*> \brief DGELSS solves overdetermined or underdetermined systems for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DGELSS + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, * WORK, LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK * DOUBLE PRECISION RCOND * .. * .. Array Arguments .. * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DGELSS 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 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 DOUBLE PRECISION array, dimension (LDA,N) *> On entry, the M-by-N matrix A. *> On exit, the first min(m,n) rows of A are overwritten with *> its right singular vectors, stored rowwise. *> \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 >= 1, and also: *> LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS ) *> 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] 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 doubleGEsolve * * ===================================================================== SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, \$ WORK, LWORK, 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 DOUBLE PRECISION RCOND * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER BDSPAC, BL, CHUNK, I, IASCL, IBSCL, IE, IL, \$ ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN, \$ MAXWRK, MINMN, MINWRK, MM, MNTHR INTEGER LWORK_DGEQRF, LWORK_DORMQR, LWORK_DGEBRD, \$ LWORK_DORMBR, LWORK_DORGBR, LWORK_DORMLQ, \$ LWORK_DGELQF DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR * .. * .. Local Arrays .. DOUBLE PRECISION DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL DBDSQR, DCOPY, DGEBRD, DGELQF, DGEMM, DGEMV, \$ DGEQRF, DLABAD, DLACPY, DLASCL, DLASET, DORGBR, \$ DORMBR, DORMLQ, DORMQR, DRSCL, XERBLA * .. * .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL ILAENV, DLAMCH, DLANGE * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. 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 IF( MINMN.GT.0 ) THEN MM = M MNTHR = ILAENV( 6, 'DGELSS', ' ', M, N, NRHS, -1 ) IF( M.GE.N .AND. M.GE.MNTHR ) THEN * * Path 1a - overdetermined, with many more rows than * columns * * Compute space needed for DGEQRF CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, INFO ) LWORK_DGEQRF=DUM(1) * Compute space needed for DORMQR CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, DUM(1), B, \$ LDB, DUM(1), -1, INFO ) LWORK_DORMQR=DUM(1) MM = N MAXWRK = MAX( MAXWRK, N + LWORK_DGEQRF ) MAXWRK = MAX( MAXWRK, N + LWORK_DORMQR ) END IF IF( M.GE.N ) THEN * * Path 1 - overdetermined or exactly determined * * Compute workspace needed for DBDSQR * BDSPAC = MAX( 1, 5*N ) * Compute space needed for DGEBRD CALL DGEBRD( MM, N, A, LDA, S, DUM(1), DUM(1), \$ DUM(1), DUM(1), -1, INFO ) LWORK_DGEBRD=DUM(1) * Compute space needed for DORMBR CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, DUM(1), \$ B, LDB, DUM(1), -1, INFO ) LWORK_DORMBR=DUM(1) * Compute space needed for DORGBR CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1), \$ DUM(1), -1, INFO ) LWORK_DORGBR=DUM(1) * Compute total workspace needed MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD ) MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORMBR ) MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR ) MAXWRK = MAX( MAXWRK, BDSPAC ) MAXWRK = MAX( MAXWRK, N*NRHS ) MINWRK = MAX( 3*N + MM, 3*N + NRHS, BDSPAC ) MAXWRK = MAX( MINWRK, MAXWRK ) END IF IF( N.GT.M ) THEN * * Compute workspace needed for DBDSQR * BDSPAC = MAX( 1, 5*M ) MINWRK = MAX( 3*M+NRHS, 3*M+N, BDSPAC ) IF( N.GE.MNTHR ) THEN * * Path 2a - underdetermined, with many more columns * than rows * * Compute space needed for DGELQF CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1), \$ -1, INFO ) LWORK_DGELQF=DUM(1) * Compute space needed for DGEBRD CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1), \$ DUM(1), DUM(1), -1, INFO ) LWORK_DGEBRD=DUM(1) * Compute space needed for DORMBR CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, \$ DUM(1), B, LDB, DUM(1), -1, INFO ) LWORK_DORMBR=DUM(1) * Compute space needed for DORGBR CALL DORGBR( 'P', M, M, M, A, LDA, DUM(1), \$ DUM(1), -1, INFO ) LWORK_DORGBR=DUM(1) * Compute space needed for DORMLQ CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, DUM(1), \$ B, LDB, DUM(1), -1, INFO ) LWORK_DORMLQ=DUM(1) * Compute total workspace needed MAXWRK = M + LWORK_DGELQF MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DGEBRD ) MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORMBR ) MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORGBR ) MAXWRK = MAX( MAXWRK, M*M + M + BDSPAC ) 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 + LWORK_DORMLQ ) ELSE * * Path 2 - underdetermined * * Compute space needed for DGEBRD CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1), \$ DUM(1), DUM(1), -1, INFO ) LWORK_DGEBRD=DUM(1) * Compute space needed for DORMBR CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, A, LDA, \$ DUM(1), B, LDB, DUM(1), -1, INFO ) LWORK_DORMBR=DUM(1) * Compute space needed for DORGBR CALL DORGBR( 'P', M, N, M, A, LDA, DUM(1), \$ DUM(1), -1, INFO ) LWORK_DORGBR=DUM(1) MAXWRK = 3*M + LWORK_DGEBRD MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORMBR ) MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR ) MAXWRK = MAX( MAXWRK, BDSPAC ) MAXWRK = MAX( MAXWRK, N*NRHS ) END IF END IF MAXWRK = MAX( MINWRK, MAXWRK ) END IF WORK( 1 ) = MAXWRK * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) \$ INFO = -12 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELSS', -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 CALL DLABAD( SMLNUM, BIGNUM ) * * Scale A if max element 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, MINMN ) RANK = 0 GO TO 70 END IF * * Scale B if max element 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 * * 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 IWORK = ITAU + N * * Compute A=Q*R * (Workspace: need 2*N, prefer N+N*NB) * CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), \$ LWORK-IWORK+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( IWORK ), LWORK-IWORK+1, INFO ) * * Zero out below R * IF( N.GT.1 ) \$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) END IF * IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = 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( IWORK ), LWORK-IWORK+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( IWORK ), LWORK-IWORK+1, INFO ) * * Generate right bidiagonalizing vectors of R in A * (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) * CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ), \$ WORK( IWORK ), LWORK-IWORK+1, INFO ) IWORK = IE + N * * Perform bidiagonal QR iteration * multiply B by transpose of left singular vectors * compute right singular vectors in A * (Workspace: need BDSPAC) * CALL DBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM, \$ 1, B, LDB, WORK( IWORK ), INFO ) IF( INFO.NE.0 ) \$ GO TO 70 * * Multiply B by reciprocals of singular values * THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) \$ THR = MAX( EPS*S( 1 ), SFMIN ) RANK = 0 DO 10 I = 1, N IF( S( I ).GT.THR ) THEN CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB ) END IF 10 CONTINUE * * Multiply B by right singular vectors * (Workspace: need N, prefer N*NRHS) * IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, A, LDA, B, LDB, ZERO, \$ WORK, LDB ) CALL DLACPY( 'G', N, NRHS, WORK, LDB, B, LDB ) ELSE IF( NRHS.GT.1 ) THEN CHUNK = LWORK / N DO 20 I = 1, NRHS, CHUNK BL = MIN( NRHS-I+1, CHUNK ) CALL DGEMM( 'T', 'N', N, BL, N, ONE, A, LDA, B( 1, I ), \$ LDB, ZERO, WORK, N ) CALL DLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB ) 20 CONTINUE ELSE CALL DGEMV( 'T', N, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 ) CALL DCOPY( N, WORK, 1, B, 1 ) END IF * 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 IWORK = M + 1 * * Compute A=L*Q * (Workspace: need 2*M, prefer M+M*NB) * CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), \$ LWORK-IWORK+1, INFO ) IL = IWORK * * Copy L to WORK(IL), zeroing out above it * 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 IWORK = 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( IWORK ), \$ LWORK-IWORK+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( IWORK ), \$ LWORK-IWORK+1, INFO ) * * Generate right bidiagonalizing vectors of R in WORK(IL) * (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB) * CALL DORGBR( 'P', M, M, M, WORK( IL ), LDWORK, WORK( ITAUP ), \$ WORK( IWORK ), LWORK-IWORK+1, INFO ) IWORK = IE + M * * Perform bidiagonal QR iteration, * computing right singular vectors of L in WORK(IL) and * multiplying B by transpose of left singular vectors * (Workspace: need M*M+M+BDSPAC) * CALL DBDSQR( 'U', M, M, 0, NRHS, S, WORK( IE ), WORK( IL ), \$ LDWORK, A, LDA, B, LDB, WORK( IWORK ), INFO ) IF( INFO.NE.0 ) \$ GO TO 70 * * Multiply B by reciprocals of singular values * THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) \$ THR = MAX( EPS*S( 1 ), SFMIN ) RANK = 0 DO 30 I = 1, M IF( S( I ).GT.THR ) THEN CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB ) END IF 30 CONTINUE IWORK = IE * * Multiply B by right singular vectors of L in WORK(IL) * (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS) * IF( LWORK.GE.LDB*NRHS+IWORK-1 .AND. NRHS.GT.1 ) THEN CALL DGEMM( 'T', 'N', M, NRHS, M, ONE, WORK( IL ), LDWORK, \$ B, LDB, ZERO, WORK( IWORK ), LDB ) CALL DLACPY( 'G', M, NRHS, WORK( IWORK ), LDB, B, LDB ) ELSE IF( NRHS.GT.1 ) THEN CHUNK = ( LWORK-IWORK+1 ) / M DO 40 I = 1, NRHS, CHUNK BL = MIN( NRHS-I+1, CHUNK ) CALL DGEMM( 'T', 'N', M, BL, M, ONE, WORK( IL ), LDWORK, \$ B( 1, I ), LDB, ZERO, WORK( IWORK ), M ) CALL DLACPY( 'G', M, BL, WORK( IWORK ), M, B( 1, I ), \$ LDB ) 40 CONTINUE ELSE CALL DGEMV( 'T', M, M, ONE, WORK( IL ), LDWORK, B( 1, 1 ), \$ 1, ZERO, WORK( IWORK ), 1 ) CALL DCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 ) END IF * * Zero out below first M rows of B * CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB ) IWORK = 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( IWORK ), LWORK-IWORK+1, INFO ) * ELSE * * Path 2 - remaining underdetermined cases * IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = 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( IWORK ), LWORK-IWORK+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( IWORK ), LWORK-IWORK+1, INFO ) * * Generate right bidiagonalizing vectors in A * (Workspace: need 4*M, prefer 3*M+M*NB) * CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ), \$ WORK( IWORK ), LWORK-IWORK+1, INFO ) IWORK = IE + M * * Perform bidiagonal QR iteration, * computing right singular vectors of A in A and * multiplying B by transpose of left singular vectors * (Workspace: need BDSPAC) * CALL DBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM, \$ 1, B, LDB, WORK( IWORK ), INFO ) IF( INFO.NE.0 ) \$ GO TO 70 * * Multiply B by reciprocals of singular values * THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) \$ THR = MAX( EPS*S( 1 ), SFMIN ) RANK = 0 DO 50 I = 1, M IF( S( I ).GT.THR ) THEN CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB ) END IF 50 CONTINUE * * Multiply B by right singular vectors of A * (Workspace: need N, prefer N*NRHS) * IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN CALL DGEMM( 'T', 'N', N, NRHS, M, ONE, A, LDA, B, LDB, ZERO, \$ WORK, LDB ) CALL DLACPY( 'F', N, NRHS, WORK, LDB, B, LDB ) ELSE IF( NRHS.GT.1 ) THEN CHUNK = LWORK / N DO 60 I = 1, NRHS, CHUNK BL = MIN( NRHS-I+1, CHUNK ) CALL DGEMM( 'T', 'N', N, BL, M, ONE, A, LDA, B( 1, I ), \$ LDB, ZERO, WORK, N ) CALL DLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB ) 60 CONTINUE ELSE CALL DGEMV( 'T', M, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 ) CALL DCOPY( N, WORK, 1, B, 1 ) END IF 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 * 70 CONTINUE WORK( 1 ) = MAXWRK RETURN * * End of DGELSS * END