CXML
        	    
LAPACK version 3.0
cbdsqr(3)

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

SYNTAX
  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, * )

DESCRIPTION
  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.