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NAME

       CBDSQR - p算一蚢 (real) NxN 上/下 (upper/lower) 三角 (bidiagonal) 矩陣
       B 的單戲 (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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