dgesdd.f

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      SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
     $                   LWORK, IWORK, 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
*     October 31, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ
      INTEGER            INFO, LDA, LDU, LDVT, LWORK, M, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), S( * ), U( LDU, * ),
     $                   VT( LDVT, * ), WORK( * )
*     ..
*     .. Common block to return operation count ..
      COMMON             / LATIME / OPS, ITCNT
*     ..
*     .. Scalars in Common ..
      DOUBLE PRECISION   ITCNT, OPS
*     ..
*
*  Purpose
*  =======
*
*  DGESDD 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 VT;
*          = '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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (min(M,N))
*          The singular values of A, sorted so that S(i) >= S(i+1).
*
*  U       (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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.
*          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 < 0 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:  DBDSDC 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 ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. 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, NB, NWORK, WRKBL
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            IDUM( 1 )
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DBDSDC, DGEBRD, DGELQF, DGEMM, DGEQRF, DLACPY,
     $                   DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR,
     $                   XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE, DOPBL3, DOPLA, DOPLA2
      EXTERNAL           DLAMCH, DLANGE, DOPBL3, DOPLA, DOPLA2, ILAENV, 
     $                   LSAME
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, INT, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      MINMN = MIN( M, N )
      MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
      WNTQA = LSAME( JOBZ, 'A' )
      WNTQS = LSAME( JOBZ, 'S' )
      WNTQAS = WNTQA .OR. WNTQS
      WNTQO = LSAME( JOBZ, 'O' )
      WNTQN = LSAME( JOBZ, 'N' )
      MINWRK = 1
      MAXWRK = 1
      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 .AND. M.GT.0 .AND. N.GT.0 ) THEN
         IF( M.GE.N ) THEN
*
*           Compute space needed for DBDSDC
*
            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, 'DGEQRF', ' ', M, N, -1,
     $                    -1 )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', 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, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
     $                    N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', '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, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
     $                    N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', '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, 'DGEQRF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
     $                    M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+2*N*
     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', '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, 'DGEBRD', ' ', 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, 'DORMBR', 'QLN', M, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', '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, 'DORMBR', 'QLN', M, N, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', '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, 'DORMBR', 'QLN', M, M, N, -1 ) )
                  WRKBL = MAX( WRKBL, 3*N+N*
     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                  MAXWRK = MAX( MAXWRK, BDSPAC+3*N )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               END IF
            END IF
         ELSE
*
*           Compute space needed for DBDSDC
*
            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, 'DGELQF', ' ', M, N, -1,
     $                    -1 )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', 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, 'DGELQF', ' ', M, N, -1, -1 )
                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
     $                    N, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+2*M*
     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
                  WRKBL = MAX( WRKBL, 3*M+M*
     $                    ILAENV( 1, 'DORMBR', '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')
*