SGESVD (3lapack)

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 orthogonal 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 bidiagonal matrix B whose diagonal is in S (not
          necessarily sorted). B satisfies A = U * B * VT, so it has the same
          singular values as A, and singular vectors 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.