forked from lijiext/lammps
337 lines
11 KiB
FortranFixed
337 lines
11 KiB
FortranFixed
|
*> \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
|