NAME

       CBDSQR  -  计算一个实  (real) NxN 上/下 (upper/lower) 三角 (bidiagonal) 矩阵 B 的单值分解 (singular value
       decomposition (SVD))

总览 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)