CBDSQR(3) MathKeisan LAPACK routine CBDSQR(3)
NAME
CBDSQR - the singular values and, optionally, the right and/or left
singular vectors from the singular value decomposition (SVD) of a real
N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-
shift QR algorithm
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 values and, optionally, the right and/or
left singular vectors from the singular value decomposition (SVD) of a
real N-by-N (upper or lower) bidiagonal matrix B using the implicit
zero-shift QR algorithm. The SVD of B has the form
B = Q * S * P**H
where S is the diagonal matrix of singular values, Q is an orthogonal
matrix of left singular vectors, and P is an orthogonal matrix of right
singular vectors. If left singular vectors are requested, this subrou-
tine actually returns U*Q instead of Q, and, if right singular vectors
are requested, this subroutine returns P**H*VT instead of P**H, for
given complex input matrices U and VT. When U and VT are the unitary
matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT,
as computed by CGEBRD, then
A = (U*Q) * S * (P**H*VT)
is the SVD of A. Optionally, the subroutine may also compute Q**H*C
for a given complex input matrix C.
See "Computing Small Singular Values of Bidiagonal Matrices With Guar-
anteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Work-
ing 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. Par-
lett and V. Fernando, Technical Report CPAM-554, Mathematics Depart-
ment, 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-1)
On entry, the N-1 offdiagonal elements of the bidiagonal matrix
B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
will contain the diagonal and superdiagonal elements of a bidi-
agonal matrix orthogonally equivalent to the one given as
input.
VT (input/output) COMPLEX array, dimension (LDVT, NCVT)
On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten
by P**H * VT. 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. 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**H * C. 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 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 ele-
ments of a bidiagonal matrix which is orthogonally similar to
the input matrix B; if INFO = i, i elements of E have not con-
verged 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(TOL-
MUL*EPS*sigma_max) is the desired absolute accuracy in the com-
puted 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 routine (version 3.1) November 2006 CBDSQR(3)