CBDSQR man page on IRIX

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CBDSQR(3F)							    CBDSQR(3F)

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

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.

									Page 1

CBDSQR(3F)							    CBDSQR(3F)

     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
	     2*N  if only singular values wanted (NCVT = NRU = NCC = 0) max(
	     1, 4*N-4 ) otherwise

     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,

									Page 2

CBDSQR(3F)							    CBDSQR(3F)

	     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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