SLASDA(3) MathKeisan LAPACK routine SLASDA(3)
NAME
SLASDA - 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 bidiagonal N-by-M matrix B with
diagonal D and offdiagonal E, where M = N + SQRE. The algorithm com-
putes 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 sin-
gular vectors in explicit form.
ARGUMENTS
ICOMPQ (input) INTEGER Specifies whether singular vectors are to be
computed in compact form, as follows = 0: Compute 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 bot-
tom 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;
= 1: The bidiagonal matrix has column dimension M = N + 1.
D (input/output) REAL array, dimension ( N )
On entry D contains the main diagonal of the bidiagonal 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 sin-
gular 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
computation 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 compo-
nents of the deflation-adjusted updating row vector for subprob-
lems on the I-th level.
POLES (output) REAL array,
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)).
IWORK (workspace) INTEGER array, dimension (7*N).
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an singular value did not converge
FURTHER DETAILS
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA
LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 SLASDA(3)