--- rpl/lapack/lapack/dgels.f 2011/07/22 07:38:04 1.8 +++ rpl/lapack/lapack/dgels.f 2011/11/21 20:42:51 1.9 @@ -1,10 +1,192 @@ +*> \brief DGELS solves overdetermined or underdetermined systems for GE matrices +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download DGELS + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, +* INFO ) +* +* .. Scalar Arguments .. +* CHARACTER TRANS +* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS +* .. +* .. Array Arguments .. +* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DGELS solves overdetermined or underdetermined real linear systems +*> involving an M-by-N matrix A, or its transpose, using a QR or LQ +*> factorization of A. It is assumed that A has full rank. +*> +*> The following options are provided: +*> +*> 1. If TRANS = 'N' and m >= n: find the least squares solution of +*> an overdetermined system, i.e., solve the least squares problem +*> minimize || B - A*X ||. +*> +*> 2. If TRANS = 'N' and m < n: find the minimum norm solution of +*> an underdetermined system A * X = B. +*> +*> 3. If TRANS = 'T' and m >= n: find the minimum norm solution of +*> an undetermined system A**T * X = B. +*> +*> 4. If TRANS = 'T' and m < n: find the least squares solution of +*> an overdetermined system, i.e., solve the least squares problem +*> minimize || B - A**T * X ||. +*> +*> Several right hand side vectors b and solution vectors x can be +*> handled in a single call; they are stored as the columns of the +*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution +*> matrix X. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] TRANS +*> \verbatim +*> TRANS is CHARACTER*1 +*> = 'N': the linear system involves A; +*> = 'T': the linear system involves A**T. +*> \endverbatim +*> +*> \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 matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of +*> columns of the matrices B and X. NRHS >=0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is DOUBLE PRECISION array, dimension (LDA,N) +*> On entry, the M-by-N matrix A. +*> On exit, +*> if M >= N, A is overwritten by details of its QR +*> factorization as returned by DGEQRF; +*> if M < N, A is overwritten by details of its LQ +*> factorization as returned by DGELQF. +*> \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,NRHS) +*> On entry, the matrix B of right hand side vectors, stored +*> columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS +*> if TRANS = 'T'. +*> On exit, if INFO = 0, B is overwritten by the solution +*> vectors, stored columnwise: +*> if TRANS = 'N' and m >= n, rows 1 to n of B contain the least +*> squares solution vectors; the residual sum of squares for the +*> solution in each column is given by the sum of squares of +*> elements N+1 to M in that column; +*> if TRANS = 'N' and m < n, rows 1 to N of B contain the +*> minimum norm solution vectors; +*> if TRANS = 'T' and m >= n, rows 1 to M of B contain the +*> minimum norm solution vectors; +*> if TRANS = 'T' and m < n, rows 1 to M of B contain the +*> least squares solution vectors; the residual sum of squares +*> for the solution in each column is given by the sum of +*> squares of elements M+1 to N in that column. +*> \endverbatim +*> +*> \param[in] LDB +*> \verbatim +*> LDB is INTEGER +*> The leading dimension of the array B. LDB >= MAX(1,M,N). +*> \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, MN + max( MN, NRHS ) ). +*> For optimal performance, +*> LWORK >= max( 1, MN + max( MN, NRHS )*NB ). +*> where MN = min(M,N) and NB is the optimum block size. +*> +*> 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 +*> > 0: if INFO = i, the i-th diagonal element of the +*> triangular factor of A is zero, so that A does not have +*> full rank; the least squares solution could not 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 doubleGEsolve +* +* ===================================================================== SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, $ INFO ) * -* -- LAPACK driver routine (version 3.3.1) -- +* -- LAPACK driver routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* -- April 2011 -- +* November 2011 * * .. Scalar Arguments .. CHARACTER TRANS @@ -14,107 +196,6 @@ DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * ) * .. * -* Purpose -* ======= -* -* DGELS solves overdetermined or underdetermined real linear systems -* involving an M-by-N matrix A, or its transpose, using a QR or LQ -* factorization of A. It is assumed that A has full rank. -* -* The following options are provided: -* -* 1. If TRANS = 'N' and m >= n: find the least squares solution of -* an overdetermined system, i.e., solve the least squares problem -* minimize || B - A*X ||. -* -* 2. If TRANS = 'N' and m < n: find the minimum norm solution of -* an underdetermined system A * X = B. -* -* 3. If TRANS = 'T' and m >= n: find the minimum norm solution of -* an undetermined system A**T * X = B. -* -* 4. If TRANS = 'T' and m < n: find the least squares solution of -* an overdetermined system, i.e., solve the least squares problem -* minimize || B - A**T * X ||. -* -* Several right hand side vectors b and solution vectors x can be -* handled in a single call; they are stored as the columns of the -* M-by-NRHS right hand side matrix B and the N-by-NRHS solution -* matrix X. -* -* Arguments -* ========= -* -* TRANS (input) CHARACTER*1 -* = 'N': the linear system involves A; -* = 'T': the linear system involves A**T. -* -* M (input) INTEGER -* The number of rows of the matrix A. M >= 0. -* -* N (input) INTEGER -* The number of columns of the matrix A. N >= 0. -* -* NRHS (input) INTEGER -* The number of right hand sides, i.e., the number of -* columns of the matrices B and X. NRHS >=0. -* -* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) -* On entry, the M-by-N matrix A. -* On exit, -* if M >= N, A is overwritten by details of its QR -* factorization as returned by DGEQRF; -* if M < N, A is overwritten by details of its LQ -* factorization as returned by DGELQF. -* -* LDA (input) INTEGER -* The leading dimension of the array A. LDA >= max(1,M). -* -* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) -* On entry, the matrix B of right hand side vectors, stored -* columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS -* if TRANS = 'T'. -* On exit, if INFO = 0, B is overwritten by the solution -* vectors, stored columnwise: -* if TRANS = 'N' and m >= n, rows 1 to n of B contain the least -* squares solution vectors; the residual sum of squares for the -* solution in each column is given by the sum of squares of -* elements N+1 to M in that column; -* if TRANS = 'N' and m < n, rows 1 to N of B contain the -* minimum norm solution vectors; -* if TRANS = 'T' and m >= n, rows 1 to M of B contain the -* minimum norm solution vectors; -* if TRANS = 'T' and m < n, rows 1 to M of B contain the -* least squares solution vectors; the residual sum of squares -* for the solution in each column is given by the sum of -* squares of elements M+1 to N in that column. -* -* LDB (input) INTEGER -* The leading dimension of the array B. LDB >= MAX(1,M,N). -* -* 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, MN + max( MN, NRHS ) ). -* For optimal performance, -* LWORK >= max( 1, MN + max( MN, NRHS )*NB ). -* where MN = min(M,N) and NB is the optimum block size. -* -* 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 -* > 0: if INFO = i, the i-th diagonal element of the -* triangular factor of A is zero, so that A does not have -* full rank; the least squares solution could not be -* computed. -* * ===================================================================== * * .. Parameters ..