491 lines
14 KiB
Fortran
491 lines
14 KiB
Fortran
*> \brief \b SGETSLS
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE SGETSLS( 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
|
|
*>
|
|
*> SGETSLS solves overdetermined or underdetermined real linear systems
|
|
*> involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
|
|
*> factorization of A. It is assumed that A has full rank.
|
|
*>
|
|
*>
|
|
*>
|
|
*> The following options are provided:
|
|
*>
|
|
*> 1. If TRANS = 'N' and m >= n: find the least squares solution of
|
|
*> an overdetermined system, i.e., solve the least squares problem
|
|
*> minimize || B - A*X ||.
|
|
*>
|
|
*> 2. If TRANS = 'N' and m < n: find the minimum norm solution of
|
|
*> an underdetermined system A * X = B.
|
|
*>
|
|
*> 3. If TRANS = 'T' and m >= n: find the minimum norm solution of
|
|
*> an undetermined system A**T * X = B.
|
|
*>
|
|
*> 4. If TRANS = 'T' and m < n: find the least squares solution of
|
|
*> an overdetermined system, i.e., solve the least squares problem
|
|
*> minimize || B - A**T * X ||.
|
|
*>
|
|
*> Several right hand side vectors b and solution vectors x can be
|
|
*> handled in a single call; they are stored as the columns of the
|
|
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
|
|
*> matrix X.
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \param[in] TRANS
|
|
*> \verbatim
|
|
*> TRANS is CHARACTER*1
|
|
*> = 'N': the linear system involves A;
|
|
*> = 'T': the linear system involves A**T.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] M
|
|
*> \verbatim
|
|
*> M is INTEGER
|
|
*> The number of rows of the matrix A. M >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] N
|
|
*> \verbatim
|
|
*> N is INTEGER
|
|
*> The number of columns of the matrix A. N >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] NRHS
|
|
*> \verbatim
|
|
*> NRHS is INTEGER
|
|
*> The number of right hand sides, i.e., the number of
|
|
*> columns of the matrices B and X. NRHS >=0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] A
|
|
*> \verbatim
|
|
*> A is REAL array, dimension (LDA,N)
|
|
*> On entry, the M-by-N matrix A.
|
|
*> On exit,
|
|
*> A is overwritten by details of its QR or LQ
|
|
*> factorization as returned by SGEQR or SGELQ.
|
|
*> \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.
|
|
*> 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.
|
|
*> \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
|
|
*> (workspace) REAL array, dimension (MAX(1,LWORK))
|
|
*> On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
|
|
*> or optimal, if query was assumed) LWORK.
|
|
*> See LWORK for details.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LWORK
|
|
*> \verbatim
|
|
*> LWORK is INTEGER
|
|
*> The dimension of the array WORK.
|
|
*> If LWORK = -1 or -2, then a workspace query is assumed.
|
|
*> If LWORK = -1, the routine calculates optimal size of WORK for the
|
|
*> optimal performance and returns this value in WORK(1).
|
|
*> If LWORK = -2, the routine calculates minimal size of WORK and
|
|
*> returns this value in WORK(1).
|
|
*> \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
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE SGETSLS( 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 = 0.0E0, ONE = 1.0E0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL LQUERY, TRAN
|
|
INTEGER I, IASCL, IBSCL, J, MAXMN, BROW,
|
|
$ SCLLEN, TSZO, TSZM, LWO, LWM, LW1, LW2,
|
|
$ WSIZEO, WSIZEM, INFO2
|
|
REAL ANRM, BIGNUM, BNRM, SMLNUM, TQ( 5 ), WORKQ( 1 )
|
|
* ..
|
|
* .. External Functions ..
|
|
LOGICAL LSAME
|
|
REAL SLAMCH, SLANGE
|
|
EXTERNAL LSAME, SLABAD, SLAMCH, SLANGE
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL SGEQR, SGEMQR, SLASCL, SLASET,
|
|
$ STRTRS, XERBLA, SGELQ, SGEMLQ
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC REAL, MAX, MIN, INT
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input arguments.
|
|
*
|
|
INFO = 0
|
|
MAXMN = MAX( M, N )
|
|
TRAN = LSAME( TRANS, 'T' )
|
|
*
|
|
LQUERY = ( LWORK.EQ.-1 .OR. LWORK.EQ.-2 )
|
|
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
|
|
END IF
|
|
*
|
|
IF( INFO.EQ.0 ) THEN
|
|
*
|
|
* Determine the optimum and minimum LWORK
|
|
*
|
|
IF( M.GE.N ) THEN
|
|
CALL SGEQR( M, N, A, LDA, TQ, -1, WORKQ, -1, INFO2 )
|
|
TSZO = INT( TQ( 1 ) )
|
|
LWO = INT( WORKQ( 1 ) )
|
|
CALL SGEMQR( 'L', TRANS, M, NRHS, N, A, LDA, TQ,
|
|
$ TSZO, B, LDB, WORKQ, -1, INFO2 )
|
|
LWO = MAX( LWO, INT( WORKQ( 1 ) ) )
|
|
CALL SGEQR( M, N, A, LDA, TQ, -2, WORKQ, -2, INFO2 )
|
|
TSZM = INT( TQ( 1 ) )
|
|
LWM = INT( WORKQ( 1 ) )
|
|
CALL SGEMQR( 'L', TRANS, M, NRHS, N, A, LDA, TQ,
|
|
$ TSZM, B, LDB, WORKQ, -1, INFO2 )
|
|
LWM = MAX( LWM, INT( WORKQ( 1 ) ) )
|
|
WSIZEO = TSZO + LWO
|
|
WSIZEM = TSZM + LWM
|
|
ELSE
|
|
CALL SGELQ( M, N, A, LDA, TQ, -1, WORKQ, -1, INFO2 )
|
|
TSZO = INT( TQ( 1 ) )
|
|
LWO = INT( WORKQ( 1 ) )
|
|
CALL SGEMLQ( 'L', TRANS, N, NRHS, M, A, LDA, TQ,
|
|
$ TSZO, B, LDB, WORKQ, -1, INFO2 )
|
|
LWO = MAX( LWO, INT( WORKQ( 1 ) ) )
|
|
CALL SGELQ( M, N, A, LDA, TQ, -2, WORKQ, -2, INFO2 )
|
|
TSZM = INT( TQ( 1 ) )
|
|
LWM = INT( WORKQ( 1 ) )
|
|
CALL SGEMLQ( 'L', TRANS, N, NRHS, M, A, LDA, TQ,
|
|
$ TSZM, B, LDB, WORKQ, -1, INFO2 )
|
|
LWM = MAX( LWM, INT( WORKQ( 1 ) ) )
|
|
WSIZEO = TSZO + LWO
|
|
WSIZEM = TSZM + LWM
|
|
END IF
|
|
*
|
|
IF( ( LWORK.LT.WSIZEM ).AND.( .NOT.LQUERY ) ) THEN
|
|
INFO = -10
|
|
END IF
|
|
*
|
|
WORK( 1 ) = REAL( WSIZEO )
|
|
*
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'SGETSLS', -INFO )
|
|
RETURN
|
|
END IF
|
|
IF( LQUERY ) THEN
|
|
IF( LWORK.EQ.-2 ) WORK( 1 ) = REAL( WSIZEM )
|
|
RETURN
|
|
END IF
|
|
IF( LWORK.LT.WSIZEO ) THEN
|
|
LW1 = TSZM
|
|
LW2 = LWM
|
|
ELSE
|
|
LW1 = TSZO
|
|
LW2 = LWO
|
|
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 )
|
|
RETURN
|
|
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, WORK )
|
|
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( 'F', MAXMN, NRHS, ZERO, ZERO, B, LDB )
|
|
GO TO 50
|
|
END IF
|
|
*
|
|
BROW = M
|
|
IF ( TRAN ) THEN
|
|
BROW = N
|
|
END IF
|
|
BNRM = SLANGE( 'M', BROW, NRHS, B, LDB, WORK )
|
|
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
|
|
*
|
|
* compute QR factorization of A
|
|
*
|
|
CALL SGEQR( M, N, A, LDA, WORK( LW2+1 ), LW1,
|
|
$ WORK( 1 ), LW2, INFO )
|
|
IF ( .NOT.TRAN ) THEN
|
|
*
|
|
* Least-Squares Problem min || A * X - B ||
|
|
*
|
|
* B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
|
|
*
|
|
CALL SGEMQR( 'L' , 'T', M, NRHS, N, A, LDA,
|
|
$ WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
|
|
$ INFO )
|
|
*
|
|
* B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
|
|
*
|
|
CALL STRTRS( 'U', 'N', 'N', N, NRHS,
|
|
$ A, LDA, B, LDB, INFO )
|
|
IF( INFO.GT.0 ) THEN
|
|
RETURN
|
|
END IF
|
|
SCLLEN = N
|
|
ELSE
|
|
*
|
|
* Overdetermined system of equations A**T * X = B
|
|
*
|
|
* B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
|
|
*
|
|
CALL STRTRS( 'U', 'T', 'N', N, NRHS,
|
|
$ A, LDA, B, LDB, INFO )
|
|
*
|
|
IF( INFO.GT.0 ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* B(N+1:M,1:NRHS) = ZERO
|
|
*
|
|
DO 20 J = 1, NRHS
|
|
DO 10 I = N + 1, M
|
|
B( I, J ) = ZERO
|
|
10 CONTINUE
|
|
20 CONTINUE
|
|
*
|
|
* B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
|
|
*
|
|
CALL SGEMQR( 'L', 'N', M, NRHS, N, A, LDA,
|
|
$ WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
|
|
$ INFO )
|
|
*
|
|
SCLLEN = M
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* Compute LQ factorization of A
|
|
*
|
|
CALL SGELQ( M, N, A, LDA, WORK( LW2+1 ), LW1,
|
|
$ WORK( 1 ), LW2, INFO )
|
|
*
|
|
* workspace at least M, optimally M*NB.
|
|
*
|
|
IF( .NOT.TRAN ) THEN
|
|
*
|
|
* underdetermined system of equations A * X = B
|
|
*
|
|
* B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
|
|
*
|
|
CALL STRTRS( 'L', 'N', 'N', M, NRHS,
|
|
$ A, LDA, B, LDB, INFO )
|
|
*
|
|
IF( INFO.GT.0 ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* B(M+1:N,1:NRHS) = 0
|
|
*
|
|
DO 40 J = 1, NRHS
|
|
DO 30 I = M + 1, N
|
|
B( I, J ) = ZERO
|
|
30 CONTINUE
|
|
40 CONTINUE
|
|
*
|
|
* B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS)
|
|
*
|
|
CALL SGEMLQ( 'L', 'T', N, NRHS, M, A, LDA,
|
|
$ WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
|
|
$ INFO )
|
|
*
|
|
* workspace at least NRHS, optimally NRHS*NB
|
|
*
|
|
SCLLEN = N
|
|
*
|
|
ELSE
|
|
*
|
|
* overdetermined system min || A**T * X - B ||
|
|
*
|
|
* B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
|
|
*
|
|
CALL SGEMLQ( 'L', 'N', N, NRHS, M, A, LDA,
|
|
$ WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
|
|
$ INFO )
|
|
*
|
|
* workspace at least NRHS, optimally NRHS*NB
|
|
*
|
|
* 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
|
|
*
|
|
50 CONTINUE
|
|
WORK( 1 ) = REAL( TSZO + LWO )
|
|
RETURN
|
|
*
|
|
* End of SGETSLS
|
|
*
|
|
END
|