SUBROUTINE SGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, \$ WORK, LWORK, INFO ) * * -- LAPACK driver routine (instrumented to count ops, version 3.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * April 25, 2001 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK REAL RCOND * .. * .. Array Arguments .. REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) * .. * Common blocks to return operation counts and timings * .. Common blocks .. COMMON / LATIME / OPS, ITCNT COMMON / LSTIME / OPCNT, TIMNG * .. * .. Scalars in Common .. REAL ITCNT, OPS * .. * .. Arrays in Common .. REAL OPCNT( 6 ), TIMNG( 6 ) * .. * * Purpose * ======= * * SGELSS 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. * * Arguments * ========= * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * N (input) INTEGER * The number of columns of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrices B and X. NRHS >= 0. * * A (input/output) REAL 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. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * B (input/output) REAL 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. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,max(M,N)). * * S (output) 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)). * * RCOND (input) 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. * * RANK (output) INTEGER * The effective rank of A, i.e., the number of singular values * which are greater than RCOND*S(1). * * WORK (workspace/output) REAL array, dimension (LWORK) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) 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, a workspace query is assumed. The optimal * size for the WORK array is calculated and stored in WORK(1), * and no other work except argument checking is performed. * * INFO (output) 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. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER BDSPAC, BDSQR, BL, CHUNK, GEBRD, GELQF, GELSS, \$ GEMM, GEMV, GEQRF, I, IASCL, IBSCL, IE, IL, \$ ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN, \$ MAXWRK, MINMN, MINWRK, MM, MNTHR, NB, \$ ORGBR, ORMBR, ORMLQ, ORMQR REAL ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR, \$ T1, T2 * .. * .. Local Arrays .. REAL VDUM( 1 ) * .. * .. External Subroutines .. EXTERNAL SBDSQR, SCOPY, SGEBRD, SGELQF, SGEMM, SGEMV, \$ SGEQRF, SLABAD, SLACPY, SLASCL, SLASET, SORGBR, \$ SORMBR, SORMLQ, SORMQR, SRSCL, XERBLA * .. * .. External Functions .. INTEGER ILAENV REAL SECOND, SLAMCH, SLANGE, SOPBL2, \$ SOPBL3, SOPLA, SOPLA2 EXTERNAL SECOND, SLAMCH, SLANGE, SOPBL2, \$ SOPBL3, SOPLA, SOPLA2, ILAENV * .. * .. Intrinsic Functions .. INTRINSIC REAL, MAX, MIN * .. * .. Data statements .. DATA BDSQR / 5 /, GEBRD / 3 /, GELQF / 2 /, \$ GELSS / 1 /, GEMM / 6 /, GEMV / 6 /, \$ GEQRF / 2 /, ORGBR / 4 /, ORMBR / 4 /, \$ ORMLQ / 6 /, ORMQR / 2 / * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 MINMN = MIN( M, N ) MAXMN = MAX( M, N ) MNTHR = ILAENV( 6, 'SGELSS', ' ', 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 * * 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 IF( INFO.EQ.0 ) THEN MAXWRK = 0 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, 'SGEQRF', ' ', M, N, \$ -1, -1 ) ) MAXWRK = MAX( MAXWRK, N+NRHS* \$ ILAENV( 1, 'SORMQR', 'LT', M, NRHS, N, -1 ) ) END IF IF( M.GE.N ) THEN * * Path 1 - overdetermined or exactly determined * * Compute workspace needed for SBDSQR * BDSPAC = MAX( 1, 5*N ) MAXWRK = MAX( MAXWRK, 3*N+( MM+N )* \$ ILAENV( 1, 'SGEBRD', ' ', MM, N, -1, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+NRHS* \$ ILAENV( 1, 'SORMBR', 'QLT', MM, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+( N-1 )* \$ ILAENV( 1, 'SORGBR', 'P', N, N, N, -1 ) ) 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 SBDSQR * 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 * MAXWRK = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 ) MAXWRK = MAX( MAXWRK, M*M+4*M+2*M* \$ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS* \$ ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )* \$ ILAENV( 1, 'SORGBR', 'P', M, M, M, -1 ) ) 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+NRHS* \$ ILAENV( 1, 'SORMLQ', 'LT', N, NRHS, M, -1 ) ) ELSE * * Path 2 - underdetermined * MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'SGEBRD', ' ', M, N, \$ -1, -1 ) MAXWRK = MAX( MAXWRK, 3*M+NRHS* \$ ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+M* \$ ILAENV( 1, 'SORGBR', 'P', M, N, M, -1 ) ) MAXWRK = MAX( MAXWRK, BDSPAC ) MAXWRK = MAX( MAXWRK, N*NRHS ) END IF END IF MAXWRK = MAX( MINWRK, MAXWRK ) WORK( 1 ) = MAXWRK IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) \$ INFO = -12 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGELSS', -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' ) OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 2 ) SMLNUM = SFMIN / EPS BIGNUM = ONE / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) * * Scale A if max element outside range [SMLNUM,BIGNUM] * ANRM = SLANGE( 'M', M, N, A, LDA, WORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM * OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( M*N ) CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM * OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( M*N ) CALL SLASCL( '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 SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) CALL SLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 ) RANK = 0 GO TO 70 END IF * * Scale B if max element outside range [SMLNUM,BIGNUM] * BNRM = SLANGE( 'M', M, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM * OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( M*NRHS ) CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM * OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( M*NRHS ) CALL SLASCL( '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) * NB = ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 ) OPCNT( GEQRF ) = OPCNT( GEQRF ) + \$ SOPLA( 'SGEQRF', M, N, 0, 0, NB ) T1 = SECOND( ) CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), \$ LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( GEQRF ) = TIMNG( GEQRF ) + ( T2-T1 ) * * Multiply B by transpose(Q) * (Workspace: need N+NRHS, prefer N+NRHS*NB) * NB = ILAENV( 1, 'SORMQR', 'LT', M, NRHS, N, -1 ) OPCNT( ORMQR ) = OPCNT( ORMQR ) + \$ SOPLA( 'SORMQR', M, NRHS, N, 0, NB ) T1 = SECOND( ) CALL SORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B, \$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMQR ) = TIMNG( ORMQR ) + ( T2-T1 ) * * Zero out below R * IF( N.GT.1 ) \$ CALL SLASET( '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) * NB = ILAENV( 1, 'SGEBRD', ' ', MM, N, -1, -1 ) OPCNT( GEBRD ) = OPCNT( GEBRD ) + \$ SOPLA( 'SGEBRD', MM, N, 0, 0, NB ) T1 = SECOND( ) CALL SGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), \$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, \$ INFO ) T2 = SECOND( ) TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 ) * * Multiply B by transpose of left bidiagonalizing vectors of R * (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB) * NB = ILAENV( 1, 'SORMBR', 'QLT', MM, NRHS, N, -1 ) OPCNT( ORMBR ) = OPCNT( ORMBR ) + \$ SOPLA2( 'SORMBR', 'QLT', MM, NRHS, N, 0, NB ) T1 = SECOND( ) CALL SORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ), \$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 ) * * Generate right bidiagonalizing vectors of R in A * (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) * NB = ILAENV( 1, 'SORGBR', 'P', N, N, N, -1 ) OPCNT( ORGBR ) = OPCNT( ORGBR ) + \$ SOPLA2( 'SORGBR', 'P', N, N, N, 0, NB ) T1 = SECOND( ) CALL SORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ), \$ WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORGBR ) = TIMNG( ORGBR ) + ( T2-T1 ) IWORK = IE + N * * Perform bidiagonal QR iteration * multiply B by transpose of left singular vectors * compute right singular vectors in A * (Workspace: need BDSPAC) * OPS = 0 T1 = SECOND( ) CALL SBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, VDUM, \$ 1, B, LDB, WORK( IWORK ), INFO ) T2 = SECOND( ) TIMNG( BDSQR ) = TIMNG( BDSQR ) + ( T2-T1 ) OPCNT( BDSQR ) = OPCNT( BDSQR ) + OPS IF( INFO.NE.0 ) \$ GO TO 70 * * Multiply B by reciprocals of singular values * OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 ) THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 ) THR = MAX( EPS*S( 1 ), SFMIN ) END IF RANK = 0 DO 10 I = 1, N IF( S( I ).GT.THR ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( NRHS + 3 ) CALL SRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL SLASET( '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 OPCNT( GEMM ) = OPCNT( GEMM ) + \$ SOPBL3( 'SGEMM ', N, NRHS, N ) T1 = SECOND( ) CALL SGEMM( 'T', 'N', N, NRHS, N, ONE, A, LDA, B, LDB, ZERO, \$ WORK, LDB ) T2 = SECOND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL SLACPY( '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 ) OPCNT( GEMM ) = OPCNT( GEMM ) + \$ SOPBL3( 'SGEMM ', N, BL, N ) T1 = SECOND( ) CALL SGEMM( 'T', 'N', N, BL, N, ONE, A, LDA, B( 1, I ), \$ LDB, ZERO, WORK, N ) T2 = SECOND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL SLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB ) 20 CONTINUE ELSE OPCNT( GEMV ) = OPCNT( GEMV ) + \$ SOPBL2( 'SGEMV ', N, N, 0, 0 ) T1 = SECOND( ) CALL SGEMV( 'T', N, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 ) T2 = SECOND( ) TIMNG( GEMV ) = TIMNG( GEMV ) + ( T2-T1 ) CALL SCOPY( 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) * NB = ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 ) OPCNT( GELQF ) = OPCNT( GELQF ) + \$ SOPLA( 'SGELQF', M, N, 0, 0, NB ) T1 = SECOND( ) CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), \$ LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( GELQF ) = TIMNG( GELQF ) + ( T2-T1 ) IL = IWORK * * Copy L to WORK(IL), zeroing out above it * CALL SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK ) CALL SLASET( '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) * NB = ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) OPCNT( GEBRD ) = OPCNT( GEBRD ) + \$ SOPLA( 'SGEBRD', M, M, 0, 0, NB ) T1 = SECOND( ) CALL SGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ), \$ WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ), \$ LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 ) * * Multiply B by transpose of left bidiagonalizing vectors of L * (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB) * NB = ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, M, -1 ) OPCNT( ORMBR ) = OPCNT( ORMBR ) + \$ SOPLA2( 'SORMBR', 'QLT', M, NRHS, M, 0, NB ) T1 = SECOND( ) CALL SORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK, \$ WORK( ITAUQ ), B, LDB, WORK( IWORK ), \$ LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 ) * * Generate right bidiagonalizing vectors of R in WORK(IL) * (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB) * NB = ILAENV( 1, 'SORGBR', 'P', M, M, M, -1 ) OPCNT( ORGBR ) = OPCNT( ORGBR ) + \$ SOPLA2( 'SORGBR', 'P', M, M, M, 0, NB ) T1 = SECOND( ) CALL SORGBR( 'P', M, M, M, WORK( IL ), LDWORK, WORK( ITAUP ), \$ WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORGBR ) = TIMNG( ORGBR ) + ( T2-T1 ) 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) * OPS = 0 T1 = SECOND( ) CALL SBDSQR( 'U', M, M, 0, NRHS, S, WORK( IE ), WORK( IL ), \$ LDWORK, A, LDA, B, LDB, WORK( IWORK ), INFO ) T2 = SECOND( ) TIMNG( BDSQR ) = TIMNG( BDSQR ) + ( T2-T1 ) OPCNT( BDSQR ) = OPCNT( BDSQR ) + OPS IF( INFO.NE.0 ) \$ GO TO 70 * * Multiply B by reciprocals of singular values * OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 ) THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 ) THR = MAX( EPS*S( 1 ), SFMIN ) END IF RANK = 0 DO 30 I = 1, M IF( S( I ).GT.THR ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( NRHS + 3 ) CALL SRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL SLASET( '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 OPCNT( GEMM ) = OPCNT( GEMM ) + \$ SOPBL3( 'SGEMM ', M, NRHS, M ) T1 = SECOND( ) CALL SGEMM( 'T', 'N', M, NRHS, M, ONE, WORK( IL ), LDWORK, \$ B, LDB, ZERO, WORK( IWORK ), LDB ) T2 = SECOND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL SLACPY( '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 ) OPCNT( GEMM ) = OPCNT( GEMM ) + \$ SOPBL3( 'SGEMM ', M, BL, M ) T1 = SECOND( ) CALL SGEMM( 'T', 'N', M, BL, M, ONE, WORK( IL ), LDWORK, \$ B( 1, I ), LDB, ZERO, WORK( IWORK ), M ) T2 = SECOND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL SLACPY( 'G', M, BL, WORK( IWORK ), M, B( 1, I ), \$ LDB ) 40 CONTINUE ELSE OPCNT( GEMV ) = OPCNT( GEMV ) + \$ SOPBL2( 'SGEMV ', M, M, 0, 0 ) T1 = SECOND( ) CALL SGEMV( 'T', M, M, ONE, WORK( IL ), LDWORK, B( 1, 1 ), \$ 1, ZERO, WORK( IWORK ), 1 ) T2 = SECOND( ) TIMNG( GEMV ) = TIMNG( GEMV ) + ( T2-T1 ) CALL SCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 ) END IF * * Zero out below first M rows of B * CALL SLASET( '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) * NB = ILAENV( 1, 'SORMLQ', 'LT', N, NRHS, M, -1 ) OPCNT( ORMLQ ) = OPCNT( ORMLQ ) + \$ SOPLA( 'SORMLQ', N, NRHS, M, 0, NB ) T1 = SECOND( ) CALL SORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B, \$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMLQ ) = TIMNG( ORMLQ ) + ( T2-T1 ) * 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) * NB = ILAENV( 1, 'SGEBRD', ' ', M, N, -1, -1 ) OPCNT( GEBRD ) = OPCNT( GEBRD ) + \$ SOPLA( 'SGEBRD', M, N, 0, 0, NB ) T1 = SECOND( ) CALL SGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), \$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, \$ INFO ) T2 = SECOND( ) TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 ) * * Multiply B by transpose of left bidiagonalizing vectors * (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB) * NB = ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, N, -1 ) OPCNT( ORMBR ) = OPCNT( ORMBR ) + \$ SOPLA2( 'SORMBR', 'QLT', M, NRHS, N, 0, NB ) T1 = SECOND( ) CALL SORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ), \$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 ) * * Generate right bidiagonalizing vectors in A * (Workspace: need 4*M, prefer 3*M+M*NB) * NB = ILAENV( 1, 'SORGBR', 'P', M, N, M, -1 ) OPCNT( ORGBR ) = OPCNT( ORGBR ) + \$ SOPLA2( 'SORGBR', 'P', M, N, M, 0, NB ) T1 = SECOND( ) CALL SORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ), \$ WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORGBR ) = TIMNG( ORGBR ) + ( T2-T1 ) 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) * OPS = 0 T1 = SECOND( ) CALL SBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, VDUM, \$ 1, B, LDB, WORK( IWORK ), INFO ) T2 = SECOND( ) TIMNG( BDSQR ) = TIMNG( BDSQR ) + ( T2-T1 ) OPCNT( BDSQR ) = OPCNT( BDSQR ) + OPS IF( INFO.NE.0 ) \$ GO TO 70 * * Multiply B by reciprocals of singular values * OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 ) THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 ) THR = MAX( EPS*S( 1 ), SFMIN ) END IF RANK = 0 DO 50 I = 1, M IF( S( I ).GT.THR ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( NRHS + 3 ) CALL SRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL SLASET( '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 OPCNT( GEMM ) = OPCNT( GEMM ) + \$ SOPBL3( 'SGEMM ', N, NRHS, M ) T1 = SECOND( ) CALL SGEMM( 'T', 'N', N, NRHS, M, ONE, A, LDA, B, LDB, ZERO, \$ WORK, LDB ) T2 = SECOND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL SLACPY( '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 ) OPCNT( GEMM ) = OPCNT( GEMM ) + \$ SOPBL3( 'SGEMM ', N, BL, M ) T1 = SECOND( ) CALL SGEMM( 'T', 'N', N, BL, M, ONE, A, LDA, B( 1, I ), \$ LDB, ZERO, WORK, N ) T2 = SECOND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL SLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB ) 60 CONTINUE ELSE OPCNT( GEMV ) = OPCNT( GEMV ) + \$ SOPBL2( 'SGEMV ', M, N, 0, 0 ) T1 = SECOND( ) CALL SGEMV( 'T', M, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 ) T2 = SECOND( ) TIMNG( GEMV ) = TIMNG( GEMV ) + ( T2-T1 ) CALL SCOPY( N, WORK, 1, B, 1 ) END IF END IF * * Undo scaling * IF( IASCL.EQ.1 ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( N*NRHS + MINMN ) CALL SLASCL( '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 OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( N*NRHS + MINMN ) CALL SLASCL( '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 OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( N*NRHS ) CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( N*NRHS ) CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF * 70 CONTINUE WORK( 1 ) = MAXWRK RETURN * * End of SGELSS * END