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489 lines
14 KiB
489 lines
14 KiB
*> \brief \b DGETSLS
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DGETSLS( TRANS, M, N, NRHS, A, LDA, B, LDB,
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* $ WORK, LWORK, 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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* DOUBLE PRECISION 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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*> DGETSLS solves overdetermined or underdetermined real linear systems
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*> involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
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*> factorization of A. It is assumed that A has full rank.
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*>
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*>
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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 undetermined 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 DOUBLE PRECISION 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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*> A is overwritten by details of its QR or LQ
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*> factorization as returned by DGEQR or DGELQ.
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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,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.
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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.
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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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*> (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
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*> On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
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*> or optimal, if query was assumed) LWORK.
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*> See LWORK for details.
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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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*> If LWORK = -1 or -2, then a workspace query is assumed.
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*> If LWORK = -1, the routine calculates optimal size of WORK for the
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*> optimal performance and returns this value in WORK(1).
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*> If LWORK = -2, the routine calculates minimal size of WORK and
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*> returns this value in WORK(1).
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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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*> \ingroup doubleGEsolve
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*
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* =====================================================================
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SUBROUTINE DGETSLS( TRANS, M, N, NRHS, A, LDA, B, LDB,
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$ WORK, LWORK, INFO )
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*
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* -- LAPACK driver routine --
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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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*
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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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DOUBLE PRECISION 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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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY, TRAN
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INTEGER I, IASCL, IBSCL, J, MAXMN, BROW,
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$ SCLLEN, TSZO, TSZM, LWO, LWM, LW1, LW2,
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$ WSIZEO, WSIZEM, INFO2
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DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM, TQ( 5 ), WORKQ( 1 )
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLAMCH, DLANGE
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EXTERNAL LSAME, DLAMCH, DLANGE
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* ..
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* .. External Subroutines ..
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EXTERNAL DGEQR, DGEMQR, DLASCL, DLASET,
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$ DTRTRS, XERBLA, DGELQ, DGEMLQ
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC DBLE, MAX, MIN, INT
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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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MAXMN = MAX( M, N )
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TRAN = LSAME( TRANS, 'T' )
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*
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LQUERY = ( LWORK.EQ.-1 .OR. LWORK.EQ.-2 )
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IF( .NOT.( LSAME( TRANS, 'N' ) .OR.
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$ 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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END IF
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*
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IF( INFO.EQ.0 ) THEN
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*
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* Determine the optimum and minimum LWORK
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*
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IF( M.GE.N ) THEN
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CALL DGEQR( M, N, A, LDA, TQ, -1, WORKQ, -1, INFO2 )
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TSZO = INT( TQ( 1 ) )
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LWO = INT( WORKQ( 1 ) )
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CALL DGEMQR( 'L', TRANS, M, NRHS, N, A, LDA, TQ,
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$ TSZO, B, LDB, WORKQ, -1, INFO2 )
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LWO = MAX( LWO, INT( WORKQ( 1 ) ) )
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CALL DGEQR( M, N, A, LDA, TQ, -2, WORKQ, -2, INFO2 )
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TSZM = INT( TQ( 1 ) )
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LWM = INT( WORKQ( 1 ) )
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CALL DGEMQR( 'L', TRANS, M, NRHS, N, A, LDA, TQ,
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$ TSZM, B, LDB, WORKQ, -1, INFO2 )
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LWM = MAX( LWM, INT( WORKQ( 1 ) ) )
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WSIZEO = TSZO + LWO
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WSIZEM = TSZM + LWM
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ELSE
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CALL DGELQ( M, N, A, LDA, TQ, -1, WORKQ, -1, INFO2 )
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TSZO = INT( TQ( 1 ) )
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LWO = INT( WORKQ( 1 ) )
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CALL DGEMLQ( 'L', TRANS, N, NRHS, M, A, LDA, TQ,
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$ TSZO, B, LDB, WORKQ, -1, INFO2 )
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LWO = MAX( LWO, INT( WORKQ( 1 ) ) )
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CALL DGELQ( M, N, A, LDA, TQ, -2, WORKQ, -2, INFO2 )
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TSZM = INT( TQ( 1 ) )
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LWM = INT( WORKQ( 1 ) )
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CALL DGEMLQ( 'L', TRANS, N, NRHS, M, A, LDA, TQ,
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$ TSZM, B, LDB, WORKQ, -1, INFO2 )
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LWM = MAX( LWM, INT( WORKQ( 1 ) ) )
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WSIZEO = TSZO + LWO
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WSIZEM = TSZM + LWM
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END IF
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*
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IF( ( LWORK.LT.WSIZEM ).AND.( .NOT.LQUERY ) ) THEN
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INFO = -10
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END IF
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*
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WORK( 1 ) = DBLE( WSIZEO )
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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( 'DGETSLS', -INFO )
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RETURN
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END IF
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IF( LQUERY ) THEN
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IF( LWORK.EQ.-2 ) WORK( 1 ) = DBLE( WSIZEM )
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RETURN
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END IF
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IF( LWORK.LT.WSIZEO ) THEN
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LW1 = TSZM
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LW2 = LWM
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ELSE
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LW1 = TSZO
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LW2 = LWO
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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 DLASET( 'FULL', MAX( M, N ), NRHS, ZERO, ZERO,
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$ 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 = DLAMCH( 'S' ) / DLAMCH( 'P' )
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BIGNUM = ONE / SMLNUM
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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 = DLANGE( 'M', M, N, A, LDA, WORK )
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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 DLASCL( '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 DLASCL( '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 DLASET( 'F', MAXMN, 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 ( TRAN ) THEN
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BROW = N
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END IF
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BNRM = DLANGE( 'M', BROW, NRHS, B, LDB, WORK )
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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 DLASCL( '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 DLASCL( '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 DGEQR( M, N, A, LDA, WORK( LW2+1 ), LW1,
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$ WORK( 1 ), LW2, INFO )
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IF ( .NOT.TRAN ) 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 DGEMQR( 'L' , 'T', M, NRHS, N, A, LDA,
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$ WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
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$ INFO )
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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 DTRTRS( 'U', 'N', 'N', N, NRHS,
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$ A, LDA, B, LDB, INFO )
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IF( INFO.GT.0 ) THEN
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RETURN
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END IF
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SCLLEN = N
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ELSE
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*
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* Overdetermined 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 DTRTRS( 'U', 'T', 'N', 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 DGEMQR( 'L', 'N', M, NRHS, N, A, LDA,
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$ WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
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$ INFO )
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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 DGELQ( M, N, A, LDA, WORK( LW2+1 ), LW1,
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$ WORK( 1 ), LW2, 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.TRAN ) 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 DTRTRS( 'L', 'N', 'N', 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 DGEMLQ( 'L', 'T', N, NRHS, M, A, LDA,
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$ WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
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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 DGEMLQ( 'L', 'N', N, NRHS, M, A, LDA,
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$ WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
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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 DTRTRS( '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 DLASCL( '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 DLASCL( '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 DLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
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$ INFO )
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ELSE IF( IBSCL.EQ.2 ) THEN
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CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
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$ INFO )
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END IF
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*
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50 CONTINUE
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WORK( 1 ) = DBLE( TSZO + LWO )
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RETURN
|
|
*
|
|
* End of DGETSLS
|
|
*
|
|
END
|
|
|