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Diffstat (limited to 'SRC/sgesdd.f')
| -rw-r--r-- | SRC/sgesdd.f | 1339 |
1 files changed, 1339 insertions, 0 deletions
diff --git a/SRC/sgesdd.f b/SRC/sgesdd.f new file mode 100644 index 00000000..de3683d8 --- /dev/null +++ b/SRC/sgesdd.f @@ -0,0 +1,1339 @@ + SUBROUTINE SGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, + $ LWORK, IWORK, INFO ) +* +* -- LAPACK driver routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* .. Scalar Arguments .. + CHARACTER JOBZ + INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N +* .. +* .. Array Arguments .. + INTEGER IWORK( * ) + REAL A( LDA, * ), S( * ), U( LDU, * ), + $ VT( LDVT, * ), WORK( * ) +* .. +* +* Purpose +* ======= +* +* SGESDD computes the singular value decomposition (SVD) of a real +* M-by-N matrix A, optionally computing the left and right singular +* vectors. If singular vectors are desired, it uses a +* divide-and-conquer algorithm. +* +* The SVD is written +* +* A = U * SIGMA * transpose(V) +* +* where SIGMA is an M-by-N matrix which is zero except for its +* min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and +* V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA +* are the singular values of A; they are real and non-negative, and +* are returned in descending order. The first min(m,n) columns of +* U and V are the left and right singular vectors of A. +* +* Note that the routine returns VT = V**T, not V. +* +* 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 +* ========= +* +* JOBZ (input) CHARACTER*1 +* Specifies options for computing all or part of the matrix U: +* = 'A': all M columns of U and all N rows of V**T are +* returned in the arrays U and VT; +* = 'S': the first min(M,N) columns of U and the first +* min(M,N) rows of V**T are returned in the arrays U +* and VT; +* = 'O': If M >= N, the first N columns of U are overwritten +* on the array A and all rows of V**T are returned in +* the array VT; +* otherwise, all columns of U are returned in the +* array U and the first M rows of V**T are overwritten +* in the array A; +* = 'N': no columns of U or rows of V**T are computed. +* +* M (input) INTEGER +* The number of rows of the input matrix A. M >= 0. +* +* N (input) INTEGER +* The number of columns of the input matrix A. N >= 0. +* +* A (input/output) REAL array, dimension (LDA,N) +* On entry, the M-by-N matrix A. +* On exit, +* if JOBZ = 'O', A is overwritten with the first N columns +* of U (the left singular vectors, stored +* columnwise) if M >= N; +* A is overwritten with the first M rows +* of V**T (the right singular vectors, stored +* rowwise) otherwise. +* if JOBZ .ne. 'O', the contents of A are destroyed. +* +* LDA (input) INTEGER +* The leading dimension of the array A. LDA >= max(1,M). +* +* S (output) REAL array, dimension (min(M,N)) +* The singular values of A, sorted so that S(i) >= S(i+1). +* +* U (output) REAL array, dimension (LDU,UCOL) +* UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N; +* UCOL = min(M,N) if JOBZ = 'S'. +* If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M +* orthogonal matrix U; +* if JOBZ = 'S', U contains the first min(M,N) columns of U +* (the left singular vectors, stored columnwise); +* if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced. +* +* LDU (input) INTEGER +* The leading dimension of the array U. LDU >= 1; if +* JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M. +* +* VT (output) REAL array, dimension (LDVT,N) +* If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the +* N-by-N orthogonal matrix V**T; +* if JOBZ = 'S', VT contains the first min(M,N) rows of +* V**T (the right singular vectors, stored rowwise); +* if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced. +* +* LDVT (input) INTEGER +* The leading dimension of the array VT. LDVT >= 1; if +* JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N; +* if JOBZ = 'S', LDVT >= min(M,N). +* +* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) +* On exit, if INFO = 0, WORK(1) returns the optimal LWORK; +* +* LWORK (input) INTEGER +* The dimension of the array WORK. LWORK >= 1. +* If JOBZ = 'N', +* LWORK >= 3*min(M,N) + max(max(M,N),6*min(M,N)). +* If JOBZ = 'O', +* LWORK >= 3*min(M,N)*min(M,N) + +* max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)). +* If JOBZ = 'S' or 'A' +* LWORK >= 3*min(M,N)*min(M,N) + +* max(max(M,N),4*min(M,N)*min(M,N)+4*min(M,N)). +* For good performance, LWORK should generally be larger. +* If LWORK = -1 but other input arguments are legal, WORK(1) +* returns the optimal LWORK. +* +* IWORK (workspace) INTEGER array, dimension (8*min(M,N)) +* +* INFO (output) INTEGER +* = 0: successful exit. +* < 0: if INFO = -i, the i-th argument had an illegal value. +* > 0: SBDSDC did not converge, updating process failed. +* +* Further Details +* =============== +* +* Based on contributions by +* Ming Gu and Huan Ren, Computer Science Division, University of +* California at Berkeley, USA +* +* ===================================================================== +* +* .. Parameters .. + REAL ZERO, ONE + PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) +* .. +* .. Local Scalars .. + LOGICAL LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS + INTEGER BDSPAC, BLK, CHUNK, I, IE, IERR, IL, + $ IR, ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT, + $ LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK, + $ MNTHR, NWORK, WRKBL + REAL ANRM, BIGNUM, EPS, SMLNUM +* .. +* .. Local Arrays .. + INTEGER IDUM( 1 ) + REAL DUM( 1 ) +* .. +* .. External Subroutines .. + EXTERNAL SBDSDC, SGEBRD, SGELQF, SGEMM, SGEQRF, SLACPY, + $ SLASCL, SLASET, SORGBR, SORGLQ, SORGQR, SORMBR, + $ XERBLA +* .. +* .. External Functions .. + LOGICAL LSAME + INTEGER ILAENV + REAL SLAMCH, SLANGE + EXTERNAL ILAENV, LSAME, SLAMCH, SLANGE +* .. +* .. Intrinsic Functions .. + INTRINSIC INT, MAX, MIN, SQRT +* .. +* .. Executable Statements .. +* +* Test the input arguments +* + INFO = 0 + MINMN = MIN( M, N ) + WNTQA = LSAME( JOBZ, 'A' ) + WNTQS = LSAME( JOBZ, 'S' ) + WNTQAS = WNTQA .OR. WNTQS + WNTQO = LSAME( JOBZ, 'O' ) + WNTQN = LSAME( JOBZ, 'N' ) + LQUERY = ( LWORK.EQ.-1 ) +* + IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN + INFO = -1 + ELSE IF( M.LT.0 ) THEN + INFO = -2 + ELSE IF( N.LT.0 ) THEN + INFO = -3 + ELSE IF( LDA.LT.MAX( 1, M ) ) THEN + INFO = -5 + ELSE IF( LDU.LT.1 .OR. ( WNTQAS .AND. LDU.LT.M ) .OR. + $ ( WNTQO .AND. M.LT.N .AND. LDU.LT.M ) ) THEN + INFO = -8 + ELSE IF( LDVT.LT.1 .OR. ( WNTQA .AND. LDVT.LT.N ) .OR. + $ ( WNTQS .AND. LDVT.LT.MINMN ) .OR. + $ ( WNTQO .AND. M.GE.N .AND. LDVT.LT.N ) ) THEN + INFO = -10 + 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( M.GE.N .AND. MINMN.GT.0 ) THEN +* +* Compute space needed for SBDSDC +* + MNTHR = INT( MINMN*11.0E0 / 6.0E0 ) + IF( WNTQN ) THEN + BDSPAC = 7*N + ELSE + BDSPAC = 3*N*N + 4*N + END IF + IF( M.GE.MNTHR ) THEN + IF( WNTQN ) THEN +* +* Path 1 (M much larger than N, JOBZ='N') +* + WRKBL = N + N*ILAENV( 1, 'SGEQRF', ' ', M, N, -1, + $ -1 ) + WRKBL = MAX( WRKBL, 3*N+2*N* + $ ILAENV( 1, 'SGEBRD', ' ', N, N, -1, -1 ) ) + MAXWRK = MAX( WRKBL, BDSPAC+N ) + MINWRK = BDSPAC + N + ELSE IF( WNTQO ) THEN +* +* Path 2 (M much larger than N, JOBZ='O') +* + WRKBL = N + N*ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 ) + WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'SORGQR', ' ', M, + $ N, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+2*N* + $ ILAENV( 1, 'SGEBRD', ' ', N, N, -1, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'QLN', N, N, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) ) + WRKBL = MAX( WRKBL, BDSPAC+3*N ) + MAXWRK = WRKBL + 2*N*N + MINWRK = BDSPAC + 2*N*N + 3*N + ELSE IF( WNTQS ) THEN +* +* Path 3 (M much larger than N, JOBZ='S') +* + WRKBL = N + N*ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 ) + WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'SORGQR', ' ', M, + $ N, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+2*N* + $ ILAENV( 1, 'SGEBRD', ' ', N, N, -1, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'QLN', N, N, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) ) + WRKBL = MAX( WRKBL, BDSPAC+3*N ) + MAXWRK = WRKBL + N*N + MINWRK = BDSPAC + N*N + 3*N + ELSE IF( WNTQA ) THEN +* +* Path 4 (M much larger than N, JOBZ='A') +* + WRKBL = N + N*ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 ) + WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'SORGQR', ' ', M, + $ M, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+2*N* + $ ILAENV( 1, 'SGEBRD', ' ', N, N, -1, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'QLN', N, N, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) ) + WRKBL = MAX( WRKBL, BDSPAC+3*N ) + MAXWRK = WRKBL + N*N + MINWRK = BDSPAC + N*N + 3*N + END IF + ELSE +* +* Path 5 (M at least N, but not much larger) +* + WRKBL = 3*N + ( M+N )*ILAENV( 1, 'SGEBRD', ' ', M, N, -1, + $ -1 ) + IF( WNTQN ) THEN + MAXWRK = MAX( WRKBL, BDSPAC+3*N ) + MINWRK = 3*N + MAX( M, BDSPAC ) + ELSE IF( WNTQO ) THEN + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'QLN', M, N, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) ) + WRKBL = MAX( WRKBL, BDSPAC+3*N ) + MAXWRK = WRKBL + M*N + MINWRK = 3*N + MAX( M, N*N+BDSPAC ) + ELSE IF( WNTQS ) THEN + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'QLN', M, N, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) ) + MAXWRK = MAX( WRKBL, BDSPAC+3*N ) + MINWRK = 3*N + MAX( M, BDSPAC ) + ELSE IF( WNTQA ) THEN + WRKBL = MAX( WRKBL, 3*N+M* + $ ILAENV( 1, 'SORMBR', 'QLN', M, M, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*N+N* + $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) ) + MAXWRK = MAX( MAXWRK, BDSPAC+3*N ) + MINWRK = 3*N + MAX( M, BDSPAC ) + END IF + END IF + ELSE IF ( MINMN.GT.0 ) THEN +* +* Compute space needed for SBDSDC +* + MNTHR = INT( MINMN*11.0E0 / 6.0E0 ) + IF( WNTQN ) THEN + BDSPAC = 7*M + ELSE + BDSPAC = 3*M*M + 4*M + END IF + IF( N.GE.MNTHR ) THEN + IF( WNTQN ) THEN +* +* Path 1t (N much larger than M, JOBZ='N') +* + WRKBL = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, + $ -1 ) + WRKBL = MAX( WRKBL, 3*M+2*M* + $ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) ) + MAXWRK = MAX( WRKBL, BDSPAC+M ) + MINWRK = BDSPAC + M + ELSE IF( WNTQO ) THEN +* +* Path 2t (N much larger than M, JOBZ='O') +* + WRKBL = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 ) + WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'SORGLQ', ' ', M, + $ N, M, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+2*M* + $ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'QLN', M, M, M, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'PRT', M, M, M, -1 ) ) + WRKBL = MAX( WRKBL, BDSPAC+3*M ) + MAXWRK = WRKBL + 2*M*M + MINWRK = BDSPAC + 2*M*M + 3*M + ELSE IF( WNTQS ) THEN +* +* Path 3t (N much larger than M, JOBZ='S') +* + WRKBL = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 ) + WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'SORGLQ', ' ', M, + $ N, M, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+2*M* + $ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'QLN', M, M, M, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'PRT', M, M, M, -1 ) ) + WRKBL = MAX( WRKBL, BDSPAC+3*M ) + MAXWRK = WRKBL + M*M + MINWRK = BDSPAC + M*M + 3*M + ELSE IF( WNTQA ) THEN +* +* Path 4t (N much larger than M, JOBZ='A') +* + WRKBL = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 ) + WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'SORGLQ', ' ', N, + $ N, M, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+2*M* + $ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'QLN', M, M, M, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'PRT', M, M, M, -1 ) ) + WRKBL = MAX( WRKBL, BDSPAC+3*M ) + MAXWRK = WRKBL + M*M + MINWRK = BDSPAC + M*M + 3*M + END IF + ELSE +* +* Path 5t (N greater than M, but not much larger) +* + WRKBL = 3*M + ( M+N )*ILAENV( 1, 'SGEBRD', ' ', M, N, -1, + $ -1 ) + IF( WNTQN ) THEN + MAXWRK = MAX( WRKBL, BDSPAC+3*M ) + MINWRK = 3*M + MAX( N, BDSPAC ) + ELSE IF( WNTQO ) THEN + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'QLN', M, M, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'PRT', M, N, M, -1 ) ) + WRKBL = MAX( WRKBL, BDSPAC+3*M ) + MAXWRK = WRKBL + M*N + MINWRK = 3*M + MAX( N, M*M+BDSPAC ) + ELSE IF( WNTQS ) THEN + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'QLN', M, M, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'PRT', M, N, M, -1 ) ) + MAXWRK = MAX( WRKBL, BDSPAC+3*M ) + MINWRK = 3*M + MAX( N, BDSPAC ) + ELSE IF( WNTQA ) THEN + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'QLN', M, M, N, -1 ) ) + WRKBL = MAX( WRKBL, 3*M+M* + $ ILAENV( 1, 'SORMBR', 'PRT', N, N, M, -1 ) ) + MAXWRK = MAX( WRKBL, BDSPAC+3*M ) + MINWRK = 3*M + MAX( N, BDSPAC ) + END IF + END IF + END IF + MAXWRK = MAX( MAXWRK, MINWRK ) + WORK( 1 ) = MAXWRK +* + IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN + INFO = -12 + END IF + END IF +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'SGESDD', -INFO ) + RETURN + ELSE IF( LQUERY ) THEN + RETURN + END IF +* +* Quick return if possible +* + IF( M.EQ.0 .OR. N.EQ.0 ) THEN + RETURN + END IF +* +* Get machine constants +* + EPS = SLAMCH( 'P' ) + SMLNUM = SQRT( SLAMCH( 'S' ) ) / EPS + BIGNUM = ONE / SMLNUM +* +* Scale A if max element outside range [SMLNUM,BIGNUM] +* + ANRM = SLANGE( 'M', M, N, A, LDA, DUM ) + ISCL = 0 + IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN + ISCL = 1 + CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR ) + ELSE IF( ANRM.GT.BIGNUM ) THEN + ISCL = 1 + CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR ) + END IF +* + IF( M.GE.N ) THEN +* +* A has at least as many rows as columns. If A has sufficiently +* more rows than columns, first reduce using the QR +* decomposition (if sufficient workspace available) +* + IF( M.GE.MNTHR ) THEN +* + IF( WNTQN ) THEN +* +* Path 1 (M much larger than N, JOBZ='N') +* No singular vectors to be computed +* + ITAU = 1 + NWORK = ITAU + N +* +* Compute A=Q*R +* (Workspace: need 2*N, prefer N+N*NB) +* + CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Zero out below R +* + CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) + IE = 1 + ITAUQ = IE + N + ITAUP = ITAUQ + N + NWORK = ITAUP + N +* +* Bidiagonalize R in A +* (Workspace: need 4*N, prefer 3*N+2*N*NB) +* + CALL SGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), + $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, + $ IERR ) + NWORK = IE + N +* +* Perform bidiagonal SVD, computing singular values only +* (Workspace: need N+BDSPAC) +* + CALL SBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1, + $ DUM, IDUM, WORK( NWORK ), IWORK, INFO ) +* + ELSE IF( WNTQO ) THEN +* +* Path 2 (M much larger than N, JOBZ = 'O') +* N left singular vectors to be overwritten on A and +* N right singular vectors to be computed in VT +* + IR = 1 +* +* WORK(IR) is LDWRKR by N +* + IF( LWORK.GE.LDA*N+N*N+3*N+BDSPAC ) THEN + LDWRKR = LDA + ELSE + LDWRKR = ( LWORK-N*N-3*N-BDSPAC ) / N + END IF + ITAU = IR + LDWRKR*N + NWORK = ITAU + N +* +* Compute A=Q*R +* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) +* + CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Copy R to WORK(IR), zeroing out below it +* + CALL SLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR ) + CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ), + $ LDWRKR ) +* +* Generate Q in A +* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) +* + CALL SORGQR( M, N, N, A, LDA, WORK( ITAU ), + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) + IE = ITAU + ITAUQ = IE + N + ITAUP = ITAUQ + N + NWORK = ITAUP + N +* +* Bidiagonalize R in VT, copying result to WORK(IR) +* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) +* + CALL SGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ), + $ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* WORK(IU) is N by N +* + IU = NWORK + NWORK = IU + N*N +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in WORK(IU) and computing right +* singular vectors of bidiagonal matrix in VT +* (Workspace: need N+N*N+BDSPAC) +* + CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N, + $ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK, + $ INFO ) +* +* Overwrite WORK(IU) by left singular vectors of R +* and VT by right singular vectors of R +* (Workspace: need 2*N*N+3*N, prefer 2*N*N+2*N+N*NB) +* + CALL SORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR, + $ WORK( ITAUQ ), WORK( IU ), N, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR, + $ WORK( ITAUP ), VT, LDVT, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Multiply Q in A by left singular vectors of R in +* WORK(IU), storing result in WORK(IR) and copying to A +* (Workspace: need 2*N*N, prefer N*N+M*N) +* + DO 10 I = 1, M, LDWRKR + CHUNK = MIN( M-I+1, LDWRKR ) + CALL SGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ), + $ LDA, WORK( IU ), N, ZERO, WORK( IR ), + $ LDWRKR ) + CALL SLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR, + $ A( I, 1 ), LDA ) + 10 CONTINUE +* + ELSE IF( WNTQS ) THEN +* +* Path 3 (M much larger than N, JOBZ='S') +* N left singular vectors to be computed in U and +* N right singular vectors to be computed in VT +* + IR = 1 +* +* WORK(IR) is N by N +* + LDWRKR = N + ITAU = IR + LDWRKR*N + NWORK = ITAU + N +* +* Compute A=Q*R +* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) +* + CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Copy R to WORK(IR), zeroing out below it +* + CALL SLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR ) + CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ), + $ LDWRKR ) +* +* Generate Q in A +* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) +* + CALL SORGQR( M, N, N, A, LDA, WORK( ITAU ), + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) + IE = ITAU + ITAUQ = IE + N + ITAUP = ITAUQ + N + NWORK = ITAUP + N +* +* Bidiagonalize R in WORK(IR) +* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) +* + CALL SGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ), + $ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagoal matrix in U and computing right singular +* vectors of bidiagonal matrix in VT +* (Workspace: need N+BDSPAC) +* + CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT, + $ LDVT, DUM, IDUM, WORK( NWORK ), IWORK, + $ INFO ) +* +* Overwrite U by left singular vectors of R and VT +* by right singular vectors of R +* (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB) +* + CALL SORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR, + $ WORK( ITAUQ ), U, LDU, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* + CALL SORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR, + $ WORK( ITAUP ), VT, LDVT, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Multiply Q in A by left singular vectors of R in +* WORK(IR), storing result in U +* (Workspace: need N*N) +* + CALL SLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR ) + CALL SGEMM( 'N', 'N', M, N, N, ONE, A, LDA, WORK( IR ), + $ LDWRKR, ZERO, U, LDU ) +* + ELSE IF( WNTQA ) THEN +* +* Path 4 (M much larger than N, JOBZ='A') +* M left singular vectors to be computed in U and +* N right singular vectors to be computed in VT +* + IU = 1 +* +* WORK(IU) is N by N +* + LDWRKU = N + ITAU = IU + LDWRKU*N + NWORK = ITAU + N +* +* Compute A=Q*R, copying result to U +* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) +* + CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SLACPY( 'L', M, N, A, LDA, U, LDU ) +* +* Generate Q in U +* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) + CALL SORGQR( M, M, N, U, LDU, WORK( ITAU ), + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) +* +* Produce R in A, zeroing out other entries +* + CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) + IE = ITAU + ITAUQ = IE + N + ITAUP = ITAUQ + N + NWORK = ITAUP + N +* +* Bidiagonalize R in A +* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) +* + CALL SGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), + $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, + $ IERR ) +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in WORK(IU) and computing right +* singular vectors of bidiagonal matrix in VT +* (Workspace: need N+N*N+BDSPAC) +* + CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N, + $ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK, + $ INFO ) +* +* Overwrite WORK(IU) by left singular vectors of R and VT +* by right singular vectors of R +* (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB) +* + CALL SORMBR( 'Q', 'L', 'N', N, N, N, A, LDA, + $ WORK( ITAUQ ), WORK( IU ), LDWRKU, + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) + CALL SORMBR( 'P', 'R', 'T', N, N, N, A, LDA, + $ WORK( ITAUP ), VT, LDVT, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Multiply Q in U by left singular vectors of R in +* WORK(IU), storing result in A +* (Workspace: need N*N) +* + CALL SGEMM( 'N', 'N', M, N, N, ONE, U, LDU, WORK( IU ), + $ LDWRKU, ZERO, A, LDA ) +* +* Copy left singular vectors of A from A to U +* + CALL SLACPY( 'F', M, N, A, LDA, U, LDU ) +* + END IF +* + ELSE +* +* M .LT. MNTHR +* +* Path 5 (M at least N, but not much larger) +* Reduce to bidiagonal form without QR decomposition +* + IE = 1 + ITAUQ = IE + N + ITAUP = ITAUQ + N + NWORK = ITAUP + N +* +* Bidiagonalize A +* (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB) +* + CALL SGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), + $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, + $ IERR ) + IF( WNTQN ) THEN +* +* Perform bidiagonal SVD, only computing singular values +* (Workspace: need N+BDSPAC) +* + CALL SBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1, + $ DUM, IDUM, WORK( NWORK ), IWORK, INFO ) + ELSE IF( WNTQO ) THEN + IU = NWORK + IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN +* +* WORK( IU ) is M by N +* + LDWRKU = M + NWORK = IU + LDWRKU*N + CALL SLASET( 'F', M, N, ZERO, ZERO, WORK( IU ), + $ LDWRKU ) + ELSE +* +* WORK( IU ) is N by N +* + LDWRKU = N + NWORK = IU + LDWRKU*N +* +* WORK(IR) is LDWRKR by N +* + IR = NWORK + LDWRKR = ( LWORK-N*N-3*N ) / N + END IF + NWORK = IU + LDWRKU*N +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in WORK(IU) and computing right +* singular vectors of bidiagonal matrix in VT +* (Workspace: need N+N*N+BDSPAC) +* + CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), + $ LDWRKU, VT, LDVT, DUM, IDUM, WORK( NWORK ), + $ IWORK, INFO ) +* +* Overwrite VT by right singular vectors of A +* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) +* + CALL SORMBR( 'P', 'R', 'T', N, N, N, A, LDA, + $ WORK( ITAUP ), VT, LDVT, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* + IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN +* +* Overwrite WORK(IU) by left singular vectors of A +* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) +* + CALL SORMBR( 'Q', 'L', 'N', M, N, N, A, LDA, + $ WORK( ITAUQ ), WORK( IU ), LDWRKU, + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) +* +* Copy left singular vectors of A from WORK(IU) to A +* + CALL SLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA ) + ELSE +* +* Generate Q in A +* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) +* + CALL SORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ), + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) +* +* Multiply Q in A by left singular vectors of +* bidiagonal matrix in WORK(IU), storing result in +* WORK(IR) and copying to A +* (Workspace: need 2*N*N, prefer N*N+M*N) +* + DO 20 I = 1, M, LDWRKR + CHUNK = MIN( M-I+1, LDWRKR ) + CALL SGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ), + $ LDA, WORK( IU ), LDWRKU, ZERO, + $ WORK( IR ), LDWRKR ) + CALL SLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR, + $ A( I, 1 ), LDA ) + 20 CONTINUE + END IF +* + ELSE IF( WNTQS ) THEN +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in U and computing right singular +* vectors of bidiagonal matrix in VT +* (Workspace: need N+BDSPAC) +* + CALL SLASET( 'F', M, N, ZERO, ZERO, U, LDU ) + CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT, + $ LDVT, DUM, IDUM, WORK( NWORK ), IWORK, + $ INFO ) +* +* Overwrite U by left singular vectors of A and VT +* by right singular vectors of A +* (Workspace: need 3*N, prefer 2*N+N*NB) +* + CALL SORMBR( 'Q', 'L', 'N', M, N, N, A, LDA, + $ WORK( ITAUQ ), U, LDU, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SORMBR( 'P', 'R', 'T', N, N, N, A, LDA, + $ WORK( ITAUP ), VT, LDVT, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + ELSE IF( WNTQA ) THEN +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in U and computing right singular +* vectors of bidiagonal matrix in VT +* (Workspace: need N+BDSPAC) +* + CALL SLASET( 'F', M, M, ZERO, ZERO, U, LDU ) + CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT, + $ LDVT, DUM, IDUM, WORK( NWORK ), IWORK, + $ INFO ) +* +* Set the right corner of U to identity matrix +* + IF( M.GT.N ) THEN + CALL SLASET( 'F', M-N, M-N, ZERO, ONE, U( N+1, N+1 ), + $ LDU ) + END IF +* +* Overwrite U by left singular vectors of A and VT +* by right singular vectors of A +* (Workspace: need N*N+2*N+M, prefer N*N+2*N+M*NB) +* + CALL SORMBR( 'Q', 'L', 'N', M, M, N, A, LDA, + $ WORK( ITAUQ ), U, LDU, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SORMBR( 'P', 'R', 'T', N, N, M, A, LDA, + $ WORK( ITAUP ), VT, LDVT, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + END IF +* + END IF +* + ELSE +* +* A has more columns than rows. If A has sufficiently more +* columns than rows, first reduce using the LQ decomposition (if +* sufficient workspace available) +* + IF( N.GE.MNTHR ) THEN +* + IF( WNTQN ) THEN +* +* Path 1t (N much larger than M, JOBZ='N') +* No singular vectors to be computed +* + ITAU = 1 + NWORK = ITAU + M +* +* Compute A=L*Q +* (Workspace: need 2*M, prefer M+M*NB) +* + CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Zero out above L +* + CALL SLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA ) + IE = 1 + ITAUQ = IE + M + ITAUP = ITAUQ + M + NWORK = ITAUP + M +* +* Bidiagonalize L in A +* (Workspace: need 4*M, prefer 3*M+2*M*NB) +* + CALL SGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ), + $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, + $ IERR ) + NWORK = IE + M +* +* Perform bidiagonal SVD, computing singular values only +* (Workspace: need M+BDSPAC) +* + CALL SBDSDC( 'U', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1, + $ DUM, IDUM, WORK( NWORK ), IWORK, INFO ) +* + ELSE IF( WNTQO ) THEN +* +* Path 2t (N much larger than M, JOBZ='O') +* M right singular vectors to be overwritten on A and +* M left singular vectors to be computed in U +* + IVT = 1 +* +* IVT is M by M +* + IL = IVT + M*M + IF( LWORK.GE.M*N+M*M+3*M+BDSPAC ) THEN +* +* WORK(IL) is M by N +* + LDWRKL = M + CHUNK = N + ELSE + LDWRKL = M + CHUNK = ( LWORK-M*M ) / M + END IF + ITAU = IL + LDWRKL*M + NWORK = ITAU + M +* +* Compute A=L*Q +* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) +* + CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Copy L to WORK(IL), zeroing about above it +* + CALL SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL ) + CALL SLASET( 'U', M-1, M-1, ZERO, ZERO, + $ WORK( IL+LDWRKL ), LDWRKL ) +* +* Generate Q in A +* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) +* + CALL SORGLQ( M, N, M, A, LDA, WORK( ITAU ), + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) + IE = ITAU + ITAUQ = IE + M + ITAUP = ITAUQ + M + NWORK = ITAUP + M +* +* Bidiagonalize L in WORK(IL) +* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) +* + CALL SGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ), + $ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in U, and computing right singular +* vectors of bidiagonal matrix in WORK(IVT) +* (Workspace: need M+M*M+BDSPAC) +* + CALL SBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, + $ WORK( IVT ), M, DUM, IDUM, WORK( NWORK ), + $ IWORK, INFO ) +* +* Overwrite U by left singular vectors of L and WORK(IVT) +* by right singular vectors of L +* (Workspace: need 2*M*M+3*M, prefer 2*M*M+2*M+M*NB) +* + CALL SORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL, + $ WORK( ITAUQ ), U, LDU, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL, + $ WORK( ITAUP ), WORK( IVT ), M, + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) +* +* Multiply right singular vectors of L in WORK(IVT) by Q +* in A, storing result in WORK(IL) and copying to A +* (Workspace: need 2*M*M, prefer M*M+M*N) +* + DO 30 I = 1, N, CHUNK + BLK = MIN( N-I+1, CHUNK ) + CALL SGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ), M, + $ A( 1, I ), LDA, ZERO, WORK( IL ), LDWRKL ) + CALL SLACPY( 'F', M, BLK, WORK( IL ), LDWRKL, + $ A( 1, I ), LDA ) + 30 CONTINUE +* + ELSE IF( WNTQS ) THEN +* +* Path 3t (N much larger than M, JOBZ='S') +* M right singular vectors to be computed in VT and +* M left singular vectors to be computed in U +* + IL = 1 +* +* WORK(IL) is M by M +* + LDWRKL = M + ITAU = IL + LDWRKL*M + NWORK = ITAU + M +* +* Compute A=L*Q +* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) +* + CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Copy L to WORK(IL), zeroing out above it +* + CALL SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL ) + CALL SLASET( 'U', M-1, M-1, ZERO, ZERO, + $ WORK( IL+LDWRKL ), LDWRKL ) +* +* Generate Q in A +* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) +* + CALL SORGLQ( M, N, M, A, LDA, WORK( ITAU ), + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) + IE = ITAU + ITAUQ = IE + M + ITAUP = ITAUQ + M + NWORK = ITAUP + M +* +* Bidiagonalize L in WORK(IU), copying result to U +* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) +* + CALL SGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ), + $ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in U and computing right singular +* vectors of bidiagonal matrix in VT +* (Workspace: need M+BDSPAC) +* + CALL SBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, VT, + $ LDVT, DUM, IDUM, WORK( NWORK ), IWORK, + $ INFO ) +* +* Overwrite U by left singular vectors of L and VT +* by right singular vectors of L +* (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB) +* + CALL SORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL, + $ WORK( ITAUQ ), U, LDU, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL, + $ WORK( ITAUP ), VT, LDVT, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* +* Multiply right singular vectors of L in WORK(IL) by +* Q in A, storing result in VT +* (Workspace: need M*M) +* + CALL SLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL ) + CALL SGEMM( 'N', 'N', M, N, M, ONE, WORK( IL ), LDWRKL, + $ A, LDA, ZERO, VT, LDVT ) +* + ELSE IF( WNTQA ) THEN +* +* Path 4t (N much larger than M, JOBZ='A') +* N right singular vectors to be computed in VT and +* M left singular vectors to be computed in U +* + IVT = 1 +* +* WORK(IVT) is M by M +* + LDWKVT = M + ITAU = IVT + LDWKVT*M + NWORK = ITAU + M +* +* Compute A=L*Q, copying result to VT +* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) +* + CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SLACPY( 'U', M, N, A, LDA, VT, LDVT ) +* +* Generate Q in VT +* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) +* + CALL SORGLQ( N, N, M, VT, LDVT, WORK( ITAU ), + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) +* +* Produce L in A, zeroing out other entries +* + CALL SLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA ) + IE = ITAU + ITAUQ = IE + M + ITAUP = ITAUQ + M + NWORK = ITAUP + M +* +* Bidiagonalize L in A +* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) +* + CALL SGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ), + $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, + $ IERR ) +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in U and computing right singular +* vectors of bidiagonal matrix in WORK(IVT) +* (Workspace: need M+M*M+BDSPAC) +* + CALL SBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, + $ WORK( IVT ), LDWKVT, DUM, IDUM, + $ WORK( NWORK ), IWORK, INFO ) +* +* Overwrite U by left singular vectors of L and WORK(IVT) +* by right singular vectors of L +* (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB) +* + CALL SORMBR( 'Q', 'L', 'N', M, M, M, A, LDA, + $ WORK( ITAUQ ), U, LDU, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SORMBR( 'P', 'R', 'T', M, M, M, A, LDA, + $ WORK( ITAUP ), WORK( IVT ), LDWKVT, + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) +* +* Multiply right singular vectors of L in WORK(IVT) by +* Q in VT, storing result in A +* (Workspace: need M*M) +* + CALL SGEMM( 'N', 'N', M, N, M, ONE, WORK( IVT ), LDWKVT, + $ VT, LDVT, ZERO, A, LDA ) +* +* Copy right singular vectors of A from A to VT +* + CALL SLACPY( 'F', M, N, A, LDA, VT, LDVT ) +* + END IF +* + ELSE +* +* N .LT. MNTHR +* +* Path 5t (N greater than M, but not much larger) +* Reduce to bidiagonal form without LQ decomposition +* + IE = 1 + ITAUQ = IE + M + ITAUP = ITAUQ + M + NWORK = ITAUP + M +* +* Bidiagonalize A +* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) +* + CALL SGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), + $ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, + $ IERR ) + IF( WNTQN ) THEN +* +* Perform bidiagonal SVD, only computing singular values +* (Workspace: need M+BDSPAC) +* + CALL SBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1, + $ DUM, IDUM, WORK( NWORK ), IWORK, INFO ) + ELSE IF( WNTQO ) THEN + LDWKVT = M + IVT = NWORK + IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN +* +* WORK( IVT ) is M by N +* + CALL SLASET( 'F', M, N, ZERO, ZERO, WORK( IVT ), + $ LDWKVT ) + NWORK = IVT + LDWKVT*N + ELSE +* +* WORK( IVT ) is M by M +* + NWORK = IVT + LDWKVT*M + IL = NWORK +* +* WORK(IL) is M by CHUNK +* + CHUNK = ( LWORK-M*M-3*M ) / M + END IF +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in U and computing right singular +* vectors of bidiagonal matrix in WORK(IVT) +* (Workspace: need M*M+BDSPAC) +* + CALL SBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, + $ WORK( IVT ), LDWKVT, DUM, IDUM, + $ WORK( NWORK ), IWORK, INFO ) +* +* Overwrite U by left singular vectors of A +* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) +* + CALL SORMBR( 'Q', 'L', 'N', M, M, N, A, LDA, + $ WORK( ITAUQ ), U, LDU, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) +* + IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN +* +* Overwrite WORK(IVT) by left singular vectors of A +* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) +* + CALL SORMBR( 'P', 'R', 'T', M, N, M, A, LDA, + $ WORK( ITAUP ), WORK( IVT ), LDWKVT, + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) +* +* Copy right singular vectors of A from WORK(IVT) to A +* + CALL SLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA ) + ELSE +* +* Generate P**T in A +* (Workspace: need M*M+2*M, prefer M*M+M+M*NB) +* + CALL SORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ), + $ WORK( NWORK ), LWORK-NWORK+1, IERR ) +* +* Multiply Q in A by right singular vectors of +* bidiagonal matrix in WORK(IVT), storing result in +* WORK(IL) and copying to A +* (Workspace: need 2*M*M, prefer M*M+M*N) +* + DO 40 I = 1, N, CHUNK + BLK = MIN( N-I+1, CHUNK ) + CALL SGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ), + $ LDWKVT, A( 1, I ), LDA, ZERO, + $ WORK( IL ), M ) + CALL SLACPY( 'F', M, BLK, WORK( IL ), M, A( 1, I ), + $ LDA ) + 40 CONTINUE + END IF + ELSE IF( WNTQS ) THEN +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in U and computing right singular +* vectors of bidiagonal matrix in VT +* (Workspace: need M+BDSPAC) +* + CALL SLASET( 'F', M, N, ZERO, ZERO, VT, LDVT ) + CALL SBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT, + $ LDVT, DUM, IDUM, WORK( NWORK ), IWORK, + $ INFO ) +* +* Overwrite U by left singular vectors of A and VT +* by right singular vectors of A +* (Workspace: need 3*M, prefer 2*M+M*NB) +* + CALL SORMBR( 'Q', 'L', 'N', M, M, N, A, LDA, + $ WORK( ITAUQ ), U, LDU, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SORMBR( 'P', 'R', 'T', M, N, M, A, LDA, + $ WORK( ITAUP ), VT, LDVT, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + ELSE IF( WNTQA ) THEN +* +* Perform bidiagonal SVD, computing left singular vectors +* of bidiagonal matrix in U and computing right singular +* vectors of bidiagonal matrix in VT +* (Workspace: need M+BDSPAC) +* + CALL SLASET( 'F', N, N, ZERO, ZERO, VT, LDVT ) + CALL SBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT, + $ LDVT, DUM, IDUM, WORK( NWORK ), IWORK, + $ INFO ) +* +* Set the right corner of VT to identity matrix +* + IF( N.GT.M ) THEN + CALL SLASET( 'F', N-M, N-M, ZERO, ONE, VT( M+1, M+1 ), + $ LDVT ) + END IF +* +* Overwrite U by left singular vectors of A and VT +* by right singular vectors of A +* (Workspace: need 2*M+N, prefer 2*M+N*NB) +* + CALL SORMBR( 'Q', 'L', 'N', M, M, N, A, LDA, + $ WORK( ITAUQ ), U, LDU, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + CALL SORMBR( 'P', 'R', 'T', N, N, M, A, LDA, + $ WORK( ITAUP ), VT, LDVT, WORK( NWORK ), + $ LWORK-NWORK+1, IERR ) + END IF +* + END IF +* + END IF +* +* Undo scaling if necessary +* + IF( ISCL.EQ.1 ) THEN + IF( ANRM.GT.BIGNUM ) + $ CALL SLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN, + $ IERR ) + IF( ANRM.LT.SMLNUM ) + $ CALL SLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN, + $ IERR ) + END IF +* +* Return optimal workspace in WORK(1) +* + WORK( 1 ) = MAXWRK +* + RETURN +* +* End of SGESDD +* + END |