version 1.5, 2010/08/07 13:22:32
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version 1.14, 2016/08/27 15:34:48
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*> \brief <b> ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices</b> |
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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 ZGGLSE + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgglse.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/zgglse.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/zgglse.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 ZGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, |
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* INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER INFO, LDA, LDB, LWORK, M, N, P |
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* .. |
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* .. Array Arguments .. |
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* COMPLEX*16 A( LDA, * ), B( LDB, * ), C( * ), D( * ), |
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* $ WORK( * ), X( * ) |
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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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*> ZGGLSE solves the linear equality-constrained least squares (LSE) |
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*> problem: |
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*> |
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*> minimize || c - A*x ||_2 subject to B*x = d |
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*> |
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*> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given |
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*> M-vector, and d is a given P-vector. It is assumed that |
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*> P <= N <= M+P, and |
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*> |
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*> rank(B) = P and rank( (A) ) = N. |
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*> ( (B) ) |
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*> |
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*> These conditions ensure that the LSE problem has a unique solution, |
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*> which is obtained using a generalized RQ factorization of the |
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*> matrices (B, A) given by |
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*> |
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*> B = (0 R)*Q, A = Z*T*Q. |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The number of rows of the matrix A. M >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The number of columns of the matrices A and B. N >= 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. 0 <= P <= N <= M+P. |
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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, the elements on and above the diagonal of the array |
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*> contain the min(M,N)-by-N upper trapezoidal matrix T. |
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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, the upper triangle of the subarray B(1:P,N-P+1:N) |
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*> contains the P-by-P upper triangular matrix R. |
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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,out] C |
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*> \verbatim |
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*> C is COMPLEX*16 array, dimension (M) |
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*> On entry, C contains the right hand side vector for the |
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*> least squares part of the LSE problem. |
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*> On exit, the residual sum of squares for the solution |
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*> is given by the sum of squares of elements N-P+1 to M of |
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*> vector C. |
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*> \endverbatim |
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*> |
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*> \param[in,out] D |
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*> \verbatim |
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*> D is COMPLEX*16 array, dimension (P) |
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*> On entry, D contains the right hand side vector for the |
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*> constrained equation. |
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*> On exit, D is destroyed. |
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*> \endverbatim |
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*> |
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*> \param[out] X |
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*> \verbatim |
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*> X is COMPLEX*16 array, dimension (N) |
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*> On exit, X is the solution of the LSE problem. |
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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(1,LWORK)) |
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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*> \endverbatim |
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*> |
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*> \param[in] LWORK |
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*> \verbatim |
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*> LWORK is INTEGER |
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*> The dimension of the array WORK. LWORK >= max(1,M+N+P). |
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*> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, |
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*> where NB is an upper bound for the optimal blocksizes for |
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*> ZGEQRF, CGERQF, ZUNMQR and CUNMRQ. |
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*> |
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*> If LWORK = -1, then a workspace query is assumed; the routine |
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*> only calculates the optimal size of the WORK array, returns |
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*> this value as the first entry of the WORK array, and no error |
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*> message related to LWORK is issued by XERBLA. |
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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 upper triangular factor R associated with B in the |
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*> generalized RQ factorization of the pair (B, A) is |
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*> singular, so that rank(B) < P; the least squares |
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*> solution could not be computed. |
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*> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor |
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*> T associated with A in the generalized RQ factorization |
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*> of the pair (B, A) is singular, so that |
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*> rank( (A) ) < N; the least squares solution could not |
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*> ( (B) ) |
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*> be computed. |
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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 complex16OTHERsolve |
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* |
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* ===================================================================== |
SUBROUTINE ZGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, |
SUBROUTINE ZGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, |
$ INFO ) |
$ INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver 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..-- |
* February 2007 |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER INFO, LDA, LDB, LWORK, M, N, P |
INTEGER INFO, LDA, LDB, LWORK, M, N, P |
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$ WORK( * ), X( * ) |
$ WORK( * ), X( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZGGLSE solves the linear equality-constrained least squares (LSE) |
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* problem: |
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* |
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* minimize || c - A*x ||_2 subject to B*x = d |
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* |
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* where A is an M-by-N matrix, B is a P-by-N matrix, c is a given |
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* M-vector, and d is a given P-vector. It is assumed that |
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* P <= N <= M+P, and |
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* |
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* rank(B) = P and rank( ( A ) ) = N. |
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* ( ( B ) ) |
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* |
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* These conditions ensure that the LSE problem has a unique solution, |
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* which is obtained using a generalized RQ factorization of the |
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* matrices (B, A) given by |
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* |
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* B = (0 R)*Q, A = Z*T*Q. |
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* |
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* Arguments |
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* ========= |
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* |
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* M (input) INTEGER |
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* The number of rows of the matrix A. M >= 0. |
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* |
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* N (input) INTEGER |
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* The number of columns of the matrices A and B. N >= 0. |
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* |
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* P (input) INTEGER |
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* The number of rows of the matrix B. 0 <= P <= N <= M+P. |
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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, the elements on and above the diagonal of the array |
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* contain the min(M,N)-by-N upper trapezoidal matrix T. |
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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, the upper triangle of the subarray B(1:P,N-P+1:N) |
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* contains the P-by-P upper triangular matrix R. |
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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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* C (input/output) COMPLEX*16 array, dimension (M) |
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* On entry, C contains the right hand side vector for the |
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* least squares part of the LSE problem. |
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* On exit, the residual sum of squares for the solution |
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* is given by the sum of squares of elements N-P+1 to M of |
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* vector C. |
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* |
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* D (input/output) COMPLEX*16 array, dimension (P) |
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* On entry, D contains the right hand side vector for the |
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* constrained equation. |
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* On exit, D is destroyed. |
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* |
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* X (output) COMPLEX*16 array, dimension (N) |
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* On exit, X is the solution of the LSE problem. |
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* |
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* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) |
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* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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* |
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* LWORK (input) INTEGER |
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* The dimension of the array WORK. LWORK >= max(1,M+N+P). |
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* For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, |
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* where NB is an upper bound for the optimal blocksizes for |
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* ZGEQRF, CGERQF, ZUNMQR and CUNMRQ. |
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* |
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* If LWORK = -1, then a workspace query is assumed; the routine |
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* only calculates the optimal size of the WORK array, returns |
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* this value as the first entry of the WORK array, and no error |
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* message related to LWORK is issued by XERBLA. |
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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 upper triangular factor R associated with B in the |
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* generalized RQ factorization of the pair (B, A) is |
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* singular, so that rank(B) < P; the least squares |
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* solution could not be computed. |
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* = 2: the (N-P) by (N-P) part of the upper trapezoidal factor |
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* T associated with A in the generalized RQ factorization |
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* of the pair (B, A) is singular, so that |
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* rank( (A) ) < N; the least squares solution could not |
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* ( (B) ) |
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* be computed. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* |
* |
* Compute the GRQ factorization of matrices B and A: |
* Compute the GRQ factorization of matrices B and A: |
* |
* |
* B*Q' = ( 0 T12 ) P Z'*A*Q' = ( R11 R12 ) N-P |
* B*Q**H = ( 0 T12 ) P Z**H*A*Q**H = ( R11 R12 ) N-P |
* N-P P ( 0 R22 ) M+P-N |
* N-P P ( 0 R22 ) M+P-N |
* N-P P |
* N-P P |
* |
* |
* where T12 and R11 are upper triangular, and Q and Z are |
* where T12 and R11 are upper triangular, and Q and Z are |
* unitary. |
* unitary. |
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$ WORK( P+MN+1 ), LWORK-P-MN, INFO ) |
$ WORK( P+MN+1 ), LWORK-P-MN, INFO ) |
LOPT = WORK( P+MN+1 ) |
LOPT = WORK( P+MN+1 ) |
* |
* |
* Update c = Z'*c = ( c1 ) N-P |
* Update c = Z**H *c = ( c1 ) N-P |
* ( c2 ) M+P-N |
* ( c2 ) M+P-N |
* |
* |
CALL ZUNMQR( 'Left', 'Conjugate Transpose', M, 1, MN, A, LDA, |
CALL ZUNMQR( 'Left', 'Conjugate Transpose', M, 1, MN, A, LDA, |
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CALL ZAXPY( NR, -CONE, D, 1, C( N-P+1 ), 1 ) |
CALL ZAXPY( NR, -CONE, D, 1, C( N-P+1 ), 1 ) |
END IF |
END IF |
* |
* |
* Backward transformation x = Q'*x |
* Backward transformation x = Q**H*x |
* |
* |
CALL ZUNMRQ( 'Left', 'Conjugate Transpose', N, 1, P, B, LDB, |
CALL ZUNMRQ( 'Left', 'Conjugate Transpose', N, 1, P, B, LDB, |
$ WORK( 1 ), X, N, WORK( P+MN+1 ), LWORK-P-MN, INFO ) |
$ WORK( 1 ), X, N, WORK( P+MN+1 ), LWORK-P-MN, INFO ) |