1*A(J,J) + ALPHA*TEMP2
JX = JX + INCX
JY = JY + INCY
80 CONTINUE
END IF
ELSE
*
* Form y when A is stored in lower triangle.
*
IF ((INCX.EQ.1) .AND. (INCY.EQ.1)) THEN
DO 100 J = 1,N
TEMP1 = ALPHA*X(J)
TEMP2 = ZERO
Y(J) = Y(J) + TEMP1*A(J,J)
DO 90 I = J + 1,N
Y(I) = Y(I) + TEMP1*A(I,J)
TEMP2 = TEMP2 + A(I,J)*X(I)
90 CONTINUE
Y(J) = Y(J) + ALPHA*TEMP2
100 CONTINUE
ELSE
JX = KX
JY = KY
DO 120 J = 1,N
TEMP1 = ALPHA*X(JX)
TEMP2 = ZERO
Y(JY) = Y(JY) + TEMP1*A(J,J)
IX = JX
IY = JY
DO 110 I = J + 1,N
IX = IX + INCX
IY = IY + INCY
Y(IY) = Y(IY) + TEMP1*A(I,J)
TEMP2 = TEMP2 + A(I,J)*X(IX)
110 CONTINUE
Y(JY) = Y(JY) + ALPHA*TEMP2
JX = JX + INCX
JY = JY + INCY
120 CONTINUE
END IF
END IF
*
RETURN
*
* End of DSYMV .
*
END

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*> \brief \b DSYR2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DSYR2(UPLO,N,ALPHA,X,INCX,Y,INCY,A,LDA)
*
* .. Scalar Arguments ..
* DOUBLE PRECISION ALPHA
* INTEGER INCX,INCY,LDA,N
* CHARACTER UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A(LDA,*),X(*),Y(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYR2 performs the symmetric rank 2 operation
*>
*> A := alpha*x*y**T + alpha*y*x**T + A,
*>
*> where alpha is a scalar, x and y are n element vectors and A is an n
*> by n symmetric matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the upper or lower
*> triangular part of the array A is to be referenced as
*> follows:
*>
*> UPLO = 'U' or 'u' Only the upper triangular part of A
*> is to be referenced.
*>
*> UPLO = 'L' or 'l' Only the lower triangular part of A
*> is to be referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of the matrix A.
*> N must be at least zero.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION.
*> On entry, ALPHA specifies the scalar alpha.
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array of dimension at least
*> ( 1 + ( n - 1 )*abs( INCX ) ).
*> Before entry, the incremented array X must contain the n
*> element vector x.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> On entry, INCX specifies the increment for the elements of
*> X. INCX must not be zero.
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*> Y is DOUBLE PRECISION array of dimension at least
*> ( 1 + ( n - 1 )*abs( INCY ) ).
*> Before entry, the incremented array Y must contain the n
*> element vector y.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*> INCY is INTEGER
*> On entry, INCY specifies the increment for the elements of
*> Y. INCY must not be zero.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array of DIMENSION ( LDA, n ).
*> Before entry with UPLO = 'U' or 'u', the leading n by n
*> upper triangular part of the array A must contain the upper
*> triangular part of the symmetric matrix and the strictly
*> lower triangular part of A is not referenced. On exit, the
*> upper triangular part of the array A is overwritten by the
*> upper triangular part of the updated matrix.
*> Before entry with UPLO = 'L' or 'l', the leading n by n
*> lower triangular part of the array A must contain the lower
*> triangular part of the symmetric matrix and the strictly
*> upper triangular part of A is not referenced. On exit, the
*> lower triangular part of the array A is overwritten by the
*> lower triangular part of the updated matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> On entry, LDA specifies the first dimension of A as declared
*> in the calling (sub) program. LDA must be at least
*> max( 1, n ).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_blas_level2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Level 2 Blas routine.
*>
*> -- Written on 22-October-1986.
*> Jack Dongarra, Argonne National Lab.
*> Jeremy Du Croz, Nag Central Office.
*> Sven Hammarling, Nag Central Office.
*> Richard Hanson, Sandia National Labs.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYR2(UPLO,N,ALPHA,X,INCX,Y,INCY,A,LDA)
*
* -- Reference BLAS level2 routine (version 3.4.0) --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA
INTEGER INCX,INCY,LDA,N
CHARACTER UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),X(*),Y(*)
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER (ZERO=0.0D+0)
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP1,TEMP2
INTEGER I,INFO,IX,IY,J,JX,JY,KX,KY
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
*
* Test the input parameters.
*
INFO = 0
IF (.NOT.LSAME(UPLO,'U') .AND. .NOT.LSAME(UPLO,'L')) THEN
INFO = 1
ELSE IF (N.LT.0) THEN
INFO = 2
ELSE IF (INCX.EQ.0) THEN
INFO = 5
ELSE IF (INCY.EQ.0) THEN
INFO = 7
ELSE IF (LDA.LT.MAX(1,N)) THEN
INFO = 9
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DSYR2 ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((N.EQ.0) .OR. (ALPHA.EQ.ZERO)) RETURN
*
* Set up the start points in X and Y if the increments are not both
* unity.
*
IF ((INCX.NE.1) .OR. (INCY.NE.1)) THEN
IF (INCX.GT.0) THEN
KX = 1
ELSE
KX = 1 - (N-1)*INCX
END IF
IF (INCY.GT.0) THEN
KY = 1
ELSE
KY = 1 - (N-1)*INCY
END IF
JX = KX
JY = KY
END IF
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through the triangular part
* of A.
*
IF (LSAME(UPLO,'U')) THEN
*
* Form A when A is stored in the upper triangle.
*
IF ((INCX.EQ.1) .AND. (INCY.EQ.1)) THEN
DO 20 J = 1,N
IF ((X(J).NE.ZERO) .OR. (Y(J).NE.ZERO)) THEN
TEMP1 = ALPHA*Y(J)
TEMP2 = ALPHA*X(J)
DO 10 I = 1,J
A(I,J) = A(I,J) + X(I)*TEMP1 + Y(I)*TEMP2
10 CONTINUE
END IF
20 CONTINUE
ELSE
DO 40 J = 1,N
IF ((X(JX).NE.ZERO) .OR. (Y(JY).NE.ZERO)) THEN
TEMP1 = ALPHA*Y(JY)
TEMP2 = ALPHA*X(JX)
IX = KX
IY = KY
DO 30 I = 1,J
A(I,J) = A(I,J) + X(IX)*TEMP1 + Y(IY)*TEMP2
IX = IX + INCX
IY = IY + INCY
30 CONTINUE
END IF
JX = JX + INCX
JY = JY + INCY
40 CONTINUE
END IF
ELSE
*
* Form A when A is stored in the lower triangle.
*
IF ((INCX.EQ.1) .AND. (INCY.EQ.1)) THEN
DO 60 J = 1,N
IF ((X(J).NE.ZERO) .OR. (Y(J).NE.ZERO)) THEN
TEMP1 = ALPHA*Y(J)
TEMP2 = ALPHA*X(J)
DO 50 I = J,N
A(I,J) = A(I,J) + X(I)*TEMP1 + Y(I)*TEMP2
50 CONTINUE
END IF
60 CONTINUE
ELSE
DO 80 J = 1,N
IF ((X(JX).NE.ZERO) .OR. (Y(JY).NE.ZERO)) THEN
TEMP1 = ALPHA*Y(JY)
TEMP2 = ALPHA*X(JX)
IX = JX
IY = JY
DO 70 I = J,N
A(I,J) = A(I,J) + X(IX)*TEMP1 + Y(IY)*TEMP2
IX = IX + INCX
IY = IY + INCY
70 CONTINUE
END IF
JX = JX + INCX
JY = JY + INCY
80 CONTINUE
END IF
END IF
*
RETURN
*
* End of DSYR2 .
*
END

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*> \brief \b DSYR2K
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DSYR2K(UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
*
* .. Scalar Arguments ..
* DOUBLE PRECISION ALPHA,BETA
* INTEGER K,LDA,LDB,LDC,N
* CHARACTER TRANS,UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A(LDA,*),B(LDB,*),C(LDC,*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYR2K performs one of the symmetric rank 2k operations
*>
*> C := alpha*A*B**T + alpha*B*A**T + beta*C,
*>
*> or
*>
*> C := alpha*A**T*B + alpha*B**T*A + beta*C,
*>
*> where alpha and beta are scalars, C is an n by n symmetric matrix
*> and A and B are n by k matrices in the first case and k by n
*> matrices in the second case.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the upper or lower
*> triangular part of the array C is to be referenced as
*> follows:
*>
*> UPLO = 'U' or 'u' Only the upper triangular part of C
*> is to be referenced.
*>
*> UPLO = 'L' or 'l' Only the lower triangular part of C
*> is to be referenced.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> On entry, TRANS specifies the operation to be performed as
*> follows:
*>
*> TRANS = 'N' or 'n' C := alpha*A*B**T + alpha*B*A**T +
*> beta*C.
*>
*> TRANS = 'T' or 't' C := alpha*A**T*B + alpha*B**T*A +
*> beta*C.
*>
*> TRANS = 'C' or 'c' C := alpha*A**T*B + alpha*B**T*A +
*> beta*C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of the matrix C. N must be
*> at least zero.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> On entry with TRANS = 'N' or 'n', K specifies the number
*> of columns of the matrices A and B, and on entry with
*> TRANS = 'T' or 't' or 'C' or 'c', K specifies the number
*> of rows of the matrices A and B. K must be at least zero.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION.
*> On entry, ALPHA specifies the scalar alpha.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is
*> k when TRANS = 'N' or 'n', and is n otherwise.
*> Before entry with TRANS = 'N' or 'n', the leading n by k
*> part of the array A must contain the matrix A, otherwise
*> the leading k by n part of the array A must contain the
*> matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> On entry, LDA specifies the first dimension of A as declared
*> in the calling (sub) program. When TRANS = 'N' or 'n'
*> then LDA must be at least max( 1, n ), otherwise LDA must
*> be at least max( 1, k ).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array of DIMENSION ( LDB, kb ), where kb is
*> k when TRANS = 'N' or 'n', and is n otherwise.
*> Before entry with TRANS = 'N' or 'n', the leading n by k
*> part of the array B must contain the matrix B, otherwise
*> the leading k by n part of the array B must contain the
*> matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> On entry, LDB specifies the first dimension of B as declared
*> in the calling (sub) program. When TRANS = 'N' or 'n'
*> then LDB must be at least max( 1, n ), otherwise LDB must
*> be at least max( 1, k ).
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION.
*> On entry, BETA specifies the scalar beta.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array of DIMENSION ( LDC, n ).
*> Before entry with UPLO = 'U' or 'u', the leading n by n
*> upper triangular part of the array C must contain the upper
*> triangular part of the symmetric matrix and the strictly
*> lower triangular part of C is not referenced. On exit, the
*> upper triangular part of the array C is overwritten by the
*> upper triangular part of the updated matrix.
*> Before entry with UPLO = 'L' or 'l', the leading n by n
*> lower triangular part of the array C must contain the lower
*> triangular part of the symmetric matrix and the strictly
*> upper triangular part of C is not referenced. On exit, the
*> lower triangular part of the array C is overwritten by the
*> lower triangular part of the updated matrix.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> On entry, LDC specifies the first dimension of C as declared
*> in the calling (sub) program. LDC must be at least
*> max( 1, n ).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_blas_level3
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Level 3 Blas routine.
*>
*>
*> -- Written on 8-February-1989.
*> Jack Dongarra, Argonne National Laboratory.
*> Iain Duff, AERE Harwell.
*> Jeremy Du Croz, Numerical Algorithms Group Ltd.
*> Sven Hammarling, Numerical Algorithms Group Ltd.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYR2K(UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
*
* -- Reference BLAS level3 routine (version 3.4.0) --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA,BETA
INTEGER K,LDA,LDB,LDC,N
CHARACTER TRANS,UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),B(LDB,*),C(LDC,*)
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP1,TEMP2
INTEGER I,INFO,J,L,NROWA
LOGICAL UPPER
* ..
* .. Parameters ..
DOUBLE PRECISION ONE,ZERO
PARAMETER (ONE=1.0D+0,ZERO=0.0D+0)
* ..
*
* Test the input parameters.
*
IF (LSAME(TRANS,'N')) THEN
NROWA = N
ELSE
NROWA = K
END IF
UPPER = LSAME(UPLO,'U')
*
INFO = 0
IF ((.NOT.UPPER) .AND. (.NOT.LSAME(UPLO,'L'))) THEN
INFO = 1
ELSE IF ((.NOT.LSAME(TRANS,'N')) .AND.
+ (.NOT.LSAME(TRANS,'T')) .AND.
+ (.NOT.LSAME(TRANS,'C'))) THEN
INFO = 2
ELSE IF (N.LT.0) THEN
INFO = 3
ELSE IF (K.LT.0) THEN
INFO = 4
ELSE IF (LDA.LT.MAX(1,NROWA)) THEN
INFO = 7
ELSE IF (LDB.LT.MAX(1,NROWA)) THEN
INFO = 9
ELSE IF (LDC.LT.MAX(1,N)) THEN
INFO = 12
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DSYR2K',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((N.EQ.0) .OR. (((ALPHA.EQ.ZERO).OR.
+ (K.EQ.0)).AND. (BETA.EQ.ONE))) RETURN
*
* And when alpha.eq.zero.
*
IF (ALPHA.EQ.ZERO) THEN
IF (UPPER) THEN
IF (BETA.EQ.ZERO) THEN
DO 20 J = 1,N
DO 10 I = 1,J
C(I,J) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1,N
DO 30 I = 1,J
C(I,J) = BETA*C(I,J)
30 CONTINUE
40 CONTINUE
END IF
ELSE
IF (BETA.EQ.ZERO) THEN
DO 60 J = 1,N
DO 50 I = J,N
C(I,J) = ZERO
50 CONTINUE
60 CONTINUE
ELSE
DO 80 J = 1,N
DO 70 I = J,N
C(I,J) = BETA*C(I,J)
70 CONTINUE
80 CONTINUE
END IF
END IF
RETURN
END IF
*
* Start the operations.
*
IF (LSAME(TRANS,'N')) THEN
*
* Form C := alpha*A*B**T + alpha*B*A**T + C.
*
IF (UPPER) THEN
DO 130 J = 1,N
IF (BETA.EQ.ZERO) THEN
DO 90 I = 1,J
C(I,J) = ZERO
90 CONTINUE
ELSE IF (BETA.NE.ONE) THEN
DO 100 I = 1,J
C(I,J) = BETA*C(I,J)
100 CONTINUE
END IF
DO 120 L = 1,K
IF ((A(J,L).NE.ZERO) .OR. (B(J,L).NE.ZERO)) THEN
TEMP1 = ALPHA*B(J,L)
TEMP2 = ALPHA*A(J,L)
DO 110 I = 1,J
C(I,J) = C(I,J) + A(I,L)*TEMP1 +
+ B(I,L)*TEMP2
110 CONTINUE
END IF
120 CONTINUE
130 CONTINUE
ELSE
DO 180 J = 1,N
IF (BETA.EQ.ZERO) THEN
DO 140 I = J,N
C(I,J) = ZERO
140 CONTINUE
ELSE IF (BETA.NE.ONE) THEN
DO 150 I = J,N
C(I,J) = BETA*C(I,J)
150 CONTINUE
END IF
DO 170 L = 1,K
IF ((A(J,L).NE.ZERO) .OR. (B(J,L).NE.ZERO)) THEN
TEMP1 = ALPHA*B(J,L)
TEMP2 = ALPHA*A(J,L)
DO 160 I = J,N
C(I,J) = C(I,J) + A(I,L)*TEMP1 +
+ B(I,L)*TEMP2
160 CONTINUE
END IF
170 CONTINUE
180 CONTINUE
END IF
ELSE
*
* Form C := alpha*A**T*B + alpha*B**T*A + C.
*
IF (UPPER) THEN
DO 210 J = 1,N
DO 200 I = 1,J
TEMP1 = ZERO
TEMP2 = ZERO
DO 190 L = 1,K
TEMP1 = TEMP1 + A(L,I)*B(L,J)
TEMP2 = TEMP2 + B(L,I)*A(L,J)
190 CONTINUE
IF (BETA.EQ.ZERO) THEN
C(I,J) = ALPHA*TEMP1 + ALPHA*TEMP2
ELSE
C(I,J) = BETA*C(I,J) + ALPHA*TEMP1 +
+ ALPHA*TEMP2
END IF
200 CONTINUE
210 CONTINUE
ELSE
DO 240 J = 1,N
DO 230 I = J,N
TEMP1 = ZERO
TEMP2 = ZERO
DO 220 L = 1,K
TEMP1 = TEMP1 + A(L,I)*B(L,J)
TEMP2 = TEMP2 + B(L,I)*A(L,J)
220 CONTINUE
IF (BETA.EQ.ZERO) THEN
C(I,J) = ALPHA*TEMP1 + ALPHA*TEMP2
ELSE
C(I,J) = BETA*C(I,J) + ALPHA*TEMP1 +
+ ALPHA*TEMP2
END IF
230 CONTINUE
240 CONTINUE
END IF
END IF
*
RETURN
*
* End of DSYR2K.
*
END

364
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*> \brief \b DSYRK
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DSYRK(UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC)
*
* .. Scalar Arguments ..
* DOUBLE PRECISION ALPHA,BETA
* INTEGER K,LDA,LDC,N
* CHARACTER TRANS,UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A(LDA,*),C(LDC,*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYRK performs one of the symmetric rank k operations
*>
*> C := alpha*A*A**T + beta*C,
*>
*> or
*>
*> C := alpha*A**T*A + beta*C,
*>
*> where alpha and beta are scalars, C is an n by n symmetric matrix
*> and A is an n by k matrix in the first case and a k by n matrix
*> in the second case.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> On entry, UPLO specifies whether the upper or lower
*> triangular part of the array C is to be referenced as
*> follows:
*>
*> UPLO = 'U' or 'u' Only the upper triangular part of C
*> is to be referenced.
*>
*> UPLO = 'L' or 'l' Only the lower triangular part of C
*> is to be referenced.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> On entry, TRANS specifies the operation to be performed as
*> follows:
*>
*> TRANS = 'N' or 'n' C := alpha*A*A**T + beta*C.
*>
*> TRANS = 'T' or 't' C := alpha*A**T*A + beta*C.
*>
*> TRANS = 'C' or 'c' C := alpha*A**T*A + beta*C.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of the matrix C. N must be
*> at least zero.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> On entry with TRANS = 'N' or 'n', K specifies the number
*> of columns of the matrix A, and on entry with
*> TRANS = 'T' or 't' or 'C' or 'c', K specifies the number
*> of rows of the matrix A. K must be at least zero.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION.
*> On entry, ALPHA specifies the scalar alpha.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is
*> k when TRANS = 'N' or 'n', and is n otherwise.
*> Before entry with TRANS = 'N' or 'n', the leading n by k
*> part of the array A must contain the matrix A, otherwise
*> the leading k by n part of the array A must contain the
*> matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> On entry, LDA specifies the first dimension of A as declared
*> in the calling (sub) program. When TRANS = 'N' or 'n'
*> then LDA must be at least max( 1, n ), otherwise LDA must
*> be at least max( 1, k ).
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION.
*> On entry, BETA specifies the scalar beta.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION array of DIMENSION ( LDC, n ).
*> Before entry with UPLO = 'U' or 'u', the leading n by n
*> upper triangular part of the array C must contain the upper
*> triangular part of the symmetric matrix and the strictly
*> lower triangular part of C is not referenced. On exit, the
*> upper triangular part of the array C is overwritten by the
*> upper triangular part of the updated matrix.
*> Before entry with UPLO = 'L' or 'l', the leading n by n
*> lower triangular part of the array C must contain the lower
*> triangular part of the symmetric matrix and the strictly
*> upper triangular part of C is not referenced. On exit, the
*> lower triangular part of the array C is overwritten by the
*> lower triangular part of the updated matrix.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> On entry, LDC specifies the first dimension of C as declared
*> in the calling (sub) program. LDC must be at least
*> max( 1, n ).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_blas_level3
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Level 3 Blas routine.
*>
*> -- Written on 8-February-1989.
*> Jack Dongarra, Argonne National Laboratory.
*> Iain Duff, AERE Harwell.
*> Jeremy Du Croz, Numerical Algorithms Group Ltd.
*> Sven Hammarling, Numerical Algorithms Group Ltd.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYRK(UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC)
*
* -- Reference BLAS level3 routine (version 3.4.0) --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA,BETA
INTEGER K,LDA,LDC,N
CHARACTER TRANS,UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),C(LDC,*)
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP
INTEGER I,INFO,J,L,NROWA
LOGICAL UPPER
* ..
* .. Parameters ..
DOUBLE PRECISION ONE,ZERO
PARAMETER (ONE=1.0D+0,ZERO=0.0D+0)
* ..
*
* Test the input parameters.
*
IF (LSAME(TRANS,'N')) THEN
NROWA = N
ELSE
NROWA = K
END IF
UPPER = LSAME(UPLO,'U')
*
INFO = 0
IF ((.NOT.UPPER) .AND. (.NOT.LSAME(UPLO,'L'))) THEN
INFO = 1
ELSE IF ((.NOT.LSAME(TRANS,'N')) .AND.
+ (.NOT.LSAME(TRANS,'T')) .AND.
+ (.NOT.LSAME(TRANS,'C'))) THEN
INFO = 2
ELSE IF (N.LT.0) THEN
INFO = 3
ELSE IF (K.LT.0) THEN
INFO = 4
ELSE IF (LDA.LT.MAX(1,NROWA)) THEN
INFO = 7
ELSE IF (LDC.LT.MAX(1,N)) THEN
INFO = 10
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DSYRK ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((N.EQ.0) .OR. (((ALPHA.EQ.ZERO).OR.
+ (K.EQ.0)).AND. (BETA.EQ.ONE))) RETURN
*
* And when alpha.eq.zero.
*
IF (ALPHA.EQ.ZERO) THEN
IF (UPPER) THEN
IF (BETA.EQ.ZERO) THEN
DO 20 J = 1,N
DO 10 I = 1,J
C(I,J) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1,N
DO 30 I = 1,J
C(I,J) = BETA*C(I,J)
30 CONTINUE
40 CONTINUE
END IF
ELSE
IF (BETA.EQ.ZERO) THEN
DO 60 J = 1,N
DO 50 I = J,N
C(I,J) = ZERO
50 CONTINUE
60 CONTINUE
ELSE
DO 80 J = 1,N
DO 70 I = J,N
C(I,J) = BETA*C(I,J)
70 CONTINUE
80 CONTINUE
END IF
END IF
RETURN
END IF
*
* Start the operations.
*
IF (LSAME(TRANS,'N')) THEN
*
* Form C := alpha*A*A**T + beta*C.
*
IF (UPPER) THEN
DO 130 J = 1,N
IF (BETA.EQ.ZERO) THEN
DO 90 I = 1,J
C(I,J) = ZERO
90 CONTINUE
ELSE IF (BETA.NE.ONE) THEN
DO 100 I = 1,J
C(I,J) = BETA*C(I,J)
100 CONTINUE
END IF
DO 120 L = 1,K
IF (A(J,L).NE.ZERO) THEN
TEMP = ALPHA*A(J,L)
DO 110 I = 1,J
C(I,J) = C(I,J) + TEMP*A(I,L)
110 CONTINUE
END IF
120 CONTINUE
130 CONTINUE
ELSE
DO 180 J = 1,N
IF (BETA.EQ.ZERO) THEN
DO 140 I = J,N
C(I,J) = ZERO
140 CONTINUE
ELSE IF (BETA.NE.ONE) THEN
DO 150 I = J,N
C(I,J) = BETA*C(I,J)
150 CONTINUE
END IF
DO 170 L = 1,K
IF (A(J,L).NE.ZERO) THEN
TEMP = ALPHA*A(J,L)
DO 160 I = J,N
C(I,J) = C(I,J) + TEMP*A(I,L)
160 CONTINUE
END IF
170 CONTINUE
180 CONTINUE
END IF
ELSE
*
* Form C := alpha*A**T*A + beta*C.
*
IF (UPPER) THEN
DO 210 J = 1,N
DO 200 I = 1,J
TEMP = ZERO
DO 190 L = 1,K
TEMP = TEMP + A(L,I)*A(L,J)
190 CONTINUE
IF (BETA.EQ.ZERO) THEN
C(I,J) = ALPHA*TEMP
ELSE
C(I,J) = ALPHA*TEMP + BETA*C(I,J)
END IF
200 CONTINUE
210 CONTINUE
ELSE
DO 240 J = 1,N
DO 230 I = J,N
TEMP = ZERO
DO 220 L = 1,K
TEMP = TEMP + A(L,I)*A(L,J)
220 CONTINUE
IF (BETA.EQ.ZERO) THEN
C(I,J) = ALPHA*TEMP
ELSE
C(I,J) = ALPHA*TEMP + BETA*C(I,J)
END IF
230 CONTINUE
240 CONTINUE
END IF
END IF
*
RETURN
*
* End of DSYRK .
*
END

323
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*> \brief \b DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
*> form T by an orthogonal similarity transformation: Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n-by-n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n-by-n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> On exit, if UPLO = 'U', the diagonal and first superdiagonal
*> of A are overwritten by the corresponding elements of the
*> tridiagonal matrix T, and the elements above the first
*> superdiagonal, with the array TAU, represent the orthogonal
*> matrix Q as a product of elementary reflectors; if UPLO
*> = 'L', the diagonal and first subdiagonal of A are over-
*> written by the corresponding elements of the tridiagonal
*> matrix T, and the elements below the first subdiagonal, with
*> the array TAU, represent the orthogonal matrix Q as a product
*> of elementary reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix T:
*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleSYcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n-1) . . . H(2) H(1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
*> A(1:i-1,i+1), and tau in TAU(i).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(n-1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
*> and tau in TAU(i).
*>
*> The contents of A on exit are illustrated by the following examples
*> with n = 5:
*>
*> if UPLO = 'U': if UPLO = 'L':
*>
*> ( d e v2 v3 v4 ) ( d )
*> ( d e v3 v4 ) ( e d )
*> ( d e v4 ) ( v1 e d )
*> ( d e ) ( v1 v2 e d )
*> ( d ) ( v1 v2 v3 e d )
*>
*> where d and e denote diagonal and off-diagonal elements of T, and vi
*> denotes an element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO, HALF
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
$ HALF = 1.0D0 / 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I
DOUBLE PRECISION ALPHA, TAUI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLARFG, DSYMV, DSYR2, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTD2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A
*
DO 10 I = N - 1, 1, -1
*
* Generate elementary reflector H(i) = I - tau * v * v**T
* to annihilate A(1:i-1,i+1)
*
CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
E( I ) = A( I, I+1 )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(1:i,1:i)
*
A( I, I+1 ) = ONE
*
* Compute x := tau * A * v storing x in TAU(1:i)
*
CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
$ TAU, 1 )
*
* Compute w := x - 1/2 * tau * (x**T * v) * v
*
ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 )
CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w**T - w * v**T
*
CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
$ LDA )
*
A( I, I+1 ) = E( I )
END IF
D( I+1 ) = A( I+1, I+1 )
TAU( I ) = TAUI
10 CONTINUE
D( 1 ) = A( 1, 1 )
ELSE
*
* Reduce the lower triangle of A
*
DO 20 I = 1, N - 1
*
* Generate elementary reflector H(i) = I - tau * v * v**T
* to annihilate A(i+2:n,i)
*
CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
$ TAUI )
E( I ) = A( I+1, I )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(i+1:n,i+1:n)
*
A( I+1, I ) = ONE
*
* Compute x := tau * A * v storing y in TAU(i:n-1)
*
CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
$ A( I+1, I ), 1, ZERO, TAU( I ), 1 )
*
* Compute w := x - 1/2 * tau * (x**T * v) * v
*
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ),
$ 1 )
CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w**T - w * v**T
*
CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
$ A( I+1, I+1 ), LDA )
*
A( I+1, I ) = E( I )
END IF
D( I ) = A( I, I )
TAU( I ) = TAUI
20 CONTINUE
D( N ) = A( N, N )
END IF
*
RETURN
*
* End of DSYTD2
*
END

376
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*> \brief \b DSYTRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTRD reduces a real symmetric matrix A to real symmetric
*> tridiagonal form T by an orthogonal similarity transformation:
*> Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> N-by-N upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading N-by-N lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> On exit, if UPLO = 'U', the diagonal and first superdiagonal
*> of A are overwritten by the corresponding elements of the
*> tridiagonal matrix T, and the elements above the first
*> superdiagonal, with the array TAU, represent the orthogonal
*> matrix Q as a product of elementary reflectors; if UPLO
*> = 'L', the diagonal and first subdiagonal of A are over-
*> written by the corresponding elements of the tridiagonal
*> matrix T, and the elements below the first subdiagonal, with
*> the array TAU, represent the orthogonal matrix Q as a product
*> of elementary reflectors. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix T:
*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1.
*> For optimum performance LWORK >= N*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleSYcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n-1) . . . H(2) H(1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
*> A(1:i-1,i+1), and tau in TAU(i).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(n-1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
*> and tau in TAU(i).
*>
*> The contents of A on exit are illustrated by the following examples
*> with n = 5:
*>
*> if UPLO = 'U': if UPLO = 'L':
*>
*> ( d e v2 v3 v4 ) ( d )
*> ( d e v3 v4 ) ( e d )
*> ( d e v4 ) ( v1 e d )
*> ( d e ) ( v1 v2 e d )
*> ( d ) ( v1 v2 v3 e d )
*>
*> where d and e denote diagonal and off-diagonal elements of T, and vi
*> denotes an element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -9
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Determine the block size.
*
NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
LWKOPT = N*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NX = N
IWS = 1
IF( NB.GT.1 .AND. NB.LT.N ) THEN
*
* Determine when to cross over from blocked to unblocked code
* (last block is always handled by unblocked code).
*
NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
IF( NX.LT.N ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: determine the
* minimum value of NB, and reduce NB or force use of
* unblocked code by setting NX = N.
*
NB = MAX( LWORK / LDWORK, 1 )
NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 )
IF( NB.LT.NBMIN )
$ NX = N
END IF
ELSE
NX = N
END IF
ELSE
NB = 1
END IF
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A.
* Columns 1:kk are handled by the unblocked method.
*
KK = N - ( ( N-NX+NB-1 ) / NB )*NB
DO 20 I = N - NB + 1, KK + 1, -NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
$ LDWORK )
*
* Update the unreduced submatrix A(1:i-1,1:i-1), using an
* update of the form: A := A - V*W**T - W*V**T
*
CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ),
$ LDA, WORK, LDWORK, ONE, A, LDA )
*
* Copy superdiagonal elements back into A, and diagonal
* elements into D
*
DO 10 J = I, I + NB - 1
A( J-1, J ) = E( J-1 )
D( J ) = A( J, J )
10 CONTINUE
20 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
ELSE
*
* Reduce the lower triangle of A
*
DO 40 I = 1, N - NX, NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
$ TAU( I ), WORK, LDWORK )
*
* Update the unreduced submatrix A(i+ib:n,i+ib:n), using
* an update of the form: A := A - V*W**T - W*V**T
*
CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
$ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
$ A( I+NB, I+NB ), LDA )
*
* Copy subdiagonal elements back into A, and diagonal
* elements into D
*
DO 30 J = I, I + NB - 1
A( J+1, J ) = E( J )
D( J ) = A( J, J )
30 CONTINUE
40 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
$ TAU( I ), IINFO )
END IF
*
WORK( 1 ) = LWKOPT
RETURN
*
* End of DSYTRD
*
END

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*> \brief \b ILADLC scans a matrix for its last non-zero column.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILADLC + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iladlc.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iladlc.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iladlc.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILADLC( M, N, A, LDA )
*
* .. Scalar Arguments ..
* INTEGER M, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ILADLC scans A for its last non-zero column.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
INTEGER FUNCTION ILADLC( M, N, A, LDA )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER M, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. Executable Statements ..
*
* Quick test for the common case where one corner is non-zero.
IF( N.EQ.0 ) THEN
ILADLC = N
ELSE IF( A(1, N).NE.ZERO .OR. A(M, N).NE.ZERO ) THEN
ILADLC = N
ELSE
* Now scan each column from the end, returning with the first non-zero.
DO ILADLC = N, 1, -1
DO I = 1, M
IF( A(I, ILADLC).NE.ZERO ) RETURN
END DO
END DO
END IF
RETURN
END

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@ -0,0 +1,121 @@
*> \brief \b ILADLR scans a matrix for its last non-zero row.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILADLR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/iladlr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/iladlr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/iladlr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILADLR( M, N, A, LDA )
*
* .. Scalar Arguments ..
* INTEGER M, N, LDA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ILADLR scans A for its last non-zero row.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
INTEGER FUNCTION ILADLR( M, N, A, LDA )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER M, N, LDA
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
* ..
* .. Executable Statements ..
*
* Quick test for the common case where one corner is non-zero.
IF( M.EQ.0 ) THEN
ILADLR = M
ELSE IF( A(M, 1).NE.ZERO .OR. A(M, N).NE.ZERO ) THEN
ILADLR = M
ELSE
* Scan up each column tracking the last zero row seen.
ILADLR = 0
DO J = 1, N
I=M
DO WHILE((A(MAX(I,1),J).EQ.ZERO).AND.(I.GE.1))
I=I-1
ENDDO
ILADLR = MAX( ILADLR, I )
END DO
END IF
RETURN
END