floraachy
/
OpenBLAS
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
OpenBLAS/lapack-netlib/SRC/cgelst.f

534 lines
16 KiB
Fortran
Raw Blame History

*> \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
Reference in New Issue View Git Blame Copy Permalink