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sbdsqr


 NAME
      SBDSQR - 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, 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-1)
              On entry, the (n-1) off-diagonal elements of the
              bidiagonal matrix B.  On normal exit, E is des-
              troyed.

      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
              (MAX( 1, 4*N-4 )) WORK is not referenced if NCVT =
              NRU = NCC = 0.

      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 con-
              tain 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 com-
              puted 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.