cbdsqr(3S)

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

SYNOPSIS
     SUBROUTINE CBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C,
                        LDC, RWORK, INFO )

         CHARACTER      UPLO

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

         REAL           D( * ), E( * ), RWORK( * )

         COMPLEX        C( LDC, * ), U( LDU, * ), VT( LDVT, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     CBDSQR 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 complex 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
             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) COMPLEX 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) COMPLEX 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) COMPLEX 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.

     RWORK   (workspace) REAL array, dimension (4*N)

     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.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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