505 lines
15 KiB
Fortran
505 lines
15 KiB
Fortran
*> \brief <b> SGELS solves overdetermined or underdetermined systems for GE 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 SGELS + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgels.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/sgels.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/sgels.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 SGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
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* INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER TRANS
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* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
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* ..
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* .. Array Arguments ..
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* REAL A( LDA, * ), B( LDB, * ), 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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*> SGELS solves overdetermined or underdetermined real linear systems
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*> involving an M-by-N matrix A, or its transpose, using a QR or LQ
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*> factorization of A. It is assumed that A has full rank.
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*>
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*> The following options are provided:
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*>
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*> 1. If TRANS = 'N' and m >= n: find the least squares solution of
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*> an overdetermined system, i.e., solve the least squares problem
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*> minimize || B - A*X ||.
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*>
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*> 2. If TRANS = 'N' and m < n: find the minimum norm solution of
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*> an underdetermined system A * X = B.
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*>
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*> 3. If TRANS = 'T' and m >= n: find the minimum norm solution of
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*> an underdetermined system A**T * X = B.
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*>
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*> 4. If TRANS = 'T' and m < n: find the least squares solution of
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*> an overdetermined system, i.e., solve the least squares problem
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*> minimize || B - A**T * X ||.
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*>
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*> Several right hand side vectors b and solution vectors x can be
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*> handled in a single call; they are stored as the columns of the
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*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
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*> matrix X.
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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] TRANS
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*> \verbatim
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*> TRANS is CHARACTER*1
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*> = 'N': the linear system involves A;
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*> = 'T': the linear system involves A**T.
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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] N
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*> \verbatim
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*> N is INTEGER
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*> The number of columns of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] NRHS
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*> \verbatim
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*> NRHS is INTEGER
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*> The number of right hand sides, i.e., the number of
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*> columns of the matrices B and X. NRHS >=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 REAL array, dimension (LDA,N)
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*> On entry, the M-by-N matrix A.
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*> On exit,
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*> if M >= N, A is overwritten by details of its QR
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*> factorization as returned by SGEQRF;
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*> if M < N, A is overwritten by details of its LQ
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*> factorization as returned by SGELQF.
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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 REAL array, dimension (LDB,NRHS)
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*> On entry, the matrix B of right hand side vectors, stored
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*> columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
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*> if TRANS = 'T'.
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*> On exit, if INFO = 0, B is overwritten by the solution
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*> vectors, stored columnwise:
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*> if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
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*> squares solution vectors; the residual sum of squares for the
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*> solution in each column is given by the sum of squares of
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*> elements N+1 to M in that column;
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*> if TRANS = 'N' and m < n, rows 1 to N of B contain the
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*> minimum norm solution vectors;
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*> if TRANS = 'T' and m >= n, rows 1 to M of B contain the
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*> minimum norm solution vectors;
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*> if TRANS = 'T' and m < n, rows 1 to M of B contain the
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*> least squares solution vectors; the residual sum of squares
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*> for the solution in each column is given by the sum of
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*> squares of elements M+1 to N in that column.
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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,M,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 REAL 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.
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*> LWORK >= max( 1, MN + max( MN, NRHS ) ).
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*> For optimal performance,
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*> LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
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*> where MN = min(M,N) and NB is the optimum block size.
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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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*> > 0: if INFO = i, the i-th diagonal element of the
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*> triangular factor of A is zero, so that A does not have
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*> full rank; the least squares solution could not be
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*> 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 December 2016
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*
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*> \ingroup realGEsolve
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*
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* =====================================================================
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SUBROUTINE SGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
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$ INFO )
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*
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* -- LAPACK driver routine (version 3.7.0) --
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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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* December 2016
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*
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* .. Scalar Arguments ..
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CHARACTER TRANS
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INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
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* ..
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* .. Array Arguments ..
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REAL A( LDA, * ), B( LDB, * ), 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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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY, TPSD
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INTEGER BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE
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REAL ANRM, BIGNUM, BNRM, SMLNUM
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* ..
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* .. Local Arrays ..
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REAL RWORK( 1 )
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV
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REAL SLAMCH, SLANGE
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EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE
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* ..
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* .. External Subroutines ..
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EXTERNAL SGELQF, SGEQRF, SLABAD, SLASCL, SLASET, SORMLQ,
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$ SORMQR, STRTRS, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, REAL
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* ..
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* .. Executable Statements ..
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*
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* Test the input arguments.
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*
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INFO = 0
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MN = MIN( M, N )
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LQUERY = ( LWORK.EQ.-1 )
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IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'T' ) ) ) THEN
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INFO = -1
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ELSE IF( M.LT.0 ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -6
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ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
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INFO = -8
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ELSE IF( LWORK.LT.MAX( 1, MN + MAX( MN, NRHS ) ) .AND.
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$ .NOT.LQUERY ) THEN
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INFO = -10
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END IF
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*
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* Figure out optimal block size
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*
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IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
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*
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TPSD = .TRUE.
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IF( LSAME( TRANS, 'N' ) )
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$ TPSD = .FALSE.
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*
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IF( M.GE.N ) THEN
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NB = ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 )
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IF( TPSD ) THEN
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NB = MAX( NB, ILAENV( 1, 'SORMQR', 'LN', M, NRHS, N,
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$ -1 ) )
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ELSE
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NB = MAX( NB, ILAENV( 1, 'SORMQR', 'LT', M, NRHS, N,
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$ -1 ) )
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END IF
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ELSE
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NB = ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 )
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IF( TPSD ) THEN
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NB = MAX( NB, ILAENV( 1, 'SORMLQ', 'LT', N, NRHS, M,
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$ -1 ) )
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ELSE
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NB = MAX( NB, ILAENV( 1, 'SORMLQ', 'LN', N, NRHS, M,
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$ -1 ) )
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END IF
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END IF
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*
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WSIZE = MAX( 1, MN + MAX( MN, NRHS )*NB )
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WORK( 1 ) = REAL( WSIZE )
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*
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SGELS ', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( MIN( M, N, NRHS ).EQ.0 ) THEN
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CALL SLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
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RETURN
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END IF
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*
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* Get machine parameters
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*
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SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'P' )
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BIGNUM = ONE / SMLNUM
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CALL SLABAD( SMLNUM, BIGNUM )
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*
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* Scale A, B if max element outside range [SMLNUM,BIGNUM]
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*
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ANRM = SLANGE( 'M', M, N, A, LDA, RWORK )
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IASCL = 0
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IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
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*
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* Scale matrix norm up to SMLNUM
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*
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CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
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IASCL = 1
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ELSE IF( ANRM.GT.BIGNUM ) THEN
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*
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* Scale matrix norm down to BIGNUM
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*
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CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
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IASCL = 2
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ELSE IF( ANRM.EQ.ZERO ) THEN
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*
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* Matrix all zero. Return zero solution.
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*
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CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
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GO TO 50
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END IF
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*
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BROW = M
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IF( TPSD )
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$ BROW = N
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BNRM = SLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
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IBSCL = 0
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IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
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*
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* Scale matrix norm up to SMLNUM
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*
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CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
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$ INFO )
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IBSCL = 1
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ELSE IF( BNRM.GT.BIGNUM ) THEN
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*
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* Scale matrix norm down to BIGNUM
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*
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CALL SLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
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$ INFO )
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IBSCL = 2
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END IF
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*
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IF( M.GE.N ) THEN
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*
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* compute QR factorization of A
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*
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CALL SGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
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$ INFO )
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*
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* workspace at least N, optimally N*NB
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*
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IF( .NOT.TPSD ) THEN
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*
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* Least-Squares Problem min || A * X - B ||
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*
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* B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
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*
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CALL SORMQR( 'Left', 'Transpose', M, NRHS, N, A, LDA,
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$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
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$ INFO )
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*
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* workspace at least NRHS, optimally NRHS*NB
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*
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* B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
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*
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CALL STRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS,
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$ A, LDA, B, LDB, INFO )
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*
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IF( INFO.GT.0 ) THEN
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RETURN
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END IF
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*
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SCLLEN = N
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*
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ELSE
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*
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* Underdetermined system of equations A**T * X = B
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*
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* B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
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*
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CALL STRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS,
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$ A, LDA, B, LDB, INFO )
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*
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IF( INFO.GT.0 ) THEN
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RETURN
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END IF
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*
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* B(N+1:M,1:NRHS) = ZERO
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*
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DO 20 J = 1, NRHS
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DO 10 I = N + 1, M
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B( I, J ) = ZERO
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10 CONTINUE
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20 CONTINUE
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*
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* B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
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*
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CALL SORMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA,
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$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
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$ INFO )
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*
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* workspace at least NRHS, optimally NRHS*NB
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*
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SCLLEN = M
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*
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END IF
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*
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ELSE
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*
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* Compute LQ factorization of A
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*
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CALL SGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
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$ INFO )
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*
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* workspace at least M, optimally M*NB.
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*
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IF( .NOT.TPSD ) THEN
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*
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* underdetermined system of equations A * X = B
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*
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* B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
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*
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CALL STRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS,
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$ A, LDA, B, LDB, INFO )
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*
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IF( INFO.GT.0 ) THEN
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RETURN
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END IF
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*
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* B(M+1:N,1:NRHS) = 0
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*
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DO 40 J = 1, NRHS
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DO 30 I = M + 1, N
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B( I, J ) = ZERO
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30 CONTINUE
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40 CONTINUE
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*
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* B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS)
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*
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CALL SORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA,
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$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
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$ INFO )
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*
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* workspace at least NRHS, optimally NRHS*NB
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*
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SCLLEN = N
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*
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ELSE
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*
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* overdetermined system min || A**T * X - B ||
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*
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* B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
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*
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CALL SORMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA,
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$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
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$ INFO )
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*
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* workspace at least NRHS, optimally NRHS*NB
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*
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* B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
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*
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CALL STRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS,
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$ A, LDA, B, LDB, INFO )
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*
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IF( INFO.GT.0 ) THEN
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RETURN
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END IF
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*
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SCLLEN = M
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*
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END IF
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*
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END IF
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*
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* Undo scaling
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*
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IF( IASCL.EQ.1 ) THEN
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CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
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$ INFO )
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ELSE IF( IASCL.EQ.2 ) THEN
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CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
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$ INFO )
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END IF
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IF( IBSCL.EQ.1 ) THEN
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CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
|
|
$ INFO )
|
|
ELSE IF( IBSCL.EQ.2 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
|
|
$ INFO )
|
|
END IF
|
|
*
|
|
50 CONTINUE
|
|
WORK( 1 ) = REAL( WSIZE )
|
|
*
|
|
RETURN
|
|
*
|
|
* End of SGELS
|
|
*
|
|
END
|