version 1.6, 2010/08/13 21:04:04
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version 1.13, 2014/01/27 09:28:33
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*> \brief \b ZGGSVP |
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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 ZGGSVP + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggsvp.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/zggsvp.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/zggsvp.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 ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, |
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* TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, |
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* IWORK, RWORK, TAU, WORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBQ, JOBU, JOBV |
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* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P |
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* DOUBLE PRECISION TOLA, TOLB |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IWORK( * ) |
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* DOUBLE PRECISION RWORK( * ) |
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* COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), |
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* $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) |
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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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*> ZGGSVP computes unitary matrices U, V and Q such that |
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*> |
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*> N-K-L K L |
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*> U**H*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; |
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*> L ( 0 0 A23 ) |
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*> M-K-L ( 0 0 0 ) |
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*> |
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*> N-K-L K L |
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*> = K ( 0 A12 A13 ) if M-K-L < 0; |
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*> M-K ( 0 0 A23 ) |
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*> |
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*> N-K-L K L |
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*> V**H*B*Q = L ( 0 0 B13 ) |
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*> P-L ( 0 0 0 ) |
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*> |
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*> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular |
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*> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, |
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*> otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective |
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*> numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H. |
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*> |
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*> This decomposition is the preprocessing step for computing the |
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*> Generalized Singular Value Decomposition (GSVD), see subroutine |
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*> ZGGSVD. |
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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] JOBU |
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*> \verbatim |
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*> JOBU is CHARACTER*1 |
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*> = 'U': Unitary matrix U is computed; |
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*> = 'N': U is not computed. |
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*> \endverbatim |
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*> |
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*> \param[in] JOBV |
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*> \verbatim |
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*> JOBV is CHARACTER*1 |
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*> = 'V': Unitary matrix V is computed; |
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*> = 'N': V is not computed. |
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*> \endverbatim |
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*> |
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*> \param[in] JOBQ |
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*> \verbatim |
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*> JOBQ is CHARACTER*1 |
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*> = 'Q': Unitary matrix Q is computed; |
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*> = 'N': Q is not computed. |
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*> \endverbatim |
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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] P |
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*> \verbatim |
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*> P is INTEGER |
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*> The number of rows of the matrix B. P >= 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 matrices A and B. N >= 0. |
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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 M-by-N matrix A. |
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*> On exit, A contains the triangular (or trapezoidal) matrix |
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*> described in the Purpose section. |
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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[in,out] B |
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*> \verbatim |
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*> B is COMPLEX*16 array, dimension (LDB,N) |
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*> On entry, the P-by-N matrix B. |
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*> On exit, B contains the triangular matrix described in |
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*> the Purpose section. |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of the array B. LDB >= max(1,P). |
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*> \endverbatim |
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*> |
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*> \param[in] TOLA |
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*> \verbatim |
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*> TOLA is DOUBLE PRECISION |
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*> \endverbatim |
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*> |
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*> \param[in] TOLB |
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*> \verbatim |
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*> TOLB is DOUBLE PRECISION |
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*> |
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*> TOLA and TOLB are the thresholds to determine the effective |
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*> numerical rank of matrix B and a subblock of A. Generally, |
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*> they are set to |
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*> TOLA = MAX(M,N)*norm(A)*MAZHEPS, |
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*> TOLB = MAX(P,N)*norm(B)*MAZHEPS. |
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*> The size of TOLA and TOLB may affect the size of backward |
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*> errors of the decomposition. |
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*> \endverbatim |
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*> |
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*> \param[out] K |
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*> \verbatim |
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*> K is INTEGER |
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*> \endverbatim |
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*> |
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*> \param[out] L |
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*> \verbatim |
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*> L is INTEGER |
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*> |
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*> On exit, K and L specify the dimension of the subblocks |
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*> described in Purpose section. |
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*> K + L = effective numerical rank of (A**H,B**H)**H. |
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*> \endverbatim |
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*> |
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*> \param[out] U |
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*> \verbatim |
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*> U is COMPLEX*16 array, dimension (LDU,M) |
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*> If JOBU = 'U', U contains the unitary matrix U. |
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*> If JOBU = 'N', U is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDU |
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*> \verbatim |
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*> LDU is INTEGER |
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*> The leading dimension of the array U. LDU >= max(1,M) if |
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*> JOBU = 'U'; LDU >= 1 otherwise. |
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*> \endverbatim |
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*> |
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*> \param[out] V |
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*> \verbatim |
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*> V is COMPLEX*16 array, dimension (LDV,P) |
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*> If JOBV = 'V', V contains the unitary matrix V. |
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*> If JOBV = 'N', V is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDV |
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*> \verbatim |
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*> LDV is INTEGER |
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*> The leading dimension of the array V. LDV >= max(1,P) if |
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*> JOBV = 'V'; LDV >= 1 otherwise. |
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*> \endverbatim |
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*> |
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*> \param[out] Q |
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*> \verbatim |
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*> Q is COMPLEX*16 array, dimension (LDQ,N) |
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*> If JOBQ = 'Q', Q contains the unitary matrix Q. |
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*> If JOBQ = 'N', Q is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDQ |
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*> \verbatim |
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*> LDQ is INTEGER |
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*> The leading dimension of the array Q. LDQ >= max(1,N) if |
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*> JOBQ = 'Q'; LDQ >= 1 otherwise. |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] RWORK |
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*> \verbatim |
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*> RWORK is DOUBLE PRECISION array, dimension (2*N) |
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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 (N) |
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*> \endverbatim |
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*> |
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*> \param[out] WORK |
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*> \verbatim |
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*> WORK is COMPLEX*16 array, dimension (max(3*N,M,P)) |
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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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*> \date November 2011 |
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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 subroutine uses LAPACK subroutine ZGEQPF for the QR factorization |
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*> with column pivoting to detect the effective numerical rank of the |
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*> a matrix. It may be replaced by a better rank determination strategy. |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, |
SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, |
$ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, |
$ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, |
$ IWORK, RWORK, TAU, WORK, INFO ) |
$ IWORK, RWORK, TAU, WORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.4.0) -- |
* -- 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 |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBQ, JOBU, JOBV |
CHARACTER JOBQ, JOBU, JOBV |
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$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) |
$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZGGSVP computes unitary matrices U, V and Q such that |
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* |
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* N-K-L K L |
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* U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; |
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* L ( 0 0 A23 ) |
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* M-K-L ( 0 0 0 ) |
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* |
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* N-K-L K L |
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* = K ( 0 A12 A13 ) if M-K-L < 0; |
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* M-K ( 0 0 A23 ) |
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* |
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* N-K-L K L |
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* V'*B*Q = L ( 0 0 B13 ) |
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* P-L ( 0 0 0 ) |
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* |
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* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular |
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* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, |
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* otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective |
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* numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the |
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* conjugate transpose of Z. |
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* |
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* This decomposition is the preprocessing step for computing the |
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* Generalized Singular Value Decomposition (GSVD), see subroutine |
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* ZGGSVD. |
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* |
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* Arguments |
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* ========= |
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* |
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* JOBU (input) CHARACTER*1 |
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* = 'U': Unitary matrix U is computed; |
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* = 'N': U is not computed. |
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* |
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* JOBV (input) CHARACTER*1 |
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* = 'V': Unitary matrix V is computed; |
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* = 'N': V is not computed. |
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* |
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* JOBQ (input) CHARACTER*1 |
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* = 'Q': Unitary matrix Q is computed; |
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* = 'N': Q is not computed. |
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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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* P (input) INTEGER |
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* The number of rows of the matrix B. P >= 0. |
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* |
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* N (input) INTEGER |
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* The number of columns of the matrices A and B. N >= 0. |
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* |
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* A (input/output) COMPLEX*16 array, dimension (LDA,N) |
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* On entry, the M-by-N matrix A. |
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* On exit, A contains the triangular (or trapezoidal) matrix |
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* described in the Purpose section. |
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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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* B (input/output) COMPLEX*16 array, dimension (LDB,N) |
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* On entry, the P-by-N matrix B. |
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* On exit, B contains the triangular matrix described in |
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* the Purpose section. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of the array B. LDB >= max(1,P). |
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* |
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* TOLA (input) DOUBLE PRECISION |
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* TOLB (input) DOUBLE PRECISION |
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* TOLA and TOLB are the thresholds to determine the effective |
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* numerical rank of matrix B and a subblock of A. Generally, |
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* they are set to |
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* TOLA = MAX(M,N)*norm(A)*MAZHEPS, |
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* TOLB = MAX(P,N)*norm(B)*MAZHEPS. |
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* The size of TOLA and TOLB may affect the size of backward |
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* errors of the decomposition. |
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* |
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* K (output) INTEGER |
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* L (output) INTEGER |
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* On exit, K and L specify the dimension of the subblocks |
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* described in Purpose section. |
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* K + L = effective numerical rank of (A',B')'. |
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* |
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* U (output) COMPLEX*16 array, dimension (LDU,M) |
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* If JOBU = 'U', U contains the unitary matrix U. |
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* If JOBU = 'N', U is not referenced. |
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* |
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* LDU (input) INTEGER |
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* The leading dimension of the array U. LDU >= max(1,M) if |
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* JOBU = 'U'; LDU >= 1 otherwise. |
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* |
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* V (output) COMPLEX*16 array, dimension (LDV,P) |
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* If JOBV = 'V', V contains the unitary matrix V. |
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* If JOBV = 'N', V is not referenced. |
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* |
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* LDV (input) INTEGER |
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* The leading dimension of the array V. LDV >= max(1,P) if |
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* JOBV = 'V'; LDV >= 1 otherwise. |
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* |
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* Q (output) COMPLEX*16 array, dimension (LDQ,N) |
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* If JOBQ = 'Q', Q contains the unitary matrix Q. |
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* If JOBQ = 'N', Q is not referenced. |
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* |
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* LDQ (input) INTEGER |
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* The leading dimension of the array Q. LDQ >= max(1,N) if |
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* JOBQ = 'Q'; LDQ >= 1 otherwise. |
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* |
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* IWORK (workspace) INTEGER array, dimension (N) |
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* |
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* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) |
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* |
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* TAU (workspace) COMPLEX*16 array, dimension (N) |
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* |
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* WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)) |
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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 subroutine uses LAPACK subroutine ZGEQPF for the QR factorization |
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* with column pivoting to detect the effective numerical rank of the |
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* a matrix. It may be replaced by a better rank determination strategy. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* |
* |
CALL ZGERQ2( L, N, B, LDB, TAU, WORK, INFO ) |
CALL ZGERQ2( L, N, B, LDB, TAU, WORK, INFO ) |
* |
* |
* Update A := A*Z' |
* Update A := A*Z**H |
* |
* |
CALL ZUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB, |
CALL ZUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB, |
$ TAU, A, LDA, WORK, INFO ) |
$ TAU, A, LDA, WORK, INFO ) |
IF( WANTQ ) THEN |
IF( WANTQ ) THEN |
* |
* |
* Update Q := Q*Z' |
* Update Q := Q*Z**H |
* |
* |
CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N, L, B, |
CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N, L, B, |
$ LDB, TAU, Q, LDQ, WORK, INFO ) |
$ LDB, TAU, Q, LDQ, WORK, INFO ) |
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* |
* |
* then the following does the complete QR decomposition of A11: |
* then the following does the complete QR decomposition of A11: |
* |
* |
* A11 = U*( 0 T12 )*P1' |
* A11 = U*( 0 T12 )*P1**H |
* ( 0 0 ) |
* ( 0 0 ) |
* |
* |
DO 70 I = 1, N - L |
DO 70 I = 1, N - L |
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$ K = K + 1 |
$ K = K + 1 |
80 CONTINUE |
80 CONTINUE |
* |
* |
* Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N ) |
* Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N ) |
* |
* |
CALL ZUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ), |
CALL ZUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ), |
$ A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO ) |
$ A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO ) |
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* |
* |
IF( WANTQ ) THEN |
IF( WANTQ ) THEN |
* |
* |
* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1' |
* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H |
* |
* |
CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A, |
CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A, |
$ LDA, TAU, Q, LDQ, WORK, INFO ) |
$ LDA, TAU, Q, LDQ, WORK, INFO ) |