You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
655 lines
22 KiB
655 lines
22 KiB
*> \brief <b> ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices</b>
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download ZGELSD + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgelsd.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgelsd.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgelsd.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
|
|
* WORK, LWORK, RWORK, IWORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
|
|
* DOUBLE PRECISION RCOND
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* INTEGER IWORK( * )
|
|
* DOUBLE PRECISION RWORK( * ), S( * )
|
|
* COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> ZGELSD computes the minimum-norm solution to a real linear least
|
|
*> squares problem:
|
|
*> minimize 2-norm(| b - A*x |)
|
|
*> using the singular value decomposition (SVD) of A. A is an M-by-N
|
|
*> matrix which may be rank-deficient.
|
|
*>
|
|
*> 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.
|
|
*>
|
|
*> The problem is solved in three steps:
|
|
*> (1) Reduce the coefficient matrix A to bidiagonal form with
|
|
*> Householder transformations, reducing the original problem
|
|
*> into a "bidiagonal least squares problem" (BLS)
|
|
*> (2) Solve the BLS using a divide and conquer approach.
|
|
*> (3) Apply back all the Householder transformations to solve
|
|
*> the original least squares problem.
|
|
*>
|
|
*> The effective rank of A is determined by treating as zero those
|
|
*> singular values which are less than RCOND times the largest singular
|
|
*> value.
|
|
*>
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \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, A has been destroyed.
|
|
*> \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 M-by-NRHS right hand side matrix B.
|
|
*> On exit, B is overwritten by the N-by-NRHS solution matrix X.
|
|
*> If m >= n and RANK = n, the residual sum-of-squares for
|
|
*> the solution in the i-th column is given by the sum of
|
|
*> squares of the modulus of elements n+1:m 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] S
|
|
*> \verbatim
|
|
*> S is DOUBLE PRECISION array, dimension (min(M,N))
|
|
*> The singular values of A in decreasing order.
|
|
*> The condition number of A in the 2-norm = S(1)/S(min(m,n)).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] RCOND
|
|
*> \verbatim
|
|
*> RCOND is DOUBLE PRECISION
|
|
*> RCOND is used to determine the effective rank of A.
|
|
*> Singular values S(i) <= RCOND*S(1) are treated as zero.
|
|
*> If RCOND < 0, machine precision is used instead.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] RANK
|
|
*> \verbatim
|
|
*> RANK is INTEGER
|
|
*> The effective rank of A, i.e., the number of singular values
|
|
*> which are greater than RCOND*S(1).
|
|
*> \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 must be at least 1.
|
|
*> The exact minimum amount of workspace needed depends on M,
|
|
*> N and NRHS. As long as LWORK is at least
|
|
*> 2*N + N*NRHS
|
|
*> if M is greater than or equal to N or
|
|
*> 2*M + M*NRHS
|
|
*> if M is less than N, the code will execute correctly.
|
|
*> For good performance, LWORK should generally be larger.
|
|
*>
|
|
*> If LWORK = -1, then a workspace query is assumed; the routine
|
|
*> only calculates the optimal size of the array WORK and the
|
|
*> minimum sizes of the arrays RWORK and IWORK, and returns
|
|
*> these values as the first entries of the WORK, RWORK and
|
|
*> IWORK arrays, and no error message related to LWORK is issued
|
|
*> by XERBLA.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] RWORK
|
|
*> \verbatim
|
|
*> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
|
|
*> LRWORK >=
|
|
*> 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
|
|
*> MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
|
|
*> if M is greater than or equal to N or
|
|
*> 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
|
|
*> MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
|
|
*> if M is less than N, the code will execute correctly.
|
|
*> SMLSIZ is returned by ILAENV and is equal to the maximum
|
|
*> size of the subproblems at the bottom of the computation
|
|
*> tree (usually about 25), and
|
|
*> NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
|
|
*> On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] IWORK
|
|
*> \verbatim
|
|
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
|
|
*> LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
|
|
*> where MINMN = MIN( M,N ).
|
|
*> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] INFO
|
|
*> \verbatim
|
|
*> INFO is INTEGER
|
|
*> = 0: successful exit
|
|
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
|
*> > 0: the algorithm for computing the SVD failed to converge;
|
|
*> if INFO = i, i off-diagonal elements of an intermediate
|
|
*> bidiagonal form did not converge to zero.
|
|
*> \endverbatim
|
|
*
|
|
* Authors:
|
|
* ========
|
|
*
|
|
*> \author Univ. of Tennessee
|
|
*> \author Univ. of California Berkeley
|
|
*> \author Univ. of Colorado Denver
|
|
*> \author NAG Ltd.
|
|
*
|
|
*> \ingroup complex16GEsolve
|
|
*
|
|
*> \par Contributors:
|
|
* ==================
|
|
*>
|
|
*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
|
|
*> California at Berkeley, USA \n
|
|
*> Osni Marques, LBNL/NERSC, USA \n
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
|
|
$ WORK, LWORK, RWORK, IWORK, 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 ..
|
|
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
|
|
DOUBLE PRECISION RCOND
|
|
* ..
|
|
* .. Array Arguments ..
|
|
INTEGER IWORK( * )
|
|
DOUBLE PRECISION RWORK( * ), S( * )
|
|
COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
DOUBLE PRECISION ZERO, ONE, TWO
|
|
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
|
|
COMPLEX*16 CZERO
|
|
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL LQUERY
|
|
INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
|
|
$ LDWORK, LIWORK, LRWORK, MAXMN, MAXWRK, MINMN,
|
|
$ MINWRK, MM, MNTHR, NLVL, NRWORK, NWORK, SMLSIZ
|
|
DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL DLASCL, DLASET, XERBLA, ZGEBRD, ZGELQF, ZGEQRF,
|
|
$ ZLACPY, ZLALSD, ZLASCL, ZLASET, ZUNMBR, ZUNMLQ,
|
|
$ ZUNMQR
|
|
* ..
|
|
* .. External Functions ..
|
|
INTEGER ILAENV
|
|
DOUBLE PRECISION DLAMCH, ZLANGE
|
|
EXTERNAL ILAENV, DLAMCH, ZLANGE
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC INT, LOG, MAX, MIN, DBLE
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input arguments.
|
|
*
|
|
INFO = 0
|
|
MINMN = MIN( M, N )
|
|
MAXMN = MAX( M, N )
|
|
LQUERY = ( LWORK.EQ.-1 )
|
|
IF( M.LT.0 ) THEN
|
|
INFO = -1
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( NRHS.LT.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
|
|
INFO = -5
|
|
ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
|
|
INFO = -7
|
|
END IF
|
|
*
|
|
* Compute workspace.
|
|
* (Note: Comments in the code beginning "Workspace:" describe the
|
|
* minimal amount of workspace needed at that point in the code,
|
|
* as well as the preferred amount for good performance.
|
|
* NB refers to the optimal block size for the immediately
|
|
* following subroutine, as returned by ILAENV.)
|
|
*
|
|
IF( INFO.EQ.0 ) THEN
|
|
MINWRK = 1
|
|
MAXWRK = 1
|
|
LIWORK = 1
|
|
LRWORK = 1
|
|
IF( MINMN.GT.0 ) THEN
|
|
SMLSIZ = ILAENV( 9, 'ZGELSD', ' ', 0, 0, 0, 0 )
|
|
MNTHR = ILAENV( 6, 'ZGELSD', ' ', M, N, NRHS, -1 )
|
|
NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ + 1 ) ) /
|
|
$ LOG( TWO ) ) + 1, 0 )
|
|
LIWORK = 3*MINMN*NLVL + 11*MINMN
|
|
MM = M
|
|
IF( M.GE.N .AND. M.GE.MNTHR ) THEN
|
|
*
|
|
* Path 1a - overdetermined, with many more rows than
|
|
* columns.
|
|
*
|
|
MM = N
|
|
MAXWRK = MAX( MAXWRK, N*ILAENV( 1, 'ZGEQRF', ' ', M, N,
|
|
$ -1, -1 ) )
|
|
MAXWRK = MAX( MAXWRK, NRHS*ILAENV( 1, 'ZUNMQR', 'LC', M,
|
|
$ NRHS, N, -1 ) )
|
|
END IF
|
|
IF( M.GE.N ) THEN
|
|
*
|
|
* Path 1 - overdetermined or exactly determined.
|
|
*
|
|
LRWORK = 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
|
|
$ MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
|
|
MAXWRK = MAX( MAXWRK, 2*N + ( MM + N )*ILAENV( 1,
|
|
$ 'ZGEBRD', ' ', MM, N, -1, -1 ) )
|
|
MAXWRK = MAX( MAXWRK, 2*N + NRHS*ILAENV( 1, 'ZUNMBR',
|
|
$ 'QLC', MM, NRHS, N, -1 ) )
|
|
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
|
|
$ 'ZUNMBR', 'PLN', N, NRHS, N, -1 ) )
|
|
MAXWRK = MAX( MAXWRK, 2*N + N*NRHS )
|
|
MINWRK = MAX( 2*N + MM, 2*N + N*NRHS )
|
|
END IF
|
|
IF( N.GT.M ) THEN
|
|
LRWORK = 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
|
|
$ MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
|
|
IF( N.GE.MNTHR ) THEN
|
|
*
|
|
* Path 2a - underdetermined, with many more columns
|
|
* than rows.
|
|
*
|
|
MAXWRK = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1,
|
|
$ -1 )
|
|
MAXWRK = MAX( MAXWRK, M*M + 4*M + 2*M*ILAENV( 1,
|
|
$ 'ZGEBRD', ' ', M, M, -1, -1 ) )
|
|
MAXWRK = MAX( MAXWRK, M*M + 4*M + NRHS*ILAENV( 1,
|
|
$ 'ZUNMBR', 'QLC', M, NRHS, M, -1 ) )
|
|
MAXWRK = MAX( MAXWRK, M*M + 4*M + ( M - 1 )*ILAENV( 1,
|
|
$ 'ZUNMLQ', 'LC', N, NRHS, M, -1 ) )
|
|
IF( NRHS.GT.1 ) THEN
|
|
MAXWRK = MAX( MAXWRK, M*M + M + M*NRHS )
|
|
ELSE
|
|
MAXWRK = MAX( MAXWRK, M*M + 2*M )
|
|
END IF
|
|
MAXWRK = MAX( MAXWRK, M*M + 4*M + M*NRHS )
|
|
! XXX: Ensure the Path 2a case below is triggered. The workspace
|
|
! calculation should use queries for all routines eventually.
|
|
MAXWRK = MAX( MAXWRK,
|
|
$ 4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
|
|
ELSE
|
|
*
|
|
* Path 2 - underdetermined.
|
|
*
|
|
MAXWRK = 2*M + ( N + M )*ILAENV( 1, 'ZGEBRD', ' ', M,
|
|
$ N, -1, -1 )
|
|
MAXWRK = MAX( MAXWRK, 2*M + NRHS*ILAENV( 1, 'ZUNMBR',
|
|
$ 'QLC', M, NRHS, M, -1 ) )
|
|
MAXWRK = MAX( MAXWRK, 2*M + M*ILAENV( 1, 'ZUNMBR',
|
|
$ 'PLN', N, NRHS, M, -1 ) )
|
|
MAXWRK = MAX( MAXWRK, 2*M + M*NRHS )
|
|
END IF
|
|
MINWRK = MAX( 2*M + N, 2*M + M*NRHS )
|
|
END IF
|
|
END IF
|
|
MINWRK = MIN( MINWRK, MAXWRK )
|
|
WORK( 1 ) = MAXWRK
|
|
IWORK( 1 ) = LIWORK
|
|
RWORK( 1 ) = LRWORK
|
|
*
|
|
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
|
|
INFO = -12
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'ZGELSD', -INFO )
|
|
RETURN
|
|
ELSE IF( LQUERY ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible.
|
|
*
|
|
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
|
|
RANK = 0
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Get machine parameters.
|
|
*
|
|
EPS = DLAMCH( 'P' )
|
|
SFMIN = DLAMCH( 'S' )
|
|
SMLNUM = SFMIN / EPS
|
|
BIGNUM = ONE / SMLNUM
|
|
*
|
|
* Scale A if max entry 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( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
|
|
CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
|
|
RANK = 0
|
|
GO TO 10
|
|
END IF
|
|
*
|
|
* Scale B if max entry outside range [SMLNUM,BIGNUM].
|
|
*
|
|
BNRM = ZLANGE( 'M', M, 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, M, 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, M, NRHS, B, LDB, INFO )
|
|
IBSCL = 2
|
|
END IF
|
|
*
|
|
* If M < N make sure B(M+1:N,:) = 0
|
|
*
|
|
IF( M.LT.N )
|
|
$ CALL ZLASET( 'F', N-M, NRHS, CZERO, CZERO, B( M+1, 1 ), LDB )
|
|
*
|
|
* Overdetermined case.
|
|
*
|
|
IF( M.GE.N ) THEN
|
|
*
|
|
* Path 1 - overdetermined or exactly determined.
|
|
*
|
|
MM = M
|
|
IF( M.GE.MNTHR ) THEN
|
|
*
|
|
* Path 1a - overdetermined, with many more rows than columns
|
|
*
|
|
MM = N
|
|
ITAU = 1
|
|
NWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R.
|
|
* (RWorkspace: need N)
|
|
* (CWorkspace: need N, prefer N*NB)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
*
|
|
* Multiply B by transpose(Q).
|
|
* (RWorkspace: need N)
|
|
* (CWorkspace: need NRHS, prefer NRHS*NB)
|
|
*
|
|
CALL ZUNMQR( 'L', 'C', M, NRHS, N, A, LDA, WORK( ITAU ), B,
|
|
$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
* Zero out below R.
|
|
*
|
|
IF( N.GT.1 ) THEN
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO, A( 2, 1 ),
|
|
$ LDA )
|
|
END IF
|
|
END IF
|
|
*
|
|
ITAUQ = 1
|
|
ITAUP = ITAUQ + N
|
|
NWORK = ITAUP + N
|
|
IE = 1
|
|
NRWORK = IE + N
|
|
*
|
|
* Bidiagonalize R in A.
|
|
* (RWorkspace: need N)
|
|
* (CWorkspace: need 2*N+MM, prefer 2*N+(MM+N)*NB)
|
|
*
|
|
CALL ZGEBRD( MM, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
|
|
$ INFO )
|
|
*
|
|
* Multiply B by transpose of left bidiagonalizing vectors of R.
|
|
* (CWorkspace: need 2*N+NRHS, prefer 2*N+NRHS*NB)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'L', 'C', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
|
|
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
* Solve the bidiagonal least squares problem.
|
|
*
|
|
CALL ZLALSD( 'U', SMLSIZ, N, NRHS, S, RWORK( IE ), B, LDB,
|
|
$ RCOND, RANK, WORK( NWORK ), RWORK( NRWORK ),
|
|
$ IWORK, INFO )
|
|
IF( INFO.NE.0 ) THEN
|
|
GO TO 10
|
|
END IF
|
|
*
|
|
* Multiply B by right bidiagonalizing vectors of R.
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ),
|
|
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
|
|
$ MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN
|
|
*
|
|
* Path 2a - underdetermined, with many more columns than rows
|
|
* and sufficient workspace for an efficient algorithm.
|
|
*
|
|
LDWORK = M
|
|
IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
|
|
$ M*LDA+M+M*NRHS ) )LDWORK = LDA
|
|
ITAU = 1
|
|
NWORK = M + 1
|
|
*
|
|
* Compute A=L*Q.
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
IL = NWORK
|
|
*
|
|
* Copy L to WORK(IL), zeroing out above its diagonal.
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO, WORK( IL+LDWORK ),
|
|
$ LDWORK )
|
|
ITAUQ = IL + LDWORK*M
|
|
ITAUP = ITAUQ + M
|
|
NWORK = ITAUP + M
|
|
IE = 1
|
|
NRWORK = IE + M
|
|
*
|
|
* Bidiagonalize L in WORK(IL).
|
|
* (RWorkspace: need M)
|
|
* (CWorkspace: need M*M+4*M, prefer M*M+4*M+2*M*NB)
|
|
*
|
|
CALL ZGEBRD( M, M, WORK( IL ), LDWORK, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
*
|
|
* Multiply B by transpose of left bidiagonalizing vectors of L.
|
|
* (CWorkspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'L', 'C', M, NRHS, M, WORK( IL ), LDWORK,
|
|
$ WORK( ITAUQ ), B, LDB, WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
*
|
|
* Solve the bidiagonal least squares problem.
|
|
*
|
|
CALL ZLALSD( 'U', SMLSIZ, M, NRHS, S, RWORK( IE ), B, LDB,
|
|
$ RCOND, RANK, WORK( NWORK ), RWORK( NRWORK ),
|
|
$ IWORK, INFO )
|
|
IF( INFO.NE.0 ) THEN
|
|
GO TO 10
|
|
END IF
|
|
*
|
|
* Multiply B by right bidiagonalizing vectors of L.
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
|
|
$ WORK( ITAUP ), B, LDB, WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
*
|
|
* Zero out below first M rows of B.
|
|
*
|
|
CALL ZLASET( 'F', N-M, NRHS, CZERO, CZERO, B( M+1, 1 ), LDB )
|
|
NWORK = ITAU + M
|
|
*
|
|
* Multiply transpose(Q) by B.
|
|
* (CWorkspace: need NRHS, prefer NRHS*NB)
|
|
*
|
|
CALL ZUNMLQ( 'L', 'C', N, NRHS, M, A, LDA, WORK( ITAU ), B,
|
|
$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
ELSE
|
|
*
|
|
* Path 2 - remaining underdetermined cases.
|
|
*
|
|
ITAUQ = 1
|
|
ITAUP = ITAUQ + M
|
|
NWORK = ITAUP + M
|
|
IE = 1
|
|
NRWORK = IE + M
|
|
*
|
|
* Bidiagonalize A.
|
|
* (RWorkspace: need M)
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
|
|
*
|
|
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
|
|
$ INFO )
|
|
*
|
|
* Multiply B by transpose of left bidiagonalizing vectors.
|
|
* (CWorkspace: need 2*M+NRHS, prefer 2*M+NRHS*NB)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'L', 'C', M, NRHS, N, A, LDA, WORK( ITAUQ ),
|
|
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
* Solve the bidiagonal least squares problem.
|
|
*
|
|
CALL ZLALSD( 'L', SMLSIZ, M, NRHS, S, RWORK( IE ), B, LDB,
|
|
$ RCOND, RANK, WORK( NWORK ), RWORK( NRWORK ),
|
|
$ IWORK, INFO )
|
|
IF( INFO.NE.0 ) THEN
|
|
GO TO 10
|
|
END IF
|
|
*
|
|
* Multiply B by right bidiagonalizing vectors of A.
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ),
|
|
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
END IF
|
|
*
|
|
* Undo scaling.
|
|
*
|
|
IF( IASCL.EQ.1 ) THEN
|
|
CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
|
|
CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
|
|
$ INFO )
|
|
ELSE IF( IASCL.EQ.2 ) THEN
|
|
CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
|
|
CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
|
|
$ INFO )
|
|
END IF
|
|
IF( IBSCL.EQ.1 ) THEN
|
|
CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
|
|
ELSE IF( IBSCL.EQ.2 ) THEN
|
|
CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
|
|
END IF
|
|
*
|
|
10 CONTINUE
|
|
WORK( 1 ) = MAXWRK
|
|
IWORK( 1 ) = LIWORK
|
|
RWORK( 1 ) = LRWORK
|
|
RETURN
|
|
*
|
|
* End of ZGELSD
|
|
*
|
|
END
|
|
|