SLASDA - a divide and conquer approach, SLASDA computes
the singular value decomposition (SVD) of a real upper
bidiagonal N-by-M matrix B with diagonal D and offdiagonal
E, where M = N + SQRE
SYNOPSIS
SUBROUTINE SLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU,
VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
GIVCOL, LDGCOL, PERM, GIVNUM, C, S,
WORK, IWORK, INFO )
INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ,
SQRE
INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ),
IWORK( * ), K( * ), PERM( LDGCOL, * )
REAL C( * ), D( * ), DIFL( LDU, * ), DIFR(
LDU, * ), E( * ), GIVNUM( LDU, * ),
POLES( LDU, * ), S( * ), U( LDU, * ),
VT( LDU, * ), WORK( * ), Z( LDU, * )
PURPOSE
Using a divide and conquer approach, SLASDA computes the
singular value decomposition (SVD) of a real upper bidiag
onal N-by-M matrix B with diagonal D and offdiagonal E,
where M = N + SQRE. The algorithm computes the singular
values in the SVD B = U * S * VT. The orthogonal matrices
U and VT are optionally computed in compact form.
A related subroutine, SLASD0, computes the singular values
and the singular vectors in explicit form.
ARGUMENTS
ICOMPQ (input) INTEGER Specifies whether singular vectors
are to be computed in compact form, as follows = 0: Com
pute singular values only.
= 1: Compute singular vectors of upper bidiagonal matrix
in compact form.
SMLSIZ (input) INTEGER The maximum size of the subproblems
at the bottom of the computation tree.
N (input) INTEGER
The row dimension of the upper bidiagonal matrix.
This is also the dimension of the main diagonal
array D.
SQRE (input) INTEGER
Specifies the column dimension of the bidiagonal
matrix. = 0: The bidiagonal matrix has column
dimension M = N;
N + 1.
D (input/output) REAL array, dimension ( N )
On entry D contains the main diagonal of the bidi
agonal matrix. On exit D, if INFO = 0, contains its
singular values.
E (input) REAL array, dimension ( M-1 )
Contains the subdiagonal entries of the bidiagonal
matrix. On exit, E has been destroyed.
U (output) REAL array,
dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not
referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, U
contains the left singular vector matrices of all
subproblems at the bottom level.
LDU (input) INTEGER, LDU = > N.
The leading dimension of arrays U, VT, DIFL, DIFR,
POLES, GIVNUM, and Z.
VT (output) REAL array,
dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not
referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit,
VT' contains the right singular vector matrices of
all subproblems at the bottom level.
K (output) INTEGER array,
dimension ( N ) if ICOMPQ = 1 and dimension 1 if
ICOMPQ = 0. If ICOMPQ = 1, on exit, K(I) is the
dimension of the I-th secular equation on the com
putation tree.
DIFL (output) REAL array, dimension ( LDU, NLVL ),
where NLVL = floor(log_2 (N/SMLSIZ))).
DIFR (output) REAL array,
dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
dimension ( N ) if ICOMPQ = 0. If ICOMPQ = 1, on
exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) record
distances between singular values on the I-th level
and singular values on the (I -1)-th level, and
DIFR(1:N, 2 * I ) contains the normalizing factors
for the right singular vector matrix. See SLASD8
for details.
Z (output) REAL array,
dimension ( LDU, NLVL ) if ICOMPQ = 1 and dimension
( N ) if ICOMPQ = 0. The first K elements of Z(1,
I) contain the components of the deflation-adjusted
updating row vector for subproblems on the I-th
level.
dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not
referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit,
POLES(1, 2*I - 1) and POLES(1, 2*I) contain the
new and old singular values involved in the secular
equations on the I-th level.
GIVPTR (output) INTEGER array, dimension ( N ) if
ICOMPQ = 1, and not referenced if ICOMPQ = 0. If
ICOMPQ = 1, on exit, GIVPTR( I ) records the number
of Givens rotations performed on the I-th problem
on the computation tree.
GIVCOL (output) INTEGER array, dimension ( LDGCOL,
2 * NLVL ) if ICOMPQ = 1, and not referenced if
ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the
locations of Givens rotations performed on the I-th
level on the computation tree.
LDGCOL (input) INTEGER, LDGCOL = > N. The leading
dimension of arrays GIVCOL and PERM.
PERM (output) INTEGER array,
dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not
referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit,
PERM(1, I) records permutations done on the I-th
level of the computation tree.
GIVNUM (output) REAL array, dimension ( LDU, 2 *
NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ
= 0. If ICOMPQ = 1, on exit, for each I, GIVNUM(1,
2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
values of Givens rotations performed on the I-th
level on the computation tree.
C (output) REAL array,
dimension ( N ) if ICOMPQ = 1, and dimension 1 if
ICOMPQ = 0. If ICOMPQ = 1 and the I-th subproblem
is not square, on exit, C( I ) contains the C-value
of a Givens rotation related to the right null
space of the I-th subproblem.
S (output) REAL array, dimension ( N ) if
ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If
ICOMPQ = 1 and the I-th subproblem is not square,
on exit, S( I ) contains the S-value of a Givens
rotation related to the right null space of the I-
th subproblem.
WORK (workspace) REAL array, dimension
(6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
Dimension must be at least (7 * N).
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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