SBDSQR (3lapack)

compute the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B

SYNOPSIS

  SUBROUTINE SBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC,
                     WORK, INFO )

      CHARACTER      UPLO

      INTEGER        INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU

      REAL           C( LDC, * ), D( * ), E( * ), U( LDU, * ), VT( LDVT, * ),
                     WORK( * )

PURPOSE

  SBDSQR computes the singular value decomposition (SVD) of a real N-by-N
  (upper or lower) bidiagonal matrix B:  B = Q * S * P' (P' denotes the
  transpose of P), where S is a diagonal matrix with non-negative diagonal
  elements (the singular values of B), and Q and P are orthogonal matrices.

  The routine computes S, and optionally computes U * Q, P' * VT, or Q' * C,
  for given real input matrices U, VT, and C.

  See "Computing  Small Singular Values of Bidiagonal Matrices With
  Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK
  Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873-
  912, Sept 1990) and
  "Accurate singular values and differential qd algorithms," by B. Parlett
  and V. Fernando, Technical Report CPAM-554, Mathematics Department,
  University of California at Berkeley, July 1992 for a detailed description
  of the algorithm.

ARGUMENTS

  UPLO    (input) CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal.

  N       (input) INTEGER
          The order of the matrix B.  N >= 0.

  NCVT    (input) INTEGER
          The number of columns of the matrix VT. NCVT >= 0.

  NRU     (input) INTEGER
          The number of rows of the matrix U. NRU >= 0.

  NCC     (input) INTEGER
          The number of columns of the matrix C. NCC >= 0.

  D       (input/output) REAL array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.  On
          exit, if INFO=0, the singular values of B in decreasing order.

  E       (input/output) REAL array, dimension (N)
          On entry, the elements of E contain the offdiagonal elements of the
          bidiagonal matrix whose SVD is desired. On normal exit (INFO = 0),
          E is destroyed.  If the algorithm does not converge (INFO > 0), D
          and E will contain the diagonal and superdiagonal elements of a
          bidiagonal matrix orthogonally equivalent to the one given as
          input. E(N) is used for workspace.

  VT      (input/output) REAL array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT.  On exit, VT is overwritten by P'
          * VT.  VT is not referenced if NCVT = 0.

  LDVT    (input) INTEGER
          The leading dimension of the array VT.  LDVT >= max(1,N) if NCVT >
          0; LDVT >= 1 if NCVT = 0.

  U       (input/output) REAL array, dimension (LDU, N)
          On entry, an NRU-by-N matrix U.  On exit, U is overwritten by U *
          Q.  U is not referenced if NRU = 0.

  LDU     (input) INTEGER
          The leading dimension of the array U.  LDU >= max(1,NRU).

  C       (input/output) REAL array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C.  On exit, C is overwritten by Q' *
          C.  C is not referenced if NCC = 0.

  LDC     (input) INTEGER
          The leading dimension of the array C.  LDC >= max(1,N) if NCC > 0;
          LDC >=1 if NCC = 0.

  WORK    (workspace) REAL array, dimension
          2*N  if only singular values wanted (NCVT = NRU = NCC = 0) max( 1,
          4*N-4 ) otherwise

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm did not converge; D and E contain the elements
          of a bidiagonal matrix which is orthogonally similar to the input
          matrix B;  if INFO = i, i elements of E have not converged to zero.

PARAMETERS

  TOLMUL  REAL, default = max(10,min(100,EPS**(-1/8)))
          TOLMUL controls the convergence criterion of the QR loop.  If it is
          positive, TOLMUL*EPS is the desired relative precision in the
          computed singular values.  If it is negative,
          abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the
          computed singular values (corresponds to relative accuracy
          abs(TOLMUL*EPS) in the largest singular value.  abs(TOLMUL) should
          be between 1 and 1/EPS, and preferably between 10 (for fast
          convergence) and .1/EPS (for there to be some accuracy in the
          results).  Default is to lose at either one eighth or 2 of the
          available decimal digits in each computed singular value (whichever
          is smaller).

  MAXITR  INTEGER, default = 6
          MAXITR controls the maximum number of passes of the algorithm
          through its inner loop. The algorithms stops (and so fails to
          converge) if the number of passes through the inner loop exceeds
          MAXITR*N**2.