Add new routines for ?GELST similar to ?GELS (Reference-LAPACK PR739)
This commit is contained in:
Martin Kroeker
GitHub
parent
63014e99ae
commit
d0afbd8d29
|
|
@ -0,0 +1,533 @@
|
|||
*> \brief <b> CGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download CGELST + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgelst.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgelst.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgelst.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE CGELST( 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 ..
|
||||
* COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> CGELST solves overdetermined or underdetermined real linear systems
|
||||
*> involving an M-by-N matrix A, or its conjugate-transpose, using a QR
|
||||
*> or LQ factorization of A with compact WY representation of Q.
|
||||
*> 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 = 'C' and m >= n: find the minimum norm solution of
|
||||
*> an underdetermined system A**T * X = B.
|
||||
*>
|
||||
*> 4. If TRANS = 'C' 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;
|
||||
*> = 'C': the linear system involves A**H.
|
||||
*> \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 COMPLEX 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 CGEQRT;
|
||||
*> if M < N, A is overwritten by details of its LQ
|
||||
*> factorization as returned by CGELQT.
|
||||
*> \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 COMPLEX 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 = 'C'.
|
||||
*> 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
|
||||
*> modulus 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 = 'C' and m >= n, rows 1 to M of B contain the
|
||||
*> minimum norm solution vectors;
|
||||
*> if TRANS = 'C' 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 the modulus 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 COMPLEX 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.
|
||||
*
|
||||
*> \ingroup complexGEsolve
|
||||
*
|
||||
*> \par Contributors:
|
||||
* ==================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> November 2022, Igor Kozachenko,
|
||||
*> Computer Science Division,
|
||||
*> University of California, Berkeley
|
||||
*> \endverbatim
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE CGELST( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
|
||||
$ INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER TRANS
|
||||
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
REAL ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
||||
COMPLEX CZERO
|
||||
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ) )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL LQUERY, TPSD
|
||||
INTEGER BROW, I, IASCL, IBSCL, J, LWOPT, MN, MNNRHS,
|
||||
$ NB, NBMIN, SCLLEN
|
||||
REAL ANRM, BIGNUM, BNRM, SMLNUM
|
||||
* ..
|
||||
* .. Local Arrays ..
|
||||
REAL RWORK( 1 )
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
REAL SLAMCH, CLANGE
|
||||
EXTERNAL LSAME, ILAENV, SLAMCH, CLANGE
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL CGELQT, CGEQRT, CGEMLQT, CGEMQRT, SLABAD,
|
||||
$ CLASCL, CLASET, CTRTRS, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC REAL, MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Test the input arguments.
|
||||
*
|
||||
INFO = 0
|
||||
MN = MIN( M, N )
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'C' ) ) ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( M.LT.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY )
|
||||
$ THEN
|
||||
INFO = -10
|
||||
END IF
|
||||
*
|
||||
* Figure out optimal block size and optimal workspace size
|
||||
*
|
||||
IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
|
||||
*
|
||||
TPSD = .TRUE.
|
||||
IF( LSAME( TRANS, 'N' ) )
|
||||
$ TPSD = .FALSE.
|
||||
*
|
||||
NB = ILAENV( 1, 'CGELST', ' ', M, N, -1, -1 )
|
||||
*
|
||||
MNNRHS = MAX( MN, NRHS )
|
||||
LWOPT = MAX( 1, (MN+MNNRHS)*NB )
|
||||
WORK( 1 ) = REAL( LWOPT )
|
||||
*
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'CGELST ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( MIN( M, N, NRHS ).EQ.0 ) THEN
|
||||
CALL CLASET( 'Full', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
|
||||
WORK( 1 ) = REAL( LWOPT )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* *GEQRT and *GELQT routines cannot accept NB larger than min(M,N)
|
||||
*
|
||||
IF( NB.GT.MN ) NB = MN
|
||||
*
|
||||
* Determine the block size from the supplied LWORK
|
||||
* ( at this stage we know that LWORK >= (minimum required workspace,
|
||||
* but it may be less than optimal)
|
||||
*
|
||||
NB = MIN( NB, LWORK/( MN + MNNRHS ) )
|
||||
*
|
||||
* The minimum value of NB, when blocked code is used
|
||||
*
|
||||
NBMIN = MAX( 2, ILAENV( 2, 'CGELST', ' ', M, N, -1, -1 ) )
|
||||
*
|
||||
IF( NB.LT.NBMIN ) THEN
|
||||
NB = 1
|
||||
END IF
|
||||
*
|
||||
* Get machine parameters
|
||||
*
|
||||
SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'P' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
CALL SLABAD( SMLNUM, BIGNUM )
|
||||
*
|
||||
* Scale A, B if max element outside range [SMLNUM,BIGNUM]
|
||||
*
|
||||
ANRM = CLANGE( 'M', M, N, A, LDA, RWORK )
|
||||
IASCL = 0
|
||||
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm up to SMLNUM
|
||||
*
|
||||
CALL CLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
|
||||
IASCL = 1
|
||||
ELSE IF( ANRM.GT.BIGNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm down to BIGNUM
|
||||
*
|
||||
CALL CLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
|
||||
IASCL = 2
|
||||
ELSE IF( ANRM.EQ.ZERO ) THEN
|
||||
*
|
||||
* Matrix all zero. Return zero solution.
|
||||
*
|
||||
CALL CLASET( 'Full', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
|
||||
WORK( 1 ) = REAL( LWOPT )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
BROW = M
|
||||
IF( TPSD )
|
||||
$ BROW = N
|
||||
BNRM = CLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
|
||||
IBSCL = 0
|
||||
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm up to SMLNUM
|
||||
*
|
||||
CALL CLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
IBSCL = 1
|
||||
ELSE IF( BNRM.GT.BIGNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm down to BIGNUM
|
||||
*
|
||||
CALL CLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
IBSCL = 2
|
||||
END IF
|
||||
*
|
||||
IF( M.GE.N ) THEN
|
||||
*
|
||||
* M > N:
|
||||
* Compute the blocked QR factorization of A,
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least N, optimally N*NB.
|
||||
*
|
||||
CALL CGEQRT( M, N, NB, A, LDA, WORK( 1 ), NB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
IF( .NOT.TPSD ) THEN
|
||||
*
|
||||
* M > N, A is not transposed:
|
||||
* Overdetermined system of equations,
|
||||
* least-squares problem, min || A * X - B ||.
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL CGEMQRT( 'Left', 'Conjugate transpose', M, NRHS, N, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
|
||||
*
|
||||
CALL CTRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
SCLLEN = N
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M > N, A is transposed:
|
||||
* Underdetermined system of equations,
|
||||
* minimum norm solution of A**T * X = B.
|
||||
*
|
||||
* Compute B := inv(R**T) * B in two row blocks of B.
|
||||
*
|
||||
* Block 1: B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
|
||||
*
|
||||
CALL CTRTRS( 'Upper', 'Conjugate transpose', 'Non-unit',
|
||||
$ N, NRHS, A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Block 2: Zero out all rows below the N-th row in B:
|
||||
* B(N+1:M,1:NRHS) = ZERO
|
||||
*
|
||||
DO J = 1, NRHS
|
||||
DO I = N + 1, M
|
||||
B( I, J ) = ZERO
|
||||
END DO
|
||||
END DO
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL CGEMQRT( 'Left', 'No transpose', M, NRHS, N, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
SCLLEN = M
|
||||
*
|
||||
END IF
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M < N:
|
||||
* Compute the blocked LQ factorization of A,
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least M, optimally M*NB.
|
||||
*
|
||||
CALL CGELQT( M, N, NB, A, LDA, WORK( 1 ), NB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
IF( .NOT.TPSD ) THEN
|
||||
*
|
||||
* M < N, A is not transposed:
|
||||
* Underdetermined system of equations,
|
||||
* minimum norm solution of A * X = B.
|
||||
*
|
||||
* Compute B := inv(L) * B in two row blocks of B.
|
||||
*
|
||||
* Block 1: B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
|
||||
*
|
||||
CALL CTRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Block 2: Zero out all rows below the M-th row in B:
|
||||
* B(M+1:N,1:NRHS) = ZERO
|
||||
*
|
||||
DO J = 1, NRHS
|
||||
DO I = M + 1, N
|
||||
B( I, J ) = ZERO
|
||||
END DO
|
||||
END DO
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL CGEMLQT( 'Left', 'Conjugate transpose', N, NRHS, M, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
SCLLEN = N
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M < N, A is transposed:
|
||||
* Overdetermined system of equations,
|
||||
* least-squares problem, min || A**T * X - B ||.
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := Q * B(1:N,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL CGEMLQT( 'Left', 'No transpose', N, NRHS, M, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1), INFO )
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
|
||||
*
|
||||
CALL CTRTRS( 'Lower', 'Conjugate transpose', 'Non-unit',
|
||||
$ M, NRHS, A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
SCLLEN = M
|
||||
*
|
||||
END IF
|
||||
*
|
||||
END IF
|
||||
*
|
||||
* Undo scaling
|
||||
*
|
||||
IF( IASCL.EQ.1 ) THEN
|
||||
CALL CLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
ELSE IF( IASCL.EQ.2 ) THEN
|
||||
CALL CLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
END IF
|
||||
IF( IBSCL.EQ.1 ) THEN
|
||||
CALL CLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
ELSE IF( IBSCL.EQ.2 ) THEN
|
||||
CALL CLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
END IF
|
||||
*
|
||||
WORK( 1 ) = REAL( LWOPT )
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of CGELST
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,531 @@
|
|||
*> \brief <b> DGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download DGELST + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelst.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelst.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelst.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE DGELST( 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
|
||||
*>
|
||||
*> DGELST 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 with compact WY representation of Q.
|
||||
*> 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 underdetermined 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 DGEQRT;
|
||||
*> if M < N, A is overwritten by details of its LQ
|
||||
*> factorization as returned by DGELQT.
|
||||
*> \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.
|
||||
*
|
||||
*> \ingroup doubleGEsolve
|
||||
*
|
||||
*> \par Contributors:
|
||||
* ==================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> November 2022, Igor Kozachenko,
|
||||
*> Computer Science Division,
|
||||
*> University of California, Berkeley
|
||||
*> \endverbatim
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE DGELST( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
|
||||
$ INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER TRANS
|
||||
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL LQUERY, TPSD
|
||||
INTEGER BROW, I, IASCL, IBSCL, J, LWOPT, MN, MNNRHS,
|
||||
$ NB, NBMIN, SCLLEN
|
||||
DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM
|
||||
* ..
|
||||
* .. Local Arrays ..
|
||||
DOUBLE PRECISION RWORK( 1 )
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
DOUBLE PRECISION DLAMCH, DLANGE
|
||||
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL DGELQT, DGEQRT, DGEMLQT, DGEMQRT, DLABAD,
|
||||
$ DLASCL, DLASET, DTRTRS, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC DBLE, MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Test the input arguments.
|
||||
*
|
||||
INFO = 0
|
||||
MN = MIN( M, N )
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'T' ) ) ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( M.LT.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY )
|
||||
$ THEN
|
||||
INFO = -10
|
||||
END IF
|
||||
*
|
||||
* Figure out optimal block size and optimal workspace size
|
||||
*
|
||||
IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
|
||||
*
|
||||
TPSD = .TRUE.
|
||||
IF( LSAME( TRANS, 'N' ) )
|
||||
$ TPSD = .FALSE.
|
||||
*
|
||||
NB = ILAENV( 1, 'DGELST', ' ', M, N, -1, -1 )
|
||||
*
|
||||
MNNRHS = MAX( MN, NRHS )
|
||||
LWOPT = MAX( 1, (MN+MNNRHS)*NB )
|
||||
WORK( 1 ) = DBLE( LWOPT )
|
||||
*
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'DGELST ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( MIN( M, N, NRHS ).EQ.0 ) THEN
|
||||
CALL DLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
|
||||
WORK( 1 ) = DBLE( LWOPT )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* *GEQRT and *GELQT routines cannot accept NB larger than min(M,N)
|
||||
*
|
||||
IF( NB.GT.MN ) NB = MN
|
||||
*
|
||||
* Determine the block size from the supplied LWORK
|
||||
* ( at this stage we know that LWORK >= (minimum required workspace,
|
||||
* but it may be less than optimal)
|
||||
*
|
||||
NB = MIN( NB, LWORK/( MN + MNNRHS ) )
|
||||
*
|
||||
* The minimum value of NB, when blocked code is used
|
||||
*
|
||||
NBMIN = MAX( 2, ILAENV( 2, 'DGELST', ' ', M, N, -1, -1 ) )
|
||||
*
|
||||
IF( NB.LT.NBMIN ) THEN
|
||||
NB = 1
|
||||
END IF
|
||||
*
|
||||
* Get machine parameters
|
||||
*
|
||||
SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
CALL DLABAD( SMLNUM, BIGNUM )
|
||||
*
|
||||
* Scale A, B if max element outside range [SMLNUM,BIGNUM]
|
||||
*
|
||||
ANRM = DLANGE( 'M', M, N, A, LDA, RWORK )
|
||||
IASCL = 0
|
||||
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm up to SMLNUM
|
||||
*
|
||||
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
|
||||
IASCL = 1
|
||||
ELSE IF( ANRM.GT.BIGNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm down to BIGNUM
|
||||
*
|
||||
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
|
||||
IASCL = 2
|
||||
ELSE IF( ANRM.EQ.ZERO ) THEN
|
||||
*
|
||||
* Matrix all zero. Return zero solution.
|
||||
*
|
||||
CALL DLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
|
||||
WORK( 1 ) = DBLE( LWOPT )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
BROW = M
|
||||
IF( TPSD )
|
||||
$ BROW = N
|
||||
BNRM = DLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
|
||||
IBSCL = 0
|
||||
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm up to SMLNUM
|
||||
*
|
||||
CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
IBSCL = 1
|
||||
ELSE IF( BNRM.GT.BIGNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm down to BIGNUM
|
||||
*
|
||||
CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
IBSCL = 2
|
||||
END IF
|
||||
*
|
||||
IF( M.GE.N ) THEN
|
||||
*
|
||||
* M > N:
|
||||
* Compute the blocked QR factorization of A,
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least N, optimally N*NB.
|
||||
*
|
||||
CALL DGEQRT( M, N, NB, A, LDA, WORK( 1 ), NB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
IF( .NOT.TPSD ) THEN
|
||||
*
|
||||
* M > N, A is not transposed:
|
||||
* Overdetermined system of equations,
|
||||
* least-squares problem, min || A * X - B ||.
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL DGEMQRT( 'Left', 'Transpose', M, NRHS, N, NB, A, LDA,
|
||||
$ WORK( 1 ), NB, B, LDB, WORK( MN*NB+1 ),
|
||||
$ INFO )
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
|
||||
*
|
||||
CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
SCLLEN = N
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M > N, A is transposed:
|
||||
* Underdetermined system of equations,
|
||||
* minimum norm solution of A**T * X = B.
|
||||
*
|
||||
* Compute B := inv(R**T) * B in two row blocks of B.
|
||||
*
|
||||
* Block 1: B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
|
||||
*
|
||||
CALL DTRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Block 2: Zero out all rows below the N-th row in B:
|
||||
* B(N+1:M,1:NRHS) = ZERO
|
||||
*
|
||||
DO J = 1, NRHS
|
||||
DO I = N + 1, M
|
||||
B( I, J ) = ZERO
|
||||
END DO
|
||||
END DO
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL DGEMQRT( 'Left', 'No transpose', M, NRHS, N, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
SCLLEN = M
|
||||
*
|
||||
END IF
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M < N:
|
||||
* Compute the blocked LQ factorization of A,
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least M, optimally M*NB.
|
||||
*
|
||||
CALL DGELQT( M, N, NB, A, LDA, WORK( 1 ), NB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
IF( .NOT.TPSD ) THEN
|
||||
*
|
||||
* M < N, A is not transposed:
|
||||
* Underdetermined system of equations,
|
||||
* minimum norm solution of A * X = B.
|
||||
*
|
||||
* Compute B := inv(L) * B in two row blocks of B.
|
||||
*
|
||||
* Block 1: B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
|
||||
*
|
||||
CALL DTRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Block 2: Zero out all rows below the M-th row in B:
|
||||
* B(M+1:N,1:NRHS) = ZERO
|
||||
*
|
||||
DO J = 1, NRHS
|
||||
DO I = M + 1, N
|
||||
B( I, J ) = ZERO
|
||||
END DO
|
||||
END DO
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL DGEMLQT( 'Left', 'Transpose', N, NRHS, M, NB, A, LDA,
|
||||
$ WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
SCLLEN = N
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M < N, A is transposed:
|
||||
* Overdetermined system of equations,
|
||||
* least-squares problem, min || A**T * X - B ||.
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := Q * B(1:N,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL DGEMLQT( 'Left', 'No transpose', N, NRHS, M, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1), INFO )
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
|
||||
*
|
||||
CALL DTRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
SCLLEN = M
|
||||
*
|
||||
END IF
|
||||
*
|
||||
END IF
|
||||
*
|
||||
* Undo scaling
|
||||
*
|
||||
IF( IASCL.EQ.1 ) THEN
|
||||
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
ELSE IF( IASCL.EQ.2 ) THEN
|
||||
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
END IF
|
||||
IF( IBSCL.EQ.1 ) THEN
|
||||
CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
ELSE IF( IBSCL.EQ.2 ) THEN
|
||||
CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
END IF
|
||||
*
|
||||
WORK( 1 ) = DBLE( LWOPT )
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of DGELST
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,531 @@
|
|||
*> \brief <b> SGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download SGELST + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgelst.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgelst.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgelst.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE SGELST( 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 ..
|
||||
* REAL A( LDA, * ), B( LDB, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> SGELST 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 with compact WY representation of Q.
|
||||
*> 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 underdetermined 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 REAL 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 SGEQRT;
|
||||
*> if M < N, A is overwritten by details of its LQ
|
||||
*> factorization as returned by SGELQT.
|
||||
*> \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 REAL 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 REAL 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.
|
||||
*
|
||||
*> \ingroup realGEsolve
|
||||
*
|
||||
*> \par Contributors:
|
||||
* ==================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> November 2022, Igor Kozachenko,
|
||||
*> Computer Science Division,
|
||||
*> University of California, Berkeley
|
||||
*> \endverbatim
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE SGELST( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
|
||||
$ INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER TRANS
|
||||
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
REAL A( LDA, * ), B( LDB, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
REAL ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL LQUERY, TPSD
|
||||
INTEGER BROW, I, IASCL, IBSCL, J, LWOPT, MN, MNNRHS,
|
||||
$ NB, NBMIN, SCLLEN
|
||||
REAL ANRM, BIGNUM, BNRM, SMLNUM
|
||||
* ..
|
||||
* .. Local Arrays ..
|
||||
REAL RWORK( 1 )
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
REAL SLAMCH, SLANGE
|
||||
EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL SGELQT, SGEQRT, SGEMLQT, SGEMQRT, SLABAD,
|
||||
$ SLASCL, SLASET, STRTRS, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC REAL, MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Test the input arguments.
|
||||
*
|
||||
INFO = 0
|
||||
MN = MIN( M, N )
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'T' ) ) ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( M.LT.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY )
|
||||
$ THEN
|
||||
INFO = -10
|
||||
END IF
|
||||
*
|
||||
* Figure out optimal block size and optimal workspace size
|
||||
*
|
||||
IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
|
||||
*
|
||||
TPSD = .TRUE.
|
||||
IF( LSAME( TRANS, 'N' ) )
|
||||
$ TPSD = .FALSE.
|
||||
*
|
||||
NB = ILAENV( 1, 'SGELST', ' ', M, N, -1, -1 )
|
||||
*
|
||||
MNNRHS = MAX( MN, NRHS )
|
||||
LWOPT = MAX( 1, (MN+MNNRHS)*NB )
|
||||
WORK( 1 ) = REAL( LWOPT )
|
||||
*
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'SGELST ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( MIN( M, N, NRHS ).EQ.0 ) THEN
|
||||
CALL SLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
|
||||
WORK( 1 ) = REAL( LWOPT )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* *GEQRT and *GELQT routines cannot accept NB larger than min(M,N)
|
||||
*
|
||||
IF( NB.GT.MN ) NB = MN
|
||||
*
|
||||
* Determine the block size from the supplied LWORK
|
||||
* ( at this stage we know that LWORK >= (minimum required workspace,
|
||||
* but it may be less than optimal)
|
||||
*
|
||||
NB = MIN( NB, LWORK/( MN + MNNRHS ) )
|
||||
*
|
||||
* The minimum value of NB, when blocked code is used
|
||||
*
|
||||
NBMIN = MAX( 2, ILAENV( 2, 'SGELST', ' ', M, N, -1, -1 ) )
|
||||
*
|
||||
IF( NB.LT.NBMIN ) THEN
|
||||
NB = 1
|
||||
END IF
|
||||
*
|
||||
* Get machine parameters
|
||||
*
|
||||
SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'P' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
CALL SLABAD( SMLNUM, BIGNUM )
|
||||
*
|
||||
* Scale A, B if max element outside range [SMLNUM,BIGNUM]
|
||||
*
|
||||
ANRM = SLANGE( 'M', M, N, A, LDA, RWORK )
|
||||
IASCL = 0
|
||||
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm up to SMLNUM
|
||||
*
|
||||
CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
|
||||
IASCL = 1
|
||||
ELSE IF( ANRM.GT.BIGNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm down to BIGNUM
|
||||
*
|
||||
CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
|
||||
IASCL = 2
|
||||
ELSE IF( ANRM.EQ.ZERO ) THEN
|
||||
*
|
||||
* Matrix all zero. Return zero solution.
|
||||
*
|
||||
CALL SLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
|
||||
WORK( 1 ) = REAL( LWOPT )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
BROW = M
|
||||
IF( TPSD )
|
||||
$ BROW = N
|
||||
BNRM = SLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
|
||||
IBSCL = 0
|
||||
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm up to SMLNUM
|
||||
*
|
||||
CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
IBSCL = 1
|
||||
ELSE IF( BNRM.GT.BIGNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm down to BIGNUM
|
||||
*
|
||||
CALL SLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
IBSCL = 2
|
||||
END IF
|
||||
*
|
||||
IF( M.GE.N ) THEN
|
||||
*
|
||||
* M > N:
|
||||
* Compute the blocked QR factorization of A,
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least N, optimally N*NB.
|
||||
*
|
||||
CALL SGEQRT( M, N, NB, A, LDA, WORK( 1 ), NB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
IF( .NOT.TPSD ) THEN
|
||||
*
|
||||
* M > N, A is not transposed:
|
||||
* Overdetermined system of equations,
|
||||
* least-squares problem, min || A * X - B ||.
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL SGEMQRT( 'Left', 'Transpose', M, NRHS, N, NB, A, LDA,
|
||||
$ WORK( 1 ), NB, B, LDB, WORK( MN*NB+1 ),
|
||||
$ INFO )
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
|
||||
*
|
||||
CALL STRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
SCLLEN = N
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M > N, A is transposed:
|
||||
* Underdetermined system of equations,
|
||||
* minimum norm solution of A**T * X = B.
|
||||
*
|
||||
* Compute B := inv(R**T) * B in two row blocks of B.
|
||||
*
|
||||
* Block 1: B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
|
||||
*
|
||||
CALL STRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Block 2: Zero out all rows below the N-th row in B:
|
||||
* B(N+1:M,1:NRHS) = ZERO
|
||||
*
|
||||
DO J = 1, NRHS
|
||||
DO I = N + 1, M
|
||||
B( I, J ) = ZERO
|
||||
END DO
|
||||
END DO
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL SGEMQRT( 'Left', 'No transpose', M, NRHS, N, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
SCLLEN = M
|
||||
*
|
||||
END IF
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M < N:
|
||||
* Compute the blocked LQ factorization of A,
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least M, optimally M*NB.
|
||||
*
|
||||
CALL SGELQT( M, N, NB, A, LDA, WORK( 1 ), NB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
IF( .NOT.TPSD ) THEN
|
||||
*
|
||||
* M < N, A is not transposed:
|
||||
* Underdetermined system of equations,
|
||||
* minimum norm solution of A * X = B.
|
||||
*
|
||||
* Compute B := inv(L) * B in two row blocks of B.
|
||||
*
|
||||
* Block 1: B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
|
||||
*
|
||||
CALL STRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Block 2: Zero out all rows below the M-th row in B:
|
||||
* B(M+1:N,1:NRHS) = ZERO
|
||||
*
|
||||
DO J = 1, NRHS
|
||||
DO I = M + 1, N
|
||||
B( I, J ) = ZERO
|
||||
END DO
|
||||
END DO
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL SGEMLQT( 'Left', 'Transpose', N, NRHS, M, NB, A, LDA,
|
||||
$ WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
SCLLEN = N
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M < N, A is transposed:
|
||||
* Overdetermined system of equations,
|
||||
* least-squares problem, min || A**T * X - B ||.
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := Q * B(1:N,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL SGEMLQT( 'Left', 'No transpose', N, NRHS, M, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1), INFO )
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
|
||||
*
|
||||
CALL STRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
SCLLEN = M
|
||||
*
|
||||
END IF
|
||||
*
|
||||
END IF
|
||||
*
|
||||
* Undo scaling
|
||||
*
|
||||
IF( IASCL.EQ.1 ) THEN
|
||||
CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
ELSE IF( IASCL.EQ.2 ) THEN
|
||||
CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
END IF
|
||||
IF( IBSCL.EQ.1 ) THEN
|
||||
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
|
||||
*
|
||||
WORK( 1 ) = REAL( LWOPT )
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of SGELST
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,533 @@
|
|||
*> \brief <b> ZGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download ZGELST + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgelst.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgelst.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgelst.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE ZGELST( 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 ..
|
||||
* COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> ZGELST solves overdetermined or underdetermined real linear systems
|
||||
*> involving an M-by-N matrix A, or its conjugate-transpose, using a QR
|
||||
*> or LQ factorization of A with compact WY representation of Q.
|
||||
*> 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 = 'C' and m >= n: find the minimum norm solution of
|
||||
*> an underdetermined system A**T * X = B.
|
||||
*>
|
||||
*> 4. If TRANS = 'C' 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;
|
||||
*> = 'C': the linear system involves A**H.
|
||||
*> \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 COMPLEX*16 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 ZGEQRT;
|
||||
*> if M < N, A is overwritten by details of its LQ
|
||||
*> factorization as returned by ZGELQT.
|
||||
*> \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 COMPLEX*16 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 = 'C'.
|
||||
*> 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
|
||||
*> modulus 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 = 'C' and m >= n, rows 1 to M of B contain the
|
||||
*> minimum norm solution vectors;
|
||||
*> if TRANS = 'C' 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 the modulus 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 COMPLEX*16 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.
|
||||
*
|
||||
*> \ingroup complex16GEsolve
|
||||
*
|
||||
*> \par Contributors:
|
||||
* ==================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> November 2022, Igor Kozachenko,
|
||||
*> Computer Science Division,
|
||||
*> University of California, Berkeley
|
||||
*> \endverbatim
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE ZGELST( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
|
||||
$ INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER TRANS
|
||||
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
|
||||
COMPLEX*16 CZERO
|
||||
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL LQUERY, TPSD
|
||||
INTEGER BROW, I, IASCL, IBSCL, J, LWOPT, MN, MNNRHS,
|
||||
$ NB, NBMIN, SCLLEN
|
||||
DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM
|
||||
* ..
|
||||
* .. Local Arrays ..
|
||||
DOUBLE PRECISION RWORK( 1 )
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
DOUBLE PRECISION DLAMCH, ZLANGE
|
||||
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL ZGELQT, ZGEQRT, ZGEMLQT, ZGEMQRT, DLABAD,
|
||||
$ ZLASCL, ZLASET, ZTRTRS, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC DBLE, MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Test the input arguments.
|
||||
*
|
||||
INFO = 0
|
||||
MN = MIN( M, N )
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'C' ) ) ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( M.LT.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY )
|
||||
$ THEN
|
||||
INFO = -10
|
||||
END IF
|
||||
*
|
||||
* Figure out optimal block size and optimal workspace size
|
||||
*
|
||||
IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
|
||||
*
|
||||
TPSD = .TRUE.
|
||||
IF( LSAME( TRANS, 'N' ) )
|
||||
$ TPSD = .FALSE.
|
||||
*
|
||||
NB = ILAENV( 1, 'ZGELST', ' ', M, N, -1, -1 )
|
||||
*
|
||||
MNNRHS = MAX( MN, NRHS )
|
||||
LWOPT = MAX( 1, (MN+MNNRHS)*NB )
|
||||
WORK( 1 ) = DBLE( LWOPT )
|
||||
*
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'ZGELST ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( MIN( M, N, NRHS ).EQ.0 ) THEN
|
||||
CALL ZLASET( 'Full', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
|
||||
WORK( 1 ) = DBLE( LWOPT )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* *GEQRT and *GELQT routines cannot accept NB larger than min(M,N)
|
||||
*
|
||||
IF( NB.GT.MN ) NB = MN
|
||||
*
|
||||
* Determine the block size from the supplied LWORK
|
||||
* ( at this stage we know that LWORK >= (minimum required workspace,
|
||||
* but it may be less than optimal)
|
||||
*
|
||||
NB = MIN( NB, LWORK/( MN + MNNRHS ) )
|
||||
*
|
||||
* The minimum value of NB, when blocked code is used
|
||||
*
|
||||
NBMIN = MAX( 2, ILAENV( 2, 'ZGELST', ' ', M, N, -1, -1 ) )
|
||||
*
|
||||
IF( NB.LT.NBMIN ) THEN
|
||||
NB = 1
|
||||
END IF
|
||||
*
|
||||
* Get machine parameters
|
||||
*
|
||||
SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
CALL DLABAD( SMLNUM, BIGNUM )
|
||||
*
|
||||
* Scale A, B if max element outside range [SMLNUM,BIGNUM]
|
||||
*
|
||||
ANRM = ZLANGE( 'M', M, N, A, LDA, RWORK )
|
||||
IASCL = 0
|
||||
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm up to SMLNUM
|
||||
*
|
||||
CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
|
||||
IASCL = 1
|
||||
ELSE IF( ANRM.GT.BIGNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm down to BIGNUM
|
||||
*
|
||||
CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
|
||||
IASCL = 2
|
||||
ELSE IF( ANRM.EQ.ZERO ) THEN
|
||||
*
|
||||
* Matrix all zero. Return zero solution.
|
||||
*
|
||||
CALL ZLASET( 'Full', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
|
||||
WORK( 1 ) = DBLE( LWOPT )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
BROW = M
|
||||
IF( TPSD )
|
||||
$ BROW = N
|
||||
BNRM = ZLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
|
||||
IBSCL = 0
|
||||
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm up to SMLNUM
|
||||
*
|
||||
CALL ZLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
IBSCL = 1
|
||||
ELSE IF( BNRM.GT.BIGNUM ) THEN
|
||||
*
|
||||
* Scale matrix norm down to BIGNUM
|
||||
*
|
||||
CALL ZLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
IBSCL = 2
|
||||
END IF
|
||||
*
|
||||
IF( M.GE.N ) THEN
|
||||
*
|
||||
* M > N:
|
||||
* Compute the blocked QR factorization of A,
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least N, optimally N*NB.
|
||||
*
|
||||
CALL ZGEQRT( M, N, NB, A, LDA, WORK( 1 ), NB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
IF( .NOT.TPSD ) THEN
|
||||
*
|
||||
* M > N, A is not transposed:
|
||||
* Overdetermined system of equations,
|
||||
* least-squares problem, min || A * X - B ||.
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL ZGEMQRT( 'Left', 'Conjugate transpose', M, NRHS, N, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
|
||||
*
|
||||
CALL ZTRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
SCLLEN = N
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M > N, A is transposed:
|
||||
* Underdetermined system of equations,
|
||||
* minimum norm solution of A**T * X = B.
|
||||
*
|
||||
* Compute B := inv(R**T) * B in two row blocks of B.
|
||||
*
|
||||
* Block 1: B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
|
||||
*
|
||||
CALL ZTRTRS( 'Upper', 'Conjugate transpose', 'Non-unit',
|
||||
$ N, NRHS, A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Block 2: Zero out all rows below the N-th row in B:
|
||||
* B(N+1:M,1:NRHS) = ZERO
|
||||
*
|
||||
DO J = 1, NRHS
|
||||
DO I = N + 1, M
|
||||
B( I, J ) = ZERO
|
||||
END DO
|
||||
END DO
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL ZGEMQRT( 'Left', 'No transpose', M, NRHS, N, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
SCLLEN = M
|
||||
*
|
||||
END IF
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M < N:
|
||||
* Compute the blocked LQ factorization of A,
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least M, optimally M*NB.
|
||||
*
|
||||
CALL ZGELQT( M, N, NB, A, LDA, WORK( 1 ), NB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
IF( .NOT.TPSD ) THEN
|
||||
*
|
||||
* M < N, A is not transposed:
|
||||
* Underdetermined system of equations,
|
||||
* minimum norm solution of A * X = B.
|
||||
*
|
||||
* Compute B := inv(L) * B in two row blocks of B.
|
||||
*
|
||||
* Block 1: B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
|
||||
*
|
||||
CALL ZTRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS,
|
||||
$ A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Block 2: Zero out all rows below the M-th row in B:
|
||||
* B(M+1:N,1:NRHS) = ZERO
|
||||
*
|
||||
DO J = 1, NRHS
|
||||
DO I = M + 1, N
|
||||
B( I, J ) = ZERO
|
||||
END DO
|
||||
END DO
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL ZGEMLQT( 'Left', 'Conjugate transpose', N, NRHS, M, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1 ), INFO )
|
||||
*
|
||||
SCLLEN = N
|
||||
*
|
||||
ELSE
|
||||
*
|
||||
* M < N, A is transposed:
|
||||
* Overdetermined system of equations,
|
||||
* least-squares problem, min || A**T * X - B ||.
|
||||
*
|
||||
* Compute B(1:N,1:NRHS) := Q * B(1:N,1:NRHS),
|
||||
* using the compact WY representation of Q,
|
||||
* workspace at least NRHS, optimally NRHS*NB.
|
||||
*
|
||||
CALL ZGEMLQT( 'Left', 'No transpose', N, NRHS, M, NB,
|
||||
$ A, LDA, WORK( 1 ), NB, B, LDB,
|
||||
$ WORK( MN*NB+1), INFO )
|
||||
*
|
||||
* Compute B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
|
||||
*
|
||||
CALL ZTRTRS( 'Lower', 'Conjugate transpose', 'Non-unit',
|
||||
$ M, NRHS, A, LDA, B, LDB, INFO )
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
SCLLEN = M
|
||||
*
|
||||
END IF
|
||||
*
|
||||
END IF
|
||||
*
|
||||
* Undo scaling
|
||||
*
|
||||
IF( IASCL.EQ.1 ) THEN
|
||||
CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
ELSE IF( IASCL.EQ.2 ) THEN
|
||||
CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
END IF
|
||||
IF( IBSCL.EQ.1 ) THEN
|
||||
CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
ELSE IF( IBSCL.EQ.2 ) THEN
|
||||
CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
|
||||
$ INFO )
|
||||
END IF
|
||||
*
|
||||
WORK( 1 ) = DBLE( LWOPT )
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of ZGELST
|
||||
*
|
||||
END
|
||||
Loading…
Reference in New Issue