SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, \$ WORK, LWORK, RWORK, IWORK, INFO ) * * -- LAPACK driver routine (version 3.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * June 30, 1999 * 8-15-00: Improve consistency of WS calculations (eca) * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK DOUBLE PRECISION RCOND * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION RWORK( * ), S( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ) * .. * * Purpose * ======= * * ZGELSD 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 tranformations, 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. * * 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. * The exact minimum amount of workspace needed depends on M, * N and NRHS. * If M >= N, LWORK >= 2*N + N*NRHS. * If M < N, LWORK >= 2*M + M*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 (LRWORK) * If M >= N, LRWORK >= 8*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS. * If M < N, LRWORK >= 8*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS. * 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 = INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 * * IWORK (workspace) INTEGER array, dimension (LIWORK) * LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, * where MINMN = 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. * * Further Details * =============== * * Based on contributions by * Ming Gu and Ren-Cang Li, Computer Science Division, University of * California at Berkeley, USA * Osni Marques, LBNL/NERSC, USA * * ===================================================================== * * .. Parameters .. INTEGER LQUERV PARAMETER ( LQUERV = -1 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) COMPLEX*16 CZERO PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ, \$ LDWORK, MAXMN, MAXWRK, MINMN, MINWRK, MM, \$ MNTHR, NRWORK, NWORK, SMLSIZ DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM * .. * .. External Subroutines .. EXTERNAL DLABAD, DLASCL, DLASET, XERBLA, ZGEBRD, ZGELQF, \$ ZGEQRF, ZLACPY, ZLALSD, ZLASCL, ZLASET, ZUNMBR, \$ ZUNMLQ, ZUNMQR * .. * .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH, ZLANGE EXTERNAL ILAENV, DLAMCH, ZLANGE * .. * .. Intrinsic Functions .. INTRINSIC DCMPLX, MAX, MIN * .. * .. Executable Statements .. * * Test the input arguments. * INFO = 0 MINMN = MIN( M, N ) MAXMN = MAX( M, N ) MNTHR = ILAENV( 6, 'ZGELSD', ' ', M, N, NRHS, -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, 'ZGELSD', ' ', 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 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*ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, \$ -1 ) ) MAXWRK = MAX( MAXWRK, NRHS*ILAENV( 1, 'ZUNMQR', 'LC', M, \$ NRHS, N, -1 ) ) END IF IF( M.GE.N ) THEN * * Path 1 - overdetermined or exactly determined. * 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, 'ZUNMBR', '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 IF( N.GE.MNTHR ) THEN * * Path 2a - underdetermined, with many more columns * than rows. * MAXWRK = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 ) MAXWRK = MAX( MAXWRK, M*M+4*M+2*M* \$ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS* \$ ILAENV( 1, 'ZUNMBR', 'QLC', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )* \$ ILAENV( 1, 'ZUNMLQ', '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 ) ELSE * * Path 2 - underdetermined. * 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, 'ZUNMBR', '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 MINWRK = MIN( MINWRK, MAXWRK ) WORK( 1 ) = DCMPLX( MAXWRK, 0 ) IF( LWORK.LT.MINWRK .AND. LWORK.NE.LQUERV ) \$ INFO = -12 END IF * * Quick returns * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGELSD', -INFO ) RETURN END IF IF( LWORK.EQ.LQUERV ) \$ RETURN 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 = ZLANGE( 'M', M, N, A, LDA, RWORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM * 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. * 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, 1 ) RANK = 0 GO TO 10 END IF * * Scale B if max entry 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. * 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. * CALL ZLASCL( '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 ZLASET( '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 ZGEQRF( 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 ZUNMQR( '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 ZLASET( '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 ZGEBRD( 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 ZUNMBR( '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 ZLALSD( '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 ZUNMBR( '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 ZGELQF( 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 ZLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK ) CALL ZLASET( '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 ZGEBRD( 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 ZUNMBR( '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 ZLALSD( '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 ZUNMBR( '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 ZLASET( '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 ZUNMLQ( '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 ZGEBRD( 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 ZUNMBR( '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 ZLALSD( '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 ZUNMBR( '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 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 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 CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF * 10 CONTINUE WORK( 1 ) = DCMPLX( MAXWRK, 0 ) RETURN * * End of ZGELSD * END