*> \brief CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CGELSD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CGELSD( 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 * REAL RCOND * .. * .. Array Arguments .. * INTEGER IWORK( * ) * REAL RWORK( * ), S( * ) * COMPLEX A( LDA, * ), B( LDB, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CGELSD 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. *> *> 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 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, 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 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 REAL 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 REAL *> 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 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 REAL 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. * *> \date December 2016 * *> \ingroup complexGEsolve * *> \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 CGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, \$ WORK, LWORK, RWORK, IWORK, INFO ) * * -- LAPACK driver routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK REAL RCOND * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL RWORK( * ), S( * ) COMPLEX A( LDA, * ), B( LDB, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) COMPLEX CZERO PARAMETER ( CZERO = ( 0.0E+0, 0.0E+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 REAL ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM * .. * .. External Subroutines .. EXTERNAL CGEBRD, CGELQF, CGEQRF, CLACPY, \$ CLALSD, CLASCL, CLASET, CUNMBR, \$ CUNMLQ, CUNMQR, SLABAD, SLASCL, \$ SLASET, XERBLA * .. * .. External Functions .. INTEGER ILAENV REAL CLANGE, SLAMCH EXTERNAL CLANGE, SLAMCH, ILAENV * .. * .. Intrinsic Functions .. INTRINSIC INT, LOG, MAX, MIN, REAL * .. * .. 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, 'CGELSD', ' ', 0, 0, 0, 0 ) MNTHR = ILAENV( 6, 'CGELSD', ' ', M, N, NRHS, -1 ) NLVL = MAX( INT( LOG( REAL( MINMN ) / REAL( 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, 'CGEQRF', ' ', M, N, \$ -1, -1 ) ) MAXWRK = MAX( MAXWRK, NRHS*ILAENV( 1, 'CUNMQR', '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, \$ 'CGEBRD', ' ', MM, N, -1, -1 ) ) MAXWRK = MAX( MAXWRK, 2*N + NRHS*ILAENV( 1, 'CUNMBR', \$ 'QLC', MM, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1, \$ 'CUNMBR', '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, 'CGELQF', ' ', M, N, -1, \$ -1 ) MAXWRK = MAX( MAXWRK, M*M + 4*M + 2*M*ILAENV( 1, \$ 'CGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( MAXWRK, M*M + 4*M + NRHS*ILAENV( 1, \$ 'CUNMBR', 'QLC', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M + 4*M + ( M - 1 )*ILAENV( 1, \$ 'CUNMLQ', '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, 'CGEBRD', ' ', M, \$ N, -1, -1 ) MAXWRK = MAX( MAXWRK, 2*M + NRHS*ILAENV( 1, 'CUNMBR', \$ 'QLC', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, 2*M + M*ILAENV( 1, 'CUNMBR', \$ '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( 'CGELSD', -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 = SLAMCH( 'P' ) SFMIN = SLAMCH( 'S' ) SMLNUM = SFMIN / EPS BIGNUM = ONE / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) * * Scale A if max entry 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( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB ) CALL SLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 ) RANK = 0 GO TO 10 END IF * * Scale B if max entry outside range [SMLNUM,BIGNUM]. * BNRM = CLANGE( 'M', M, 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, M, 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, M, NRHS, B, LDB, INFO ) IBSCL = 2 END IF * * If M < N make sure B(M+1:N,:) = 0 * IF( M.LT.N ) \$ CALL CLASET( '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 CGEQRF( 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 CUNMQR( '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 CLASET( '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 CGEBRD( 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 CUNMBR( '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 CLALSD( '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 CUNMBR( '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 CGELQF( 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 CLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK ) CALL CLASET( '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 CGEBRD( 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 CUNMBR( '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 CLALSD( '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 CUNMBR( '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 CLASET( '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 CUNMLQ( '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 CGEBRD( 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 CUNMBR( '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 CLALSD( '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 CUNMBR( '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 CLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) CALL SLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN, \$ INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL CLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) CALL SLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN, \$ INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL CLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL CLASCL( '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 CGELSD * END