sbdsqr(3)

NAME

SBDSQR - compute the singular value decomposition (SVD) of

a real N-by-N (upper or lower) bidiagonal matrix B

SYNOPSIS

SUBROUTINE  SBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO )
    CHARACTER      UPLO
    INTEGER        INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
    REAL            C( LDC, * ), D( * ), E( * ), U( LDU, *
), VT( LDVT, * ), WORK( * )

PURPOSE

SBDSQR 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 real input matrices U, VT, and C.

See "Computing Small Singular Values of Bidiagonal Matri

ces 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, Mathe

matics 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 bidiago

nal 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 offdiago

nal elements 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 equiv

alent to the one given as input. E(N) is used for workspace.

VT (input/output) REAL 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) REAL 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) REAL 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.

WORK (workspace) REAL array, dimension (4*N)

INFO (output) INTEGER

= 0: successful exit
< 0: If INFO = -i, the i-th argument had an ille

gal value
> 0: the algorithm did not converge; D and E con

tain 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.

LAPACK version 3.0 15 June 2000

SBDSQR(3)