version 1.6, 2010/08/13 21:04:05
|
version 1.15, 2017/06/17 10:54:15
|
Line 1
|
Line 1
|
|
*> \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 December 2016 |
|
* |
|
*> \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 ) |
SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.7.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* December 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO |
CHARACTER UPLO |
Line 14
|
Line 206
|
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) |
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* ZHETRD reduces a complex Hermitian matrix A to real symmetric |
|
* tridiagonal form T by a unitary similarity transformation: |
|
* Q**H * A * Q = T. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
|
* UPLO (input) CHARACTER*1 |
|
* = 'U': Upper triangle of A is stored; |
|
* = 'L': Lower triangle of A is stored. |
|
* |
|
* N (input) INTEGER |
|
* The order of the matrix A. N >= 0. |
|
* |
|
* A (input/output) 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. |
|
* |
|
* LDA (input) INTEGER |
|
* The leading dimension of the array A. LDA >= max(1,N). |
|
* |
|
* D (output) DOUBLE PRECISION array, dimension (N) |
|
* The diagonal elements of the tridiagonal matrix T: |
|
* D(i) = A(i,i). |
|
* |
|
* E (output) 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'. |
|
* |
|
* TAU (output) COMPLEX*16 array, dimension (N-1) |
|
* The scalar factors of the elementary reflectors (see Further |
|
* Details). |
|
* |
|
* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) |
|
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
|
* |
|
* LWORK (input) 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. |
|
* |
|
* INFO (output) INTEGER |
|
* = 0: successful exit |
|
* < 0: if INFO = -i, the i-th argument had an illegal value |
|
* |
|
* Further Details |
|
* =============== |
|
* |
|
* 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' |
|
* |
|
* 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' |
|
* |
|
* 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). |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
Line 237
|
Line 318
|
$ LDWORK ) |
$ LDWORK ) |
* |
* |
* Update the unreduced submatrix A(1:i-1,1:i-1), using an |
* Update the unreduced submatrix A(1:i-1,1:i-1), using an |
* update of the form: A := A - V*W' - W*V' |
* update of the form: A := A - V*W**H - W*V**H |
* |
* |
CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE, |
CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE, |
$ A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA ) |
$ A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA ) |
Line 268
|
Line 349
|
$ TAU( I ), WORK, LDWORK ) |
$ TAU( I ), WORK, LDWORK ) |
* |
* |
* Update the unreduced submatrix A(i+nb:n,i+nb:n), using |
* Update the unreduced submatrix A(i+nb:n,i+nb:n), using |
* an update of the form: A := A - V*W' - W*V' |
* an update of the form: A := A - V*W**H - W*V**H |
* |
* |
CALL ZHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE, |
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 ), LDA, WORK( NB+1 ), LDWORK, ONE, |