SUBROUTINE ZGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, \$ WORK, LWORK, RWORK, 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 DOUBLE PRECISION RCOND * .. * .. Array Arguments .. DOUBLE PRECISION RWORK( * ), S( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ) * .. * Common blocks to return operation counts and timings * .. Common blocks .. COMMON / LATIME / OPS, ITCNT COMMON / LSTIME / OPCNT, TIMNG * .. * .. Scalars in Common .. DOUBLE PRECISION ITCNT, OPS * .. * .. Arrays in Common .. DOUBLE PRECISION OPCNT( 6 ), TIMNG( 6 ) * .. * * Purpose * ======= * * ZGELSS computes the minimum norm solution to a complex 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) COMPLEX*16 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) 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 elements n+1:m in that column. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,M,N). * * S (output) 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)). * * RCOND (input) 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. * * 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) COMPLEX*16 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 >= 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. * * RWORK (workspace) DOUBLE PRECISION array, dimension (5*min(M,N)) * * 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 .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), \$ CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER BDSQR, BL, CHUNK, GEBRD, GELQF, GELSS, GEMM, \$ GEMV, GEQRF, I, IASCL, IBSCL, IE, IL, IRWORK, \$ ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN, \$ MAXWRK, MINMN, MINWRK, MM, MNTHR, NB, UNGBR, \$ UNMBR, UNMLQ, UNMQR DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, T1, T2, \$ THR * .. * .. Local Arrays .. COMPLEX*16 VDUM( 1 ) * .. * .. External Subroutines .. EXTERNAL DLABAD, DLASCL, DLASET, XERBLA, ZBDSQR, ZCOPY, \$ ZDRSCL, ZGEBRD, ZGELQF, ZGEMM, ZGEMV, ZGEQRF, \$ ZLACPY, ZLASCL, ZLASET, ZUNGBR, ZUNMBR, ZUNMLQ, \$ ZUNMQR * .. * .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH, DOPBL2, DOPBL3, DOPLA, DSECND, DOPLA2, \$ ZLANGE EXTERNAL ILAENV, DLAMCH, DOPBL2, DOPBL3, DOPLA, DSECND, \$ DOPLA2, ZLANGE * .. * .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN * .. * .. Data statements .. DATA BDSQR / 5 / , GEBRD / 3 / , GELQF / 2 / , \$ GELSS / 1 / , GEMM / 6 / , GEMV / 6 / , \$ GEQRF / 2 / , UNGBR / 4 / , UNMBR / 4 / , \$ UNMLQ / 6 / , UNMQR / 2 / * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 MINMN = MIN( M, N ) MAXMN = MAX( M, N ) MNTHR = ILAENV( 6, 'ZGELSS', ' ', 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. * CWorkspace refers to complex workspace, and RWorkspace refers * to real workspace. 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 * * Space needed for ZBDSQR is BDSPAC = 5*N * MM = N MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'ZGEQRF', ' ', M, N, \$ -1, -1 ) ) MAXWRK = MAX( MAXWRK, N+NRHS* \$ ILAENV( 1, 'ZUNMQR', 'LC', M, NRHS, N, -1 ) ) END IF IF( M.GE.N ) THEN * * Path 1 - overdetermined or exactly determined * * Space needed for ZBDSQR is BDSPC = 7*N+12 * 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, 'ZUNGBR', 'P', N, N, N, -1 ) ) MAXWRK = MAX( MAXWRK, N*NRHS ) MINWRK = 2*N + MAX( NRHS, M ) END IF IF( N.GT.M ) THEN MINWRK = 2*M + MAX( NRHS, N ) IF( N.GE.MNTHR ) THEN * * Path 2a - underdetermined, with many more columns * than rows * * Space needed for ZBDSQR is BDSPAC = 5*M * MAXWRK = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 ) MAXWRK = MAX( MAXWRK, 3*M+M*M+2*M* \$ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+M*M+NRHS* \$ ILAENV( 1, 'ZUNMBR', 'QLC', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+M*M+( M-1 )* \$ ILAENV( 1, 'ZUNGBR', 'P', M, M, 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, 'ZUNMLQ', 'LC', N, NRHS, M, -1 ) ) ELSE * * Path 2 - underdetermined * * Space needed for ZBDSQR is BDSPAC = 5*M * 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, 'ZUNGBR', 'P', M, N, M, -1 ) ) 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( 'ZGELSS', -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' ) OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 2 ) SMLNUM = SFMIN / EPS BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) * * Scale A if max element 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 * OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 6*M*N ) 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 * OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 6*M*N ) 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, MINMN ) RANK = 0 GO TO 70 END IF * * Scale B if max element 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 * OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 6*M*NRHS ) 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 * OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 6*M*NRHS ) CALL ZLASCL( '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 * (CWorkspace: need 2*N, prefer N+N*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 ) OPCNT( GEQRF ) = OPCNT( GEQRF ) + \$ DOPLA( 'ZGEQRF', M, N, 0, 0, NB ) T1 = DSECND( ) CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), \$ LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( GEQRF ) = TIMNG( GEQRF ) + ( T2-T1 ) * * Multiply B by transpose(Q) * (CWorkspace: need N+NRHS, prefer N+NRHS*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZUNMQR', 'LC', M, NRHS, N, -1 ) OPCNT( UNMQR ) = OPCNT( UNMQR ) + \$ DOPLA( 'ZUNMQR', M, NRHS, N, 0, NB ) T1 = DSECND( ) CALL ZUNMQR( 'L', 'C', M, NRHS, N, A, LDA, WORK( ITAU ), B, \$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( UNMQR ) = TIMNG( UNMQR ) + ( T2-T1 ) * * Zero out below R * IF( N.GT.1 ) \$ CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO, A( 2, 1 ), \$ LDA ) END IF * IE = 1 ITAUQ = 1 ITAUP = ITAUQ + N IWORK = ITAUP + N * * Bidiagonalize R in A * (CWorkspace: need 2*N+MM, prefer 2*N+(MM+N)*NB) * (RWorkspace: need N) * NB = ILAENV( 1, 'ZGEBRD', ' ', MM, N, -1, -1 ) OPCNT( GEBRD ) = OPCNT( GEBRD ) + \$ DOPLA( 'ZGEBRD', MM, N, 0, 0, NB ) T1 = DSECND( ) CALL ZGEBRD( MM, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ), \$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, \$ INFO ) T2 = DSECND( ) TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 ) * * Multiply B by transpose of left bidiagonalizing vectors of R * (CWorkspace: need 2*N+NRHS, prefer 2*N+NRHS*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZUNMBR', 'QLC', MM, NRHS, N, -1 ) OPCNT( UNMBR ) = OPCNT( UNMBR ) + \$ DOPLA2( 'ZUNMBR', 'QLC', MM, NRHS, N, 0, NB ) T1 = DSECND( ) CALL ZUNMBR( 'Q', 'L', 'C', MM, NRHS, N, A, LDA, WORK( ITAUQ ), \$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( UNMBR ) = TIMNG( UNMBR ) + ( T2-T1 ) * * Generate right bidiagonalizing vectors of R in A * (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZUNGBR', 'P', N, N, N, -1 ) OPCNT( UNGBR ) = OPCNT( UNGBR ) + \$ DOPLA2( 'ZUNGBR', 'P', N, N, N, 0, NB ) T1 = DSECND( ) CALL ZUNGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ), \$ WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( UNGBR ) = TIMNG( UNGBR ) + ( T2-T1 ) IRWORK = IE + N * * Perform bidiagonal QR iteration * multiply B by transpose of left singular vectors * compute right singular vectors in A * (CWorkspace: none) * (RWorkspace: need BDSPAC) * OPS = 0 T1 = DSECND( ) CALL ZBDSQR( 'U', N, N, 0, NRHS, S, RWORK( IE ), A, LDA, VDUM, \$ 1, B, LDB, RWORK( IRWORK ), INFO ) T2 = DSECND( ) 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 ) + DBLE( 1 ) THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 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 ) + DBLE( 6*NRHS+3 ) CALL ZDRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL ZLASET( 'F', 1, NRHS, CZERO, CZERO, B( I, 1 ), LDB ) END IF 10 CONTINUE * * Multiply B by right singular vectors * (CWorkspace: need N, prefer N*NRHS) * (RWorkspace: none) * IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN OPCNT( GEMM ) = OPCNT( GEMM ) + \$ DOPBL3( 'ZGEMM ', N, NRHS, N ) T1 = DSECND( ) CALL ZGEMM( 'C', 'N', N, NRHS, N, CONE, A, LDA, B, LDB, \$ CZERO, WORK, LDB ) T2 = DSECND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL ZLACPY( '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 ) + \$ DOPBL3( 'ZGEMM ', N, BL, N ) T1 = DSECND( ) CALL ZGEMM( 'C', 'N', N, BL, N, CONE, A, LDA, B( 1, I ), \$ LDB, CZERO, WORK, N ) T2 = DSECND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL ZLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB ) 20 CONTINUE ELSE OPCNT( GEMV ) = OPCNT( GEMV ) + \$ DOPBL2( 'ZGEMV ', N, N, 0, 0 ) T1 = DSECND( ) CALL ZGEMV( 'C', N, N, CONE, A, LDA, B, 1, CZERO, WORK, 1 ) T2 = DSECND( ) TIMNG( GEMV ) = TIMNG( GEMV ) + ( T2-T1 ) CALL ZCOPY( N, WORK, 1, B, 1 ) END IF * ELSE IF( N.GE.MNTHR .AND. LWORK.GE.3*M+M*M+MAX( M, NRHS, N-2*M ) ) \$ THEN * * Underdetermined case, M much less than N * * Path 2a - underdetermined, with many more columns than rows * and sufficient workspace for an efficient algorithm * LDWORK = M IF( LWORK.GE.3*M+M*LDA+MAX( M, NRHS, N-2*M ) ) \$ LDWORK = LDA ITAU = 1 IWORK = M + 1 * * Compute A=L*Q * (CWorkspace: need 2*M, prefer M+M*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 ) OPCNT( GELQF ) = OPCNT( GELQF ) + \$ DOPLA( 'ZGELQF', M, N, 0, 0, NB ) T1 = DSECND( ) CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), \$ LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( GELQF ) = TIMNG( GELQF ) + ( T2-T1 ) IL = IWORK * * Copy L to WORK(IL), zeroing out above it * CALL ZLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK ) CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO, WORK( IL+LDWORK ), \$ LDWORK ) IE = 1 ITAUQ = IL + LDWORK*M ITAUP = ITAUQ + M IWORK = ITAUP + M * * Bidiagonalize L in WORK(IL) * (CWorkspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) * (RWorkspace: need M) * NB = ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) OPCNT( GEBRD ) = OPCNT( GEBRD ) + \$ DOPLA( 'ZGEBRD', M, M, 0, 0, NB ) T1 = DSECND( ) CALL ZGEBRD( M, M, WORK( IL ), LDWORK, S, RWORK( IE ), \$ WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ), \$ LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 ) * * Multiply B by transpose of left bidiagonalizing vectors of L * (CWorkspace: need M*M+3*M+NRHS, prefer M*M+3*M+NRHS*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZUNMBR', 'QLC', M, NRHS, M, -1 ) OPCNT( UNMBR ) = OPCNT( UNMBR ) + \$ DOPLA2( 'ZUNMBR', 'QLC', M, NRHS, M, 0, NB ) T1 = DSECND( ) CALL ZUNMBR( 'Q', 'L', 'C', M, NRHS, M, WORK( IL ), LDWORK, \$ WORK( ITAUQ ), B, LDB, WORK( IWORK ), \$ LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( UNMBR ) = TIMNG( UNMBR ) + ( T2-T1 ) * * Generate right bidiagonalizing vectors of R in WORK(IL) * (CWorkspace: need M*M+4*M-1, prefer M*M+3*M+(M-1)*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZUNGBR', 'P', M, M, M, -1 ) OPCNT( UNGBR ) = OPCNT( UNGBR ) + \$ DOPLA2( 'ZUNGBR', 'P', M, M, M, 0, NB ) T1 = DSECND( ) CALL ZUNGBR( 'P', M, M, M, WORK( IL ), LDWORK, WORK( ITAUP ), \$ WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( UNGBR ) = TIMNG( UNGBR ) + ( T2-T1 ) IRWORK = IE + M * * Perform bidiagonal QR iteration, computing right singular * vectors of L in WORK(IL) and multiplying B by transpose of * left singular vectors * (CWorkspace: need M*M) * (RWorkspace: need BDSPAC) * OPS = 0 T1 = DSECND( ) CALL ZBDSQR( 'U', M, M, 0, NRHS, S, RWORK( IE ), WORK( IL ), \$ LDWORK, A, LDA, B, LDB, RWORK( IRWORK ), INFO ) T2 = DSECND( ) 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 ) + DBLE( 1 ) THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 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 ) + DBLE( 6*NRHS+3 ) CALL ZDRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL ZLASET( 'F', 1, NRHS, CZERO, CZERO, B( I, 1 ), LDB ) END IF 30 CONTINUE IWORK = IL + M*LDWORK * * Multiply B by right singular vectors of L in WORK(IL) * (CWorkspace: need M*M+2*M, prefer M*M+M+M*NRHS) * (RWorkspace: none) * IF( LWORK.GE.LDB*NRHS+IWORK-1 .AND. NRHS.GT.1 ) THEN OPCNT( GEMM ) = OPCNT( GEMM ) + \$ DOPBL3( 'ZGEMM ', M, NRHS, M ) CALL ZGEMM( 'C', 'N', M, NRHS, M, CONE, WORK( IL ), LDWORK, \$ B, LDB, CZERO, WORK( IWORK ), LDB ) CALL ZLACPY( '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 ) + \$ DOPBL3( 'ZGEMM ', M, BL, M ) T1 = DSECND( ) CALL ZGEMM( 'C', 'N', M, BL, M, CONE, WORK( IL ), LDWORK, \$ B( 1, I ), LDB, CZERO, WORK( IWORK ), M ) T2 = DSECND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL ZLACPY( 'G', M, BL, WORK( IWORK ), M, B( 1, I ), \$ LDB ) 40 CONTINUE ELSE OPCNT( GEMV ) = OPCNT( GEMV ) + \$ DOPBL2( 'ZGEMV ', M, M, 0, 0 ) T1 = DSECND( ) CALL ZGEMV( 'C', M, M, CONE, WORK( IL ), LDWORK, B( 1, 1 ), \$ 1, CZERO, WORK( IWORK ), 1 ) T2 = DSECND( ) TIMNG( GEMV ) = TIMNG( GEMV ) + ( T2-T1 ) CALL ZCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 ) END IF * * Zero out below first M rows of B * CALL ZLASET( 'F', N-M, NRHS, CZERO, CZERO, B( M+1, 1 ), LDB ) IWORK = ITAU + M * * Multiply transpose(Q) by B * (CWorkspace: need M+NRHS, prefer M+NHRS*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZUNMLQ', 'LC', N, NRHS, M, -1 ) OPCNT( UNMLQ ) = OPCNT( UNMLQ ) + \$ DOPLA( 'ZUNMLQ', N, NRHS, M, 0, NB ) T1 = DSECND( ) CALL ZUNMLQ( 'L', 'C', N, NRHS, M, A, LDA, WORK( ITAU ), B, \$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( UNMLQ ) = TIMNG( UNMLQ ) + ( T2-T1 ) * ELSE * * Path 2 - remaining underdetermined cases * IE = 1 ITAUQ = 1 ITAUP = ITAUQ + M IWORK = ITAUP + M * * Bidiagonalize A * (CWorkspace: need 3*M, prefer 2*M+(M+N)*NB) * (RWorkspace: need N) * NB = ILAENV( 1, 'ZGEBRD', ' ', M, N, -1, -1 ) OPCNT( GEBRD ) = OPCNT( GEBRD ) + \$ DOPLA( 'ZGEBRD', M, N, 0, 0, NB ) T1 = DSECND( ) CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ), \$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, \$ INFO ) T2 = DSECND( ) TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 ) * * Multiply B by transpose of left bidiagonalizing vectors * (CWorkspace: need 2*M+NRHS, prefer 2*M+NRHS*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZUNMBR', 'QLC', M, NRHS, N, -1 ) OPCNT( UNMBR ) = OPCNT( UNMBR ) + \$ DOPLA2( 'ZUNMBR', 'QLC', M, NRHS, N, 0, NB ) T1 = DSECND( ) CALL ZUNMBR( 'Q', 'L', 'C', M, NRHS, N, A, LDA, WORK( ITAUQ ), \$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( UNMBR ) = TIMNG( UNMBR ) + ( T2-T1 ) * * Generate right bidiagonalizing vectors in A * (CWorkspace: need 3*M, prefer 2*M+M*NB) * (RWorkspace: none) * NB = ILAENV( 1, 'ZUNGBR', 'P', M, N, M, -1 ) OPCNT( UNGBR ) = OPCNT( UNGBR ) + \$ DOPLA2( 'ZUNGBR', 'P', M, N, M, 0, NB ) T1 = DSECND( ) CALL ZUNGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ), \$ WORK( IWORK ), LWORK-IWORK+1, INFO ) T2 = DSECND( ) TIMNG( UNGBR ) = TIMNG( UNGBR ) + ( T2-T1 ) IRWORK = IE + M * * Perform bidiagonal QR iteration, * computing right singular vectors of A in A and * multiplying B by transpose of left singular vectors * (CWorkspace: none) * (RWorkspace: need BDSPAC) * OPS = 0 T1 = DSECND( ) CALL ZBDSQR( 'L', M, N, 0, NRHS, S, RWORK( IE ), A, LDA, VDUM, \$ 1, B, LDB, RWORK( IRWORK ), INFO ) T2 = DSECND( ) 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 ) + DBLE( 1 ) THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 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 ) + DBLE( 6*NRHS+3 ) CALL ZDRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL ZLASET( 'F', 1, NRHS, CZERO, CZERO, B( I, 1 ), LDB ) END IF 50 CONTINUE * * Multiply B by right singular vectors of A * (CWorkspace: need N, prefer N*NRHS) * (RWorkspace: none) * IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN OPCNT( GEMM ) = OPCNT( GEMM ) + \$ DOPBL3( 'ZGEMM ', N, NRHS, M ) T1 = DSECND( ) CALL ZGEMM( 'C', 'N', N, NRHS, M, CONE, A, LDA, B, LDB, \$ CZERO, WORK, LDB ) T2 = DSECND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL ZLACPY( 'G', 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 ) + \$ DOPBL3( 'ZGEMM ', N, BL, N ) T1 = DSECND( ) CALL ZGEMM( 'C', 'N', N, BL, M, CONE, A, LDA, B( 1, I ), \$ LDB, CZERO, WORK, N ) T2 = DSECND( ) TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 ) CALL ZLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB ) 60 CONTINUE ELSE OPCNT( GELSS ) = OPCNT( GELSS ) + \$ DOPBL2( 'ZGEMV ', M, N, 0, 0 ) T1 = DSECND( ) CALL ZGEMV( 'C', M, N, CONE, A, LDA, B, 1, CZERO, WORK, 1 ) T2 = DSECND( ) TIMNG( GEMV ) = TIMNG( GEMV ) + ( T2-T1 ) CALL ZCOPY( N, WORK, 1, B, 1 ) END IF END IF * * Undo scaling * IF( IASCL.EQ.1 ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 6*( N*NRHS+MINMN ) ) 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 OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 6*( N*NRHS+MINMN ) ) 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 OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 6*N*NRHS ) CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN OPCNT( GELSS ) = OPCNT( GELSS ) + DBLE( 6*N*NRHS ) CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF 70 CONTINUE WORK( 1 ) = MAXWRK RETURN * * End of ZGELSS * END