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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 Matrices
With Guaranteed High Relative Accuracy," by J. Demmel and W.
Kahan, LAPACK Working Note #3, 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-1)
On entry, the (n-1) off-diagonal elements of the
bidiagonal matrix B. On normal exit, E is des-
troyed.
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
(MAX( 1, 4*N-4 )) WORK is not referenced if NCVT =
NRU = NCC = 0.
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 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 com-
puted 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.