version 1.3, 2010/08/06 15:28:38
|
version 1.11, 2012/08/22 09:48:14
|
Line 1
|
Line 1
|
|
*> \brief <b> DGGLSE solves overdetermined or underdetermined systems for OTHER matrices</b> |
|
* |
|
* =========== DOCUMENTATION =========== |
|
* |
|
* Online html documentation available at |
|
* http://www.netlib.org/lapack/explore-html/ |
|
* |
|
*> \htmlonly |
|
*> Download DGGLSE + dependencies |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgglse.f"> |
|
*> [TGZ]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgglse.f"> |
|
*> [ZIP]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgglse.f"> |
|
*> [TXT]</a> |
|
*> \endhtmlonly |
|
* |
|
* Definition: |
|
* =========== |
|
* |
|
* SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, |
|
* INFO ) |
|
* |
|
* .. Scalar Arguments .. |
|
* INTEGER INFO, LDA, LDB, LWORK, M, N, P |
|
* .. |
|
* .. Array Arguments .. |
|
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( * ), D( * ), |
|
* $ WORK( * ), X( * ) |
|
* .. |
|
* |
|
* |
|
*> \par Purpose: |
|
* ============= |
|
*> |
|
*> \verbatim |
|
*> |
|
*> DGGLSE solves the linear equality-constrained least squares (LSE) |
|
*> problem: |
|
*> |
|
*> minimize || c - A*x ||_2 subject to B*x = d |
|
*> |
|
*> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given |
|
*> M-vector, and d is a given P-vector. It is assumed that |
|
*> P <= N <= M+P, and |
|
*> |
|
*> rank(B) = P and rank( (A) ) = N. |
|
*> ( (B) ) |
|
*> |
|
*> These conditions ensure that the LSE problem has a unique solution, |
|
*> which is obtained using a generalized RQ factorization of the |
|
*> matrices (B, A) given by |
|
*> |
|
*> B = (0 R)*Q, A = Z*T*Q. |
|
*> \endverbatim |
|
* |
|
* Arguments: |
|
* ========== |
|
* |
|
*> \param[in] M |
|
*> \verbatim |
|
*> M is INTEGER |
|
*> The number of rows of the matrix A. M >= 0. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] N |
|
*> \verbatim |
|
*> N is INTEGER |
|
*> The number of columns of the matrices A and B. N >= 0. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] P |
|
*> \verbatim |
|
*> P is INTEGER |
|
*> The number of rows of the matrix B. 0 <= P <= N <= M+P. |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] A |
|
*> \verbatim |
|
*> A is DOUBLE PRECISION array, dimension (LDA,N) |
|
*> On entry, the M-by-N matrix A. |
|
*> On exit, the elements on and above the diagonal of the array |
|
*> contain the min(M,N)-by-N upper trapezoidal matrix T. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDA |
|
*> \verbatim |
|
*> LDA is INTEGER |
|
*> The leading dimension of the array A. LDA >= max(1,M). |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] B |
|
*> \verbatim |
|
*> B is DOUBLE PRECISION array, dimension (LDB,N) |
|
*> On entry, the P-by-N matrix B. |
|
*> On exit, the upper triangle of the subarray B(1:P,N-P+1:N) |
|
*> contains the P-by-P upper triangular matrix R. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDB |
|
*> \verbatim |
|
*> LDB is INTEGER |
|
*> The leading dimension of the array B. LDB >= max(1,P). |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] C |
|
*> \verbatim |
|
*> C is DOUBLE PRECISION array, dimension (M) |
|
*> On entry, C contains the right hand side vector for the |
|
*> least squares part of the LSE problem. |
|
*> On exit, the residual sum of squares for the solution |
|
*> is given by the sum of squares of elements N-P+1 to M of |
|
*> vector C. |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] D |
|
*> \verbatim |
|
*> D is DOUBLE PRECISION array, dimension (P) |
|
*> On entry, D contains the right hand side vector for the |
|
*> constrained equation. |
|
*> On exit, D is destroyed. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] X |
|
*> \verbatim |
|
*> X is DOUBLE PRECISION array, dimension (N) |
|
*> On exit, X is the solution of the LSE problem. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] WORK |
|
*> \verbatim |
|
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) |
|
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LWORK |
|
*> \verbatim |
|
*> LWORK is INTEGER |
|
*> The dimension of the array WORK. LWORK >= max(1,M+N+P). |
|
*> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, |
|
*> where NB is an upper bound for the optimal blocksizes for |
|
*> DGEQRF, SGERQF, DORMQR and SORMRQ. |
|
*> |
|
*> If LWORK = -1, then a workspace query is assumed; the routine |
|
*> only calculates the optimal size of the WORK array, returns |
|
*> this value as the first entry of the WORK array, and no error |
|
*> message related to LWORK is issued by XERBLA. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] INFO |
|
*> \verbatim |
|
*> INFO is INTEGER |
|
*> = 0: successful exit. |
|
*> < 0: if INFO = -i, the i-th argument had an illegal value. |
|
*> = 1: the upper triangular factor R associated with B in the |
|
*> generalized RQ factorization of the pair (B, A) is |
|
*> singular, so that rank(B) < P; the least squares |
|
*> solution could not be computed. |
|
*> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor |
|
*> T associated with A in the generalized RQ factorization |
|
*> of the pair (B, A) is singular, so that |
|
*> rank( (A) ) < N; the least squares solution could not |
|
*> ( (B) ) |
|
*> be computed. |
|
*> \endverbatim |
|
* |
|
* Authors: |
|
* ======== |
|
* |
|
*> \author Univ. of Tennessee |
|
*> \author Univ. of California Berkeley |
|
*> \author Univ. of Colorado Denver |
|
*> \author NAG Ltd. |
|
* |
|
*> \date November 2011 |
|
* |
|
*> \ingroup doubleOTHERsolve |
|
* |
|
* ===================================================================== |
SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, |
SUBROUTINE DGGLSE( 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..-- |
* November 2006 |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER INFO, LDA, LDB, LWORK, M, N, P |
INTEGER INFO, LDA, LDB, LWORK, M, N, P |
Line 14
|
Line 193
|
$ WORK( * ), X( * ) |
$ WORK( * ), X( * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* DGGLSE solves the linear equality-constrained least squares (LSE) |
|
* problem: |
|
* |
|
* minimize || c - A*x ||_2 subject to B*x = d |
|
* |
|
* where A is an M-by-N matrix, B is a P-by-N matrix, c is a given |
|
* M-vector, and d is a given P-vector. It is assumed that |
|
* P <= N <= M+P, and |
|
* |
|
* rank(B) = P and rank( (A) ) = N. |
|
* ( (B) ) |
|
* |
|
* These conditions ensure that the LSE problem has a unique solution, |
|
* which is obtained using a generalized RQ factorization of the |
|
* matrices (B, A) given by |
|
* |
|
* B = (0 R)*Q, A = Z*T*Q. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
|
* M (input) INTEGER |
|
* The number of rows of the matrix A. M >= 0. |
|
* |
|
* N (input) INTEGER |
|
* The number of columns of the matrices A and B. N >= 0. |
|
* |
|
* P (input) INTEGER |
|
* The number of rows of the matrix B. 0 <= P <= N <= M+P. |
|
* |
|
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
|
* On entry, the M-by-N matrix A. |
|
* On exit, the elements on and above the diagonal of the array |
|
* contain the min(M,N)-by-N upper trapezoidal matrix T. |
|
* |
|
* LDA (input) INTEGER |
|
* The leading dimension of the array A. LDA >= max(1,M). |
|
* |
|
* B (input/output) DOUBLE PRECISION array, dimension (LDB,N) |
|
* On entry, the P-by-N matrix B. |
|
* On exit, the upper triangle of the subarray B(1:P,N-P+1:N) |
|
* contains the P-by-P upper triangular matrix R. |
|
* |
|
* LDB (input) INTEGER |
|
* The leading dimension of the array B. LDB >= max(1,P). |
|
* |
|
* C (input/output) DOUBLE PRECISION array, dimension (M) |
|
* On entry, C contains the right hand side vector for the |
|
* least squares part of the LSE problem. |
|
* On exit, the residual sum of squares for the solution |
|
* is given by the sum of squares of elements N-P+1 to M of |
|
* vector C. |
|
* |
|
* D (input/output) DOUBLE PRECISION array, dimension (P) |
|
* On entry, D contains the right hand side vector for the |
|
* constrained equation. |
|
* On exit, D is destroyed. |
|
* |
|
* X (output) DOUBLE PRECISION array, dimension (N) |
|
* On exit, X is the solution of the LSE problem. |
|
* |
|
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) |
|
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
|
* |
|
* LWORK (input) INTEGER |
|
* The dimension of the array WORK. LWORK >= max(1,M+N+P). |
|
* For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, |
|
* where NB is an upper bound for the optimal blocksizes for |
|
* DGEQRF, SGERQF, DORMQR and SORMRQ. |
|
* |
|
* If LWORK = -1, then a workspace query is assumed; the routine |
|
* only calculates the optimal size of the WORK array, returns |
|
* this value as the first entry of the WORK array, and no error |
|
* message related to LWORK is issued by XERBLA. |
|
* |
|
* INFO (output) INTEGER |
|
* = 0: successful exit. |
|
* < 0: if INFO = -i, the i-th argument had an illegal value. |
|
* = 1: the upper triangular factor R associated with B in the |
|
* generalized RQ factorization of the pair (B, A) is |
|
* singular, so that rank(B) < P; the least squares |
|
* solution could not be computed. |
|
* = 2: the (N-P) by (N-P) part of the upper trapezoidal factor |
|
* T associated with A in the generalized RQ factorization |
|
* of the pair (B, A) is singular, so that |
|
* rank( (A) ) < N; the least squares solution could not |
|
* ( (B) ) |
|
* be computed. |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
Line 183
|
Line 270
|
* |
* |
* 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**T = ( 0 T12 ) P Z**T*A*Q**T = ( 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 |
* orthogonal. |
* orthogonal. |
Line 194
|
Line 281
|
$ 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**T *c = ( c1 ) N-P |
* ( c2 ) M+P-N |
* ( c2 ) M+P-N |
* |
* |
CALL DORMQR( 'Left', 'Transpose', M, 1, MN, A, LDA, WORK( P+1 ), |
CALL DORMQR( 'Left', 'Transpose', M, 1, MN, A, LDA, WORK( P+1 ), |
$ C, MAX( 1, M ), WORK( P+MN+1 ), LWORK-P-MN, INFO ) |
$ C, MAX( 1, M ), WORK( P+MN+1 ), LWORK-P-MN, INFO ) |
Line 254
|
Line 341
|
CALL DAXPY( NR, -ONE, D, 1, C( N-P+1 ), 1 ) |
CALL DAXPY( NR, -ONE, D, 1, C( N-P+1 ), 1 ) |
END IF |
END IF |
* |
* |
* Backward transformation x = Q'*x |
* Backward transformation x = Q**T*x |
* |
* |
CALL DORMRQ( 'Left', 'Transpose', N, 1, P, B, LDB, WORK( 1 ), X, |
CALL DORMRQ( 'Left', 'Transpose', N, 1, P, B, LDB, WORK( 1 ), X, |
$ N, WORK( P+MN+1 ), LWORK-P-MN, INFO ) |
$ N, WORK( P+MN+1 ), LWORK-P-MN, INFO ) |