forked from lijiext/lammps
379 lines
11 KiB
Fortran
379 lines
11 KiB
Fortran
*> \brief \b ZHETRD
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download ZHETRD + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetrd.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetrd.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetrd.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* CHARACTER UPLO
|
|
* INTEGER INFO, LDA, LWORK, N
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* DOUBLE PRECISION D( * ), E( * )
|
|
* COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> ZHETRD reduces a complex Hermitian matrix A to real symmetric
|
|
*> tridiagonal form T by a unitary similarity transformation:
|
|
*> Q**H * 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 COMPLEX*16 array, dimension (LDA,N)
|
|
*> On entry, the Hermitian 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 unitary
|
|
*> 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 unitary 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 COMPLEX*16 array, dimension (N-1)
|
|
*> The scalar factors of the elementary reflectors (see Further
|
|
*> Details).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] WORK
|
|
*> \verbatim
|
|
*> WORK is COMPLEX*16 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 complex16HEcomputational
|
|
*
|
|
*> \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**H
|
|
*>
|
|
*> where tau is a complex scalar, and v is a complex 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**H
|
|
*>
|
|
*> where tau is a complex scalar, and v is a complex 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 ZHETRD( 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 D( * ), E( * )
|
|
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
DOUBLE PRECISION ONE
|
|
PARAMETER ( ONE = 1.0D+0 )
|
|
COMPLEX*16 CONE
|
|
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL LQUERY, UPPER
|
|
INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
|
|
$ NBMIN, NX
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL XERBLA, ZHER2K, ZHETD2, ZLATRD
|
|
* ..
|
|
* .. 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, 'ZHETRD', UPLO, N, -1, -1, -1 )
|
|
LWKOPT = N*NB
|
|
WORK( 1 ) = LWKOPT
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'ZHETRD', -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, 'ZHETRD', 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, 'ZHETRD', 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 ZLATRD( 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**H - W*V**H
|
|
*
|
|
CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE,
|
|
$ 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 ZHETD2( 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 ZLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
|
|
$ TAU( I ), WORK, LDWORK )
|
|
*
|
|
* Update the unreduced submatrix A(i+nb:n,i+nb:n), using
|
|
* an update of the form: A := A - V*W**H - W*V**H
|
|
*
|
|
CALL ZHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE,
|
|
$ 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 ZHETD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
|
|
$ TAU( I ), IINFO )
|
|
END IF
|
|
*
|
|
WORK( 1 ) = LWKOPT
|
|
RETURN
|
|
*
|
|
* End of ZHETRD
|
|
*
|
|
END
|