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sgesvd

 NAME
      SGESVD - compute the singular value decomposition (SVD) of a
      real M-by-N matrix A, optionally computing the left and/or
      right singular vectors

 SYNOPSIS
      SUBROUTINE SGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT,
                         LDVT, WORK, LWORK, INFO )

          CHARACTER      JOBU, JOBVT

          INTEGER        INFO, LDA, LDU, LDVT, LWORK, M, N

          REAL           A( LDA, * ), S( * ), U( LDU, * ), VT(
                         LDVT, * ), WORK( * )

 PURPOSE
      SGESVD computes the singular value decomposition (SVD) of a
      real M-by-N matrix A, optionally computing the left and/or
      right singular vectors. 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 V**T, not V.

 ARGUMENTS
      JOBU    (input) CHARACTER*1
              Specifies options for computing all or part of the
              matrix U:
              = 'A':  all M columns of U are returned in array U:
              = 'S':  the first min(m,n) columns of U (the left
              singular vectors) are returned in the array U; =
              'O':  the first min(m,n) columns of U (the left
              singular vectors) are overwritten on the array A; =
              'N':  no columns of U (no left singular vectors) are
              computed.

      JOBVT   (input) CHARACTER*1
              Specifies options for computing all or part of the
              matrix V**T:
              = 'A':  all N rows of V**T are returned in the array
              VT;
              = 'S':  the first min(m,n) rows of V**T (the right

              singular vectors) are returned in the array VT; =
              'O':  the first min(m,n) rows of V**T (the right
              singular vectors) are overwritten on the array A; =
              'N':  no rows of V**T (no right singular vectors)
              are computed.

              JOBVT and JOBU cannot both be 'O'.

      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 JOBU =
              'O',  A is overwritten with the first min(m,n)
              columns of U (the left singular vectors, stored
              columnwise); if JOBVT = 'O', A is overwritten with
              the first min(m,n) rows of V**T (the right singular
              vectors, stored rowwise); if JOBU .ne. 'O' and JOBVT
              .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)
              (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU =
              'S'.  If JOBU = 'A', U contains the M-by-M orthogo-
              nal matrix U; if JOBU = 'S', U contains the first
              min(m,n) columns of U (the left singular vectors,
              stored columnwise); if JOBU = 'N' or 'O', U is not
              referenced.

      LDU     (input) INTEGER
              The leading dimension of the array U.  LDU >= 1; if
              JOBU = 'S' or 'A', LDU >= M.

      VT      (output) REAL array, dimension (LDVT,N)
              If JOBVT = 'A', VT contains the N-by-N orthogonal
              matrix V**T; if JOBVT = 'S', VT contains the first
              min(m,n) rows of V**T (the right singular vectors,
              stored rowwise); if JOBVT = 'N' or 'O', VT is not
              referenced.

      LDVT    (input) INTEGER

              The leading dimension of the array VT.  LDVT >= 1;
              if JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >=
              min(M,N).

      WORK    (workspace/output) REAL array, dimension (LWORK)
              On exit, if INFO = 0, WORK(1) returns the optimal
              LWORK; if INFO > 0, WORK(2:MIN(M,N)) contains the
              unconverged superdiagonal elements of an upper bidi-
              agonal matrix B whose diagonal is in S (not neces-
              sarily sorted). B satisfies A = U * B * VT, so it
              has the same singular values as A, and singular vec-
              tors related by U and VT.

      LWORK   (input) INTEGER
              The dimension of the array WORK. LWORK >= 1.  LWORK
              >= MAX(3*MIN(M,N)+MAX(M,N),5*MIN(M,N)-4).  For good
              performance, LWORK should generally be larger.

      INFO    (output) INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal
              value.
              > 0:  if SBDSQR did not converge, INFO specifies how
              many superdiagonals of an intermediate bidiagonal
              form B did not converge to zero. See the description
              of WORK above for details.