CBDSQR(3)                  Library Functions Manual                  CBDSQR(3)

NAME
       CBDSQR - <?><?><?><?><?> (real) NxN <?>/<?> (upper/lower) <?><?>
       (bidiagonal) <?><?> B <?><?><?><?><?> (singular value decomposition
       (SVD))

<?><?> 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, * )

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.


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LAPACK version 3.0               15 June 2000                        CBDSQR(3)