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version 1.19, 2023/08/07 08:39:42
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*> \brief \b ZTZRQF |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download ZTZRQF + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztzrqf.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztzrqf.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztzrqf.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER INFO, LDA, M, N |
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* .. |
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* .. Array Arguments .. |
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* COMPLEX*16 A( LDA, * ), TAU( * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> This routine is deprecated and has been replaced by routine ZTZRZF. |
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*> |
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*> ZTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A |
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*> to upper triangular form by means of unitary transformations. |
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*> |
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*> The upper trapezoidal matrix A is factored as |
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*> |
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*> A = ( R 0 ) * Z, |
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*> |
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*> where Z is an N-by-N unitary matrix and R is an M-by-M upper |
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*> triangular matrix. |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The number of rows of the matrix A. M >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The number of columns of the matrix A. N >= M. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is COMPLEX*16 array, dimension (LDA,N) |
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*> On entry, the leading M-by-N upper trapezoidal part of the |
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*> array A must contain the matrix to be factorized. |
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*> On exit, the leading M-by-M upper triangular part of A |
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*> contains the upper triangular matrix R, and elements M+1 to |
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*> N of the first M rows of A, with the array TAU, represent the |
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*> unitary matrix Z as a product of M elementary reflectors. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max(1,M). |
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*> \endverbatim |
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*> |
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*> \param[out] TAU |
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*> \verbatim |
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*> TAU is COMPLEX*16 array, dimension (M) |
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*> The scalar factors of the elementary reflectors. |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit |
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*> < 0: if INFO = -i, the i-th argument had an illegal value |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \ingroup complex16OTHERcomputational |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> The factorization is obtained by Householder's method. The kth |
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*> transformation matrix, Z( k ), whose conjugate transpose is used to |
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*> introduce zeros into the (m - k + 1)th row of A, is given in the form |
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*> |
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*> Z( k ) = ( I 0 ), |
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*> ( 0 T( k ) ) |
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*> |
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*> where |
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*> |
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*> T( k ) = I - tau*u( k )*u( k )**H, u( k ) = ( 1 ), |
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*> ( 0 ) |
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*> ( z( k ) ) |
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*> |
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*> tau is a scalar and z( k ) is an ( n - m ) element vector. |
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*> tau and z( k ) are chosen to annihilate the elements of the kth row |
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*> of X. |
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*> |
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*> The scalar tau is returned in the kth element of TAU and the vector |
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*> u( k ) in the kth row of A, such that the elements of z( k ) are |
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*> in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in |
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*> the upper triangular part of A. |
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*> |
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*> Z is given by |
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*> |
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*> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO ) |
SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine -- |
* -- 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 |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER INFO, LDA, M, N |
INTEGER INFO, LDA, M, N |
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COMPLEX*16 A( LDA, * ), TAU( * ) |
COMPLEX*16 A( LDA, * ), TAU( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* This routine is deprecated and has been replaced by routine ZTZRZF. |
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* |
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* ZTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A |
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* to upper triangular form by means of unitary transformations. |
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* |
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* The upper trapezoidal matrix A is factored as |
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* |
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* A = ( R 0 ) * Z, |
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* |
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* where Z is an N-by-N unitary matrix and R is an M-by-M upper |
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* triangular matrix. |
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* |
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* Arguments |
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* ========= |
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* |
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* M (input) INTEGER |
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* The number of rows of the matrix A. M >= 0. |
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* |
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* N (input) INTEGER |
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* The number of columns of the matrix A. N >= M. |
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* |
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* A (input/output) COMPLEX*16 array, dimension (LDA,N) |
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* On entry, the leading M-by-N upper trapezoidal part of the |
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* array A must contain the matrix to be factorized. |
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* On exit, the leading M-by-M upper triangular part of A |
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* contains the upper triangular matrix R, and elements M+1 to |
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* N of the first M rows of A, with the array TAU, represent the |
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* unitary matrix Z as a product of M elementary reflectors. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,M). |
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* |
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* TAU (output) COMPLEX*16 array, dimension (M) |
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* The scalar factors of the elementary reflectors. |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -i, the i-th argument had an illegal value |
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* |
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* Further Details |
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* =============== |
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* |
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* The factorization is obtained by Householder's method. The kth |
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* transformation matrix, Z( k ), whose conjugate transpose is used to |
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* introduce zeros into the (m - k + 1)th row of A, is given in the form |
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* |
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* Z( k ) = ( I 0 ), |
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* ( 0 T( k ) ) |
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* |
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* where |
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* |
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* T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), |
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* ( 0 ) |
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* ( z( k ) ) |
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* |
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* tau is a scalar and z( k ) is an ( n - m ) element vector. |
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* tau and z( k ) are chosen to annihilate the elements of the kth row |
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* of X. |
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* |
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* The scalar tau is returned in the kth element of TAU and the vector |
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* u( k ) in the kth row of A, such that the elements of z( k ) are |
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* in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in |
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* the upper triangular part of A. |
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* |
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* Z is given by |
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* |
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* Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* .. |
* .. |
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGERC, ZLACGV, |
EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGERC, ZLACGV, |
$ ZLARFP |
$ ZLARFG |
* .. |
* .. |
* .. Executable Statements .. |
* .. Executable Statements .. |
* |
* |
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A( K, K ) = DCONJG( A( K, K ) ) |
A( K, K ) = DCONJG( A( K, K ) ) |
CALL ZLACGV( N-M, A( K, M1 ), LDA ) |
CALL ZLACGV( N-M, A( K, M1 ), LDA ) |
ALPHA = A( K, K ) |
ALPHA = A( K, K ) |
CALL ZLARFP( N-M+1, ALPHA, A( K, M1 ), LDA, TAU( K ) ) |
CALL ZLARFG( N-M+1, ALPHA, A( K, M1 ), LDA, TAU( K ) ) |
A( K, K ) = ALPHA |
A( K, K ) = ALPHA |
TAU( K ) = DCONJG( TAU( K ) ) |
TAU( K ) = DCONJG( TAU( K ) ) |
* |
* |
IF( TAU( K ).NE.CZERO .AND. K.GT.1 ) THEN |
IF( TAU( K ).NE.CZERO .AND. K.GT.1 ) THEN |
* |
* |
* We now perform the operation A := A*P( k )'. |
* We now perform the operation A := A*P( k )**H. |
* |
* |
* Use the first ( k - 1 ) elements of TAU to store a( k ), |
* Use the first ( k - 1 ) elements of TAU to store a( k ), |
* where a( k ) consists of the first ( k - 1 ) elements of |
* where a( k ) consists of the first ( k - 1 ) elements of |
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$ LDA, A( K, M1 ), LDA, CONE, TAU, 1 ) |
$ LDA, A( K, M1 ), LDA, CONE, TAU, 1 ) |
* |
* |
* Now form a( k ) := a( k ) - conjg(tau)*w |
* Now form a( k ) := a( k ) - conjg(tau)*w |
* and B := B - conjg(tau)*w*z( k )'. |
* and B := B - conjg(tau)*w*z( k )**H. |
* |
* |
CALL ZAXPY( K-1, -DCONJG( TAU( K ) ), TAU, 1, A( 1, K ), |
CALL ZAXPY( K-1, -DCONJG( TAU( K ) ), TAU, 1, A( 1, K ), |
$ 1 ) |
$ 1 ) |