521 lines
14 KiB
Fortran
521 lines
14 KiB
Fortran
*> \brief \b DGGSVP
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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 DGGSVP + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvp.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/dggsvp.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/dggsvp.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 DGGSVP( 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, 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 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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*> DGGSVP computes orthogonal 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**T*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**T*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**T,B**T)**T.
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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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*> DGGSVD.
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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': Orthogonal 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': Orthogonal 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': Orthogonal 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 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 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 DOUBLE PRECISION 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)*MACHEPS,
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*> TOLB = MAX(P,N)*norm(B)*MACHEPS.
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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**T,B**T)**T.
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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 DOUBLE PRECISION array, dimension (LDU,M)
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*> If JOBU = 'U', U contains the orthogonal 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 DOUBLE PRECISION array, dimension (LDV,P)
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*> If JOBV = 'V', V contains the orthogonal 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 DOUBLE PRECISION array, dimension (LDQ,N)
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*> If JOBQ = 'Q', Q contains the orthogonal 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] TAU
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*> \verbatim
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*> TAU is DOUBLE PRECISION 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 DOUBLE PRECISION 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 doubleOTHERcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> The subroutine uses LAPACK subroutine DGEQPF 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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* =====================================================================
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SUBROUTINE DGGSVP( 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, TAU, WORK, INFO )
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*
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* -- LAPACK computational routine (version 3.4.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* November 2011
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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 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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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL FORWRD, WANTQ, WANTU, WANTV
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INTEGER I, J
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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* ..
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* .. External Subroutines ..
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EXTERNAL DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET,
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$ DORG2R, DORM2R, DORMR2, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters
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*
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WANTU = LSAME( JOBU, 'U' )
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WANTV = LSAME( JOBV, 'V' )
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WANTQ = LSAME( JOBQ, 'Q' )
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FORWRD = .TRUE.
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*
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INFO = 0
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IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
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INFO = -1
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ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
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INFO = -2
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ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
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INFO = -3
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ELSE IF( M.LT.0 ) THEN
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INFO = -4
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ELSE IF( P.LT.0 ) THEN
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INFO = -5
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ELSE IF( N.LT.0 ) THEN
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INFO = -6
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -8
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ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
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INFO = -10
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ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
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INFO = -16
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ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
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INFO = -18
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ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
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INFO = -20
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'DGGSVP', -INFO )
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RETURN
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END IF
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*
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* QR with column pivoting of B: B*P = V*( S11 S12 )
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* ( 0 0 )
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*
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DO 10 I = 1, N
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IWORK( I ) = 0
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10 CONTINUE
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CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO )
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*
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* Update A := A*P
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*
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CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK )
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*
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* Determine the effective rank of matrix B.
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*
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L = 0
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DO 20 I = 1, MIN( P, N )
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IF( ABS( B( I, I ) ).GT.TOLB )
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$ L = L + 1
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20 CONTINUE
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*
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IF( WANTV ) THEN
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*
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* Copy the details of V, and form V.
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*
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CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
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IF( P.GT.1 )
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$ CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
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$ LDV )
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CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
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END IF
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*
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* Clean up B
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*
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DO 40 J = 1, L - 1
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DO 30 I = J + 1, L
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B( I, J ) = ZERO
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30 CONTINUE
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40 CONTINUE
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IF( P.GT.L )
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$ CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
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*
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IF( WANTQ ) THEN
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*
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* Set Q = I and Update Q := Q*P
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*
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CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
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CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
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END IF
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*
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IF( P.GE.L .AND. N.NE.L ) THEN
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*
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* RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
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*
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CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO )
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*
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* Update A := A*Z**T
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*
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CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
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$ LDA, WORK, INFO )
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*
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IF( WANTQ ) THEN
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*
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* Update Q := Q*Z**T
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*
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CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q,
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$ LDQ, WORK, INFO )
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END IF
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*
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* Clean up B
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*
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CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
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DO 60 J = N - L + 1, N
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DO 50 I = J - N + L + 1, L
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B( I, J ) = ZERO
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50 CONTINUE
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60 CONTINUE
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*
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END IF
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*
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* Let N-L L
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* A = ( A11 A12 ) M,
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*
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* then the following does the complete QR decomposition of A11:
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*
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* A11 = U*( 0 T12 )*P1**T
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* ( 0 0 )
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*
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DO 70 I = 1, N - L
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IWORK( I ) = 0
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70 CONTINUE
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CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO )
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*
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* Determine the effective rank of A11
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*
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K = 0
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DO 80 I = 1, MIN( M, N-L )
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IF( ABS( A( I, I ) ).GT.TOLA )
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$ K = K + 1
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80 CONTINUE
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*
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* Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
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*
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CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
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$ TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
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*
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IF( WANTU ) THEN
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*
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* Copy the details of U, and form U
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*
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CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
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IF( M.GT.1 )
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$ CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
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$ LDU )
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CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
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END IF
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*
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IF( WANTQ ) THEN
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*
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* Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
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*
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CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
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END IF
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*
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* Clean up A: set the strictly lower triangular part of
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* A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
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*
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DO 100 J = 1, K - 1
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DO 90 I = J + 1, K
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A( I, J ) = ZERO
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90 CONTINUE
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100 CONTINUE
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IF( M.GT.K )
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$ CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA )
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*
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IF( N-L.GT.K ) THEN
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*
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* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
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*
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CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
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*
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IF( WANTQ ) THEN
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*
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* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
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*
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CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU,
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$ Q, LDQ, WORK, INFO )
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END IF
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*
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* Clean up A
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*
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CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
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DO 120 J = N - L - K + 1, N - L
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DO 110 I = J - N + L + K + 1, K
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A( I, J ) = ZERO
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110 CONTINUE
|
|
120 CONTINUE
|
|
*
|
|
END IF
|
|
*
|
|
IF( M.GT.K ) THEN
|
|
*
|
|
* QR factorization of A( K+1:M,N-L+1:N )
|
|
*
|
|
CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
|
|
*
|
|
IF( WANTU ) THEN
|
|
*
|
|
* Update U(:,K+1:M) := U(:,K+1:M)*U1
|
|
*
|
|
CALL DORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
|
|
$ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
|
|
$ WORK, INFO )
|
|
END IF
|
|
*
|
|
* Clean up
|
|
*
|
|
DO 140 J = N - L + 1, N
|
|
DO 130 I = J - N + K + L + 1, M
|
|
A( I, J ) = ZERO
|
|
130 CONTINUE
|
|
140 CONTINUE
|
|
*
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of DGGSVP
|
|
*
|
|
END
|