#include "ESMF_LapackBlas.inc" !> \brief <b> DGELSD 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 DGELSD + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsd.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsd.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsd.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, ! WORK, LWORK, IWORK, INFO ) ! ! .. Scalar Arguments .. ! INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK ! DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. ! INTEGER IWORK( * ) ! DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DGELSD 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 tranformations 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. !> !> The divide and conquer algorithm makes very mild assumptions about !> floating point arithmetic. It will work on machines with a guard !> digit in add/subtract, or on those binary machines without guard !> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or !> Cray-2. It could conceivably fail on hexadecimal or decimal machines !> without guard digits, but we know of none. !> \endverbatim ! ! Arguments: ! ========== ! !> \param[in] M !> \verbatim !> M is INTEGER !> The number of rows of A. M >= 0. !> \endverbatim !> !> \param[in] N !> \verbatim !> N is INTEGER !> The number of columns of 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] A !> \verbatim !> A is DOUBLE PRECISION 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 DOUBLE PRECISION 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 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,max(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 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 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 !> 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, !> if M is greater than or equal to N or !> 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, !> 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 ) !> 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 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] 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. ! !> \date November 2011 ! !> \ingroup doubleGEsolve ! !> \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 DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, & & WORK, LWORK, IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.4.0) -- ! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- ! November 2011 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ, & & LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK, & & MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM ! .. ! .. External Subroutines .. EXTERNAL DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD, & & DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, INT, LOG, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments. ! INFO = 0 MINMN = MIN( M, N ) MAXMN = MAX( M, N ) MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 ) 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 ! SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 ) ! ! 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.) ! MINWRK = 1 LIWORK = 1 MINMN = MAX( 1, MINMN ) NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) / & & LOG( TWO ) ) + 1, 0 ) ! IF( INFO.EQ.0 ) THEN MAXWRK = 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+N*ILAENV( 1, 'DGEQRF', ' ', M, N, & & -1, -1 ) ) MAXWRK = MAX( MAXWRK, N+NRHS* & & ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) ) END IF IF( M.GE.N ) THEN ! ! Path 1 - overdetermined or exactly determined. ! MAXWRK = MAX( MAXWRK, 3*N+( MM+N )* & & ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+NRHS* & & ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+( N-1 )* & & ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) ) WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2 MAXWRK = MAX( MAXWRK, 3*N+WLALSD ) MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD ) END IF IF( N.GT.M ) THEN WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2 IF( N.GE.MNTHR ) THEN ! ! Path 2a - underdetermined, with many more columns ! than rows. ! MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) MAXWRK = MAX( MAXWRK, M*M+4*M+2*M* & & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS* & & ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )* & & ILAENV( 1, 'DORMBR', 'PLN', M, 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+NRHS* & & ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD ) ! 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 - remaining underdetermined cases. ! MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N, & & -1, -1 ) MAXWRK = MAX( MAXWRK, 3*M+NRHS* & & ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+M* & & ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+WLALSD ) END IF MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD ) END IF MINWRK = MIN( MINWRK, MAXWRK ) WORK( 1 ) = MAXWRK IWORK( 1 ) = LIWORK IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELSD', -INFO ) RETURN ELSE IF( LQUERY ) THEN GO TO 10 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 CALL DLABAD( SMLNUM, BIGNUM ) ! ! Scale A if max entry outside range [SMLNUM,BIGNUM]. ! ANRM = DLANGE( 'M', M, N, A, LDA, WORK ) 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( 'F', MAX( M, N ), NRHS, ZERO, ZERO, 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 = DLANGE( 'M', M, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM. ! CALL DLASCL( '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 DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO ) IBSCL = 2 END IF ! ! If M < N make sure certain entries of B are zero. ! IF( M.LT.N ) & & CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, 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. ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & & LWORK-NWORK+1, INFO ) ! ! Multiply B by transpose(Q). ! (Workspace: need N+NRHS, prefer N+NRHS*NB) ! CALL DORMQR( 'L', 'T', 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 DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) END IF END IF ! IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N NWORK = ITAUP + N ! ! Bidiagonalize R in A. ! (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB) ! CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & & INFO ) ! ! Multiply B by transpose of left bidiagonalizing vectors of R. ! (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB) ! CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ), & & B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) ! ! Solve the bidiagonal least squares problem. ! CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB, & & RCOND, RANK, WORK( NWORK ), IWORK, INFO ) IF( INFO.NE.0 ) THEN GO TO 10 END IF ! ! Multiply B by right bidiagonalizing vectors of R. ! CALL DORMBR( '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, WLALSD ) ) 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, 4*M+M*LDA+WLALSD ) )LDWORK = LDA ITAU = 1 NWORK = M + 1 ! ! Compute A=L*Q. ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( 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 DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ), & & LDWORK ) IE = IL + LDWORK*M ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IL). ! (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ), & & WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), & & LWORK-NWORK+1, INFO ) ! ! Multiply B by transpose of left bidiagonalizing vectors of L. ! (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB) ! CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK, & & WORK( ITAUQ ), B, LDB, WORK( NWORK ), & & LWORK-NWORK+1, INFO ) ! ! Solve the bidiagonal least squares problem. ! CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB, & & RCOND, RANK, WORK( NWORK ), IWORK, INFO ) IF( INFO.NE.0 ) THEN GO TO 10 END IF ! ! Multiply B by right bidiagonalizing vectors of L. ! CALL DORMBR( '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 DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB ) NWORK = ITAU + M ! ! Multiply transpose(Q) by B. ! (Workspace: need M+NRHS, prefer M+NRHS*NB) ! CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B, & & LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) ! ELSE ! ! Path 2 - remaining underdetermined cases. ! IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M ! ! Bidiagonalize A. ! (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) ! CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & & INFO ) ! ! Multiply B by transpose of left bidiagonalizing vectors. ! (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB) ! CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ), & & B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) ! ! Solve the bidiagonal least squares problem. ! CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB, & & RCOND, RANK, WORK( NWORK ), IWORK, INFO ) IF( INFO.NE.0 ) THEN GO TO 10 END IF ! ! Multiply B by right bidiagonalizing vectors of A. ! CALL DORMBR( '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 DLASCL( '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 DLASCL( '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 DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF ! 10 CONTINUE WORK( 1 ) = MAXWRK IWORK( 1 ) = LIWORK RETURN ! ! End of DGELSD ! END