SLASD1 - compute the SVD of an upper bidiagonal N-by-M
matrix B,
SYNOPSIS
SUBROUTINE SLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU,
VT, LDVT, IDXQ, IWORK, WORK, INFO )
INTEGER INFO, LDU, LDVT, NL, NR, SQRE
REAL ALPHA, BETA
INTEGER IDXQ( * ), IWORK( * )
REAL D( * ), U( LDU, * ), VT( LDVT, * ),
WORK( * )
PURPOSE
SLASD1 computes the SVD of an upper bidiagonal N-by-M
matrix B, where N = NL + NR + 1 and M = N + SQRE. SLASD1
is called from SLASD0.
A related subroutine SLASD7 handles the case in which the
singular values (and the singular vectors in factored
form) are desired.
SLASD1 computes the SVD as follows:
( D1(in) 0 0 0 )
B = U(in) * ( Z1' a Z2' b ) * VT(in)
( 0 0 D2(in) 0 )
= U(out) * ( D(out) 0) * VT(out)
where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of
dimension M with ALPHA and BETA in the NL+1 and NL+2 th
entries and zeros elsewhere; and the entry b is empty if
SQRE = 0.
The left singular vectors of the original matrix are
stored in U, and the transpose of the right singular vec
tors are stored in VT, and the singular values are in D.
The algorithm consists of three stages:
The first stage consists of deflating the size of the
problem
when there are multiple singular values or when there
are zeros in
the Z vector. For each such occurence the dimension of
the
secular equation problem is reduced by one. This stage
is
performed by the routine SLASD2.
singular values. This is done by finding the square
roots of the
roots of the secular equation via the routine SLASD4
(as called
by SLASD3). This routine also calculates the singular
vectors of
the current problem.
The final stage consists of computing the updated sin
gular vectors
directly using the updated singular values. The singu
lar vectors
for the current problem are multiplied with the singu
lar vectors
from the overall problem.
ARGUMENTS
NL (input) INTEGER
The row dimension of the upper block. NL >= 1.
NR (input) INTEGER
The row dimension of the lower block. NR >= 1.
SQRE (input) INTEGER
= 0: the lower block is an NR-by-NR square matrix.
= 1: the lower block is an NR-by-(NR+1) rectangular
matrix.
The bidiagonal matrix has row dimension N = NL + NR
+ 1, and column dimension M = N + SQRE.
D (input/output) REAL array,
dimension (N = NL+NR+1). On entry D(1:NL,1:NL)
contains the singular values of the
upper block; and D(NL+2:N) contains the singular
values of
the lower block. On exit D(1:N) contains the singu
lar values of the modified matrix.
ALPHA (input) REAL
Contains the diagonal element associated with the
added row.
BETA (input) REAL
Contains the off-diagonal element associated with
the added row.
U (input/output) REAL array, dimension(LDU,N)
On entry U(1:NL, 1:NL) contains the left singular
vectors of
the upper block; U(NL+2:N, NL+2:N) contains the
left singular vectors of the lower block. On exit U
nal matrix.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(
1, N ).
VT (input/output) REAL array, dimension(LDVT,M)
where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)'
contains the right singular
vectors of the upper block; VT(NL+2:M, NL+2:M)'
contains the right singular vectors of the lower
block. On exit VT' contains the right singular vec
tors of the bidiagonal matrix.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >=
max( 1, M ).
IDXQ (output) INTEGER array, dimension(N)
This contains the permutation which will reintegrate
the subproblem just solved back into sorted order,
i.e. D( IDXQ( I = 1, N ) ) will be in ascending
order.
IWORK (workspace) INTEGER array, dimension( 4 * N )
WORK (workspace) REAL array, dimension( 3*M**2 + 2*M )
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille
gal value.
> 0: if INFO = 1, an singular value did not con
verge
FURTHER DETAILS
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, Uni
versity of
California at Berkeley, USA
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