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NAME

       CBDSQR  -  計算一個實  (real)  NxN 上/下 (upper/lower) 三角 (bidiagonal) 矩陣 B 的單值分解
       (singular value decomposition (SVD))

總覽 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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