version 1.1, 2010/01/26 15:22:45
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version 1.10, 2011/11/21 22:19:42
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*> \brief \b DTGSJA |
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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 DTGSJA + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsja.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/dtgsja.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/dtgsja.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 DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, |
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* LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, |
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* Q, LDQ, WORK, NCYCLE, 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, |
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* $ NCYCLE, P |
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* DOUBLE PRECISION TOLA, TOLB |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), |
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* $ BETA( * ), Q( LDQ, * ), U( LDU, * ), |
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* $ 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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*> DTGSJA computes the generalized singular value decomposition (GSVD) |
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*> of two real upper triangular (or trapezoidal) matrices A and B. |
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*> |
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*> On entry, it is assumed that matrices A and B have the following |
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*> forms, which may be obtained by the preprocessing subroutine DGGSVP |
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*> from a general M-by-N matrix A and P-by-N matrix B: |
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*> |
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*> N-K-L K L |
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*> A = 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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*> A = 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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*> B = 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. |
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*> |
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*> On exit, |
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*> |
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*> U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ), |
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*> |
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*> where U, V and Q are orthogonal matrices. |
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*> R is a nonsingular upper triangular matrix, and D1 and D2 are |
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*> ``diagonal'' matrices, which are of the following structures: |
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*> |
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*> If M-K-L >= 0, |
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*> |
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*> K L |
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*> D1 = K ( I 0 ) |
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*> L ( 0 C ) |
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*> M-K-L ( 0 0 ) |
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*> |
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*> K L |
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*> D2 = L ( 0 S ) |
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*> P-L ( 0 0 ) |
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*> |
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*> N-K-L K L |
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*> ( 0 R ) = K ( 0 R11 R12 ) K |
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*> L ( 0 0 R22 ) L |
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*> |
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*> where |
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*> |
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*> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), |
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*> S = diag( BETA(K+1), ... , BETA(K+L) ), |
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*> C**2 + S**2 = I. |
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*> |
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*> R is stored in A(1:K+L,N-K-L+1:N) on exit. |
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*> |
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*> If M-K-L < 0, |
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*> |
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*> K M-K K+L-M |
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*> D1 = K ( I 0 0 ) |
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*> M-K ( 0 C 0 ) |
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*> |
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*> K M-K K+L-M |
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*> D2 = M-K ( 0 S 0 ) |
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*> K+L-M ( 0 0 I ) |
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*> P-L ( 0 0 0 ) |
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*> |
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*> N-K-L K M-K K+L-M |
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*> ( 0 R ) = K ( 0 R11 R12 R13 ) |
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*> M-K ( 0 0 R22 R23 ) |
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*> K+L-M ( 0 0 0 R33 ) |
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*> |
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*> where |
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*> C = diag( ALPHA(K+1), ... , ALPHA(M) ), |
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*> S = diag( BETA(K+1), ... , BETA(M) ), |
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*> C**2 + S**2 = I. |
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*> |
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*> R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored |
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*> ( 0 R22 R23 ) |
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*> in B(M-K+1:L,N+M-K-L+1:N) on exit. |
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*> |
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*> The computation of the orthogonal transformation matrices U, V or Q |
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*> is optional. These matrices may either be formed explicitly, or they |
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*> may be postmultiplied into input matrices U1, V1, or Q1. |
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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': U must contain an orthogonal matrix U1 on entry, and |
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*> the product U1*U is returned; |
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*> = 'I': U is initialized to the unit matrix, and the |
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*> orthogonal matrix U is returned; |
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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': V must contain an orthogonal matrix V1 on entry, and |
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*> the product V1*V is returned; |
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*> = 'I': V is initialized to the unit matrix, and the |
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*> orthogonal matrix V is returned; |
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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': Q must contain an orthogonal matrix Q1 on entry, and |
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*> the product Q1*Q is returned; |
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*> = 'I': Q is initialized to the unit matrix, and the |
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*> orthogonal matrix Q is returned; |
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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] 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[in] L |
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*> \verbatim |
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*> L is INTEGER |
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*> |
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*> K and L specify the subblocks in the input matrices A and B: |
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*> A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N) |
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*> of A and B, whose GSVD is going to be computed by DTGSJA. |
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*> See Further Details. |
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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 DOUBLE PRECISION array, dimension (LDA,N) |
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*> On entry, the M-by-N matrix A. |
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*> On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular |
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*> matrix R or part of R. See Purpose for details. |
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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 DOUBLE PRECISION array, dimension (LDB,N) |
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*> On entry, the P-by-N matrix B. |
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*> On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains |
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*> a part of R. See Purpose for details. |
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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 convergence criteria for the Jacobi- |
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*> Kogbetliantz iteration procedure. Generally, they are the |
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*> same as used in the preprocessing step, say |
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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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*> \endverbatim |
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*> |
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*> \param[out] ALPHA |
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*> \verbatim |
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*> ALPHA is DOUBLE PRECISION array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] BETA |
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*> \verbatim |
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*> BETA is DOUBLE PRECISION array, dimension (N) |
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*> |
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*> On exit, ALPHA and BETA contain the generalized singular |
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*> value pairs of A and B; |
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*> ALPHA(1:K) = 1, |
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*> BETA(1:K) = 0, |
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*> and if M-K-L >= 0, |
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*> ALPHA(K+1:K+L) = diag(C), |
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*> BETA(K+1:K+L) = diag(S), |
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*> or if M-K-L < 0, |
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*> ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0 |
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*> BETA(K+1:M) = S, BETA(M+1:K+L) = 1. |
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*> Furthermore, if K+L < N, |
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*> ALPHA(K+L+1:N) = 0 and |
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*> BETA(K+L+1:N) = 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] U |
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*> \verbatim |
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*> U is DOUBLE PRECISION array, dimension (LDU,M) |
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*> On entry, if JOBU = 'U', U must contain a matrix U1 (usually |
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*> the orthogonal matrix returned by DGGSVP). |
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*> On exit, |
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*> if JOBU = 'I', U contains the orthogonal matrix U; |
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*> if JOBU = 'U', U contains the product U1*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[in,out] V |
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*> \verbatim |
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*> V is DOUBLE PRECISION array, dimension (LDV,P) |
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*> On entry, if JOBV = 'V', V must contain a matrix V1 (usually |
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*> the orthogonal matrix returned by DGGSVP). |
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*> On exit, |
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*> if JOBV = 'I', V contains the orthogonal matrix V; |
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*> if JOBV = 'V', V contains the product V1*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[in,out] Q |
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*> \verbatim |
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*> Q is DOUBLE PRECISION array, dimension (LDQ,N) |
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*> On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually |
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*> the orthogonal matrix returned by DGGSVP). |
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*> On exit, |
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*> if JOBQ = 'I', Q contains the orthogonal matrix Q; |
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*> if JOBQ = 'Q', Q contains the product Q1*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] WORK |
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*> \verbatim |
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*> WORK is DOUBLE PRECISION array, dimension (2*N) |
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*> \endverbatim |
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*> |
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*> \param[out] NCYCLE |
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*> \verbatim |
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*> NCYCLE is INTEGER |
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*> The number of cycles required for convergence. |
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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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*> = 1: the procedure does not converge after MAXIT cycles. |
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*> \endverbatim |
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*> |
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*> \verbatim |
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*> Internal Parameters |
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*> =================== |
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*> |
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*> MAXIT INTEGER |
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*> MAXIT specifies the total loops that the iterative procedure |
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*> may take. If after MAXIT cycles, the routine fails to |
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*> converge, we return INFO = 1. |
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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 doubleOTHERcomputational |
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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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*> DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce |
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*> min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L |
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*> matrix B13 to the form: |
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*> |
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*> U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1, |
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*> |
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*> where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose |
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*> of Z. C1 and S1 are diagonal matrices satisfying |
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*> |
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*> C1**2 + S1**2 = I, |
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*> |
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*> and R1 is an L-by-L nonsingular upper triangular matrix. |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, |
SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, |
$ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, |
$ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, |
$ Q, LDQ, WORK, NCYCLE, INFO ) |
$ Q, LDQ, WORK, NCYCLE, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2.1) -- |
* -- 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..-- |
* -- April 2009 -- |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBQ, JOBU, JOBV |
CHARACTER JOBQ, JOBU, JOBV |
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$ V( LDV, * ), WORK( * ) |
$ V( LDV, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DTGSJA computes the generalized singular value decomposition (GSVD) |
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* of two real upper triangular (or trapezoidal) matrices A and B. |
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* |
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* On entry, it is assumed that matrices A and B have the following |
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* forms, which may be obtained by the preprocessing subroutine DGGSVP |
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* from a general M-by-N matrix A and P-by-N matrix B: |
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* |
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* N-K-L K L |
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* A = 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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* A = 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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* B = 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. |
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* |
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* On exit, |
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* |
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* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ), |
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* |
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* where U, V and Q are orthogonal matrices, Z' denotes the transpose |
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* of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are |
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* ``diagonal'' matrices, which are of the following structures: |
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* |
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* If M-K-L >= 0, |
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* |
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* K L |
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* D1 = K ( I 0 ) |
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* L ( 0 C ) |
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* M-K-L ( 0 0 ) |
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* |
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* K L |
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* D2 = L ( 0 S ) |
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* P-L ( 0 0 ) |
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* |
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* N-K-L K L |
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* ( 0 R ) = K ( 0 R11 R12 ) K |
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* L ( 0 0 R22 ) L |
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* |
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* where |
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* |
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* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), |
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* S = diag( BETA(K+1), ... , BETA(K+L) ), |
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* C**2 + S**2 = I. |
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* |
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* R is stored in A(1:K+L,N-K-L+1:N) on exit. |
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* |
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* If M-K-L < 0, |
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* |
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* K M-K K+L-M |
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* D1 = K ( I 0 0 ) |
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* M-K ( 0 C 0 ) |
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* |
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* K M-K K+L-M |
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* D2 = M-K ( 0 S 0 ) |
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* K+L-M ( 0 0 I ) |
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* P-L ( 0 0 0 ) |
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* |
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* N-K-L K M-K K+L-M |
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* ( 0 R ) = K ( 0 R11 R12 R13 ) |
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* M-K ( 0 0 R22 R23 ) |
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* K+L-M ( 0 0 0 R33 ) |
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* |
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* where |
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* C = diag( ALPHA(K+1), ... , ALPHA(M) ), |
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* S = diag( BETA(K+1), ... , BETA(M) ), |
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* C**2 + S**2 = I. |
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* |
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* R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored |
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* ( 0 R22 R23 ) |
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* in B(M-K+1:L,N+M-K-L+1:N) on exit. |
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* |
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* The computation of the orthogonal transformation matrices U, V or Q |
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* is optional. These matrices may either be formed explicitly, or they |
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* may be postmultiplied into input matrices U1, V1, or Q1. |
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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': U must contain an orthogonal matrix U1 on entry, and |
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* the product U1*U is returned; |
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* = 'I': U is initialized to the unit matrix, and the |
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* orthogonal matrix U is returned; |
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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': V must contain an orthogonal matrix V1 on entry, and |
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* the product V1*V is returned; |
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* = 'I': V is initialized to the unit matrix, and the |
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* orthogonal matrix V is returned; |
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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': Q must contain an orthogonal matrix Q1 on entry, and |
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* the product Q1*Q is returned; |
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* = 'I': Q is initialized to the unit matrix, and the |
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* orthogonal matrix Q is returned; |
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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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* K (input) INTEGER |
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* L (input) INTEGER |
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* K and L specify the subblocks in the input matrices A and B: |
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* A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N) |
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* of A and B, whose GSVD is going to be computed by DTGSJA. |
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* See Further Details. |
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* |
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* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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* On entry, the M-by-N matrix A. |
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* On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular |
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* matrix R or part of R. See Purpose for details. |
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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) DOUBLE PRECISION array, dimension (LDB,N) |
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* On entry, the P-by-N matrix B. |
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* On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains |
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* a part of R. See Purpose for details. |
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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 convergence criteria for the Jacobi- |
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* Kogbetliantz iteration procedure. Generally, they are the |
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* same as used in the preprocessing step, say |
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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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* |
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* ALPHA (output) DOUBLE PRECISION array, dimension (N) |
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* BETA (output) DOUBLE PRECISION array, dimension (N) |
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* On exit, ALPHA and BETA contain the generalized singular |
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* value pairs of A and B; |
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* ALPHA(1:K) = 1, |
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* BETA(1:K) = 0, |
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* and if M-K-L >= 0, |
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* ALPHA(K+1:K+L) = diag(C), |
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* BETA(K+1:K+L) = diag(S), |
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* or if M-K-L < 0, |
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* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0 |
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* BETA(K+1:M) = S, BETA(M+1:K+L) = 1. |
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* Furthermore, if K+L < N, |
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* ALPHA(K+L+1:N) = 0 and |
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* BETA(K+L+1:N) = 0. |
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* |
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* U (input/output) DOUBLE PRECISION array, dimension (LDU,M) |
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* On entry, if JOBU = 'U', U must contain a matrix U1 (usually |
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* the orthogonal matrix returned by DGGSVP). |
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* On exit, |
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* if JOBU = 'I', U contains the orthogonal matrix U; |
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* if JOBU = 'U', U contains the product U1*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 (input/output) DOUBLE PRECISION array, dimension (LDV,P) |
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* On entry, if JOBV = 'V', V must contain a matrix V1 (usually |
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* the orthogonal matrix returned by DGGSVP). |
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* On exit, |
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* if JOBV = 'I', V contains the orthogonal matrix V; |
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* if JOBV = 'V', V contains the product V1*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 (input/output) DOUBLE PRECISION array, dimension (LDQ,N) |
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* On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually |
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* the orthogonal matrix returned by DGGSVP). |
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* On exit, |
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* if JOBQ = 'I', Q contains the orthogonal matrix Q; |
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* if JOBQ = 'Q', Q contains the product Q1*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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* WORK (workspace) DOUBLE PRECISION array, dimension (2*N) |
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* |
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* NCYCLE (output) INTEGER |
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* The number of cycles required for convergence. |
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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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* = 1: the procedure does not converge after MAXIT cycles. |
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* |
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* Internal Parameters |
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* =================== |
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* |
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* MAXIT INTEGER |
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* MAXIT specifies the total loops that the iterative procedure |
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* may take. If after MAXIT cycles, the routine fails to |
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* converge, we return INFO = 1. |
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* |
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* Further Details |
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* =============== |
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* |
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* DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce |
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* min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L |
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* matrix B13 to the form: |
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* |
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* U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1, |
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* |
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* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose |
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* of Z. C1 and S1 are diagonal matrices satisfying |
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* |
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* C1**2 + S1**2 = I, |
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* |
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* and R1 is an L-by-L nonsingular upper triangular matrix. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
Line 367
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Line 506
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CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, |
CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, |
$ CSV, SNV, CSQ, SNQ ) |
$ CSV, SNV, CSQ, SNQ ) |
* |
* |
* Update (K+I)-th and (K+J)-th rows of matrix A: U'*A |
* Update (K+I)-th and (K+J)-th rows of matrix A: U**T *A |
* |
* |
IF( K+J.LE.M ) |
IF( K+J.LE.M ) |
$ CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ), |
$ CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ), |
$ LDA, CSU, SNU ) |
$ LDA, CSU, SNU ) |
* |
* |
* Update I-th and J-th rows of matrix B: V'*B |
* Update I-th and J-th rows of matrix B: V**T *B |
* |
* |
CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB, |
CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB, |
$ CSV, SNV ) |
$ CSV, SNV ) |