SLASD6 - compute the SVD of an updated upper bidiagonal
matrix B obtained by merging two smaller ones by appending
a row
SYNOPSIS
SUBROUTINE SLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA,
BETA, IDXQ, PERM, GIVPTR, GIVCOL, LDG
COL, GIVNUM, LDGNUM, POLES, DIFL, DIFR,
Z, K, C, S, WORK, IWORK, INFO )
INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL,
LDGNUM, NL, NR, SQRE
REAL ALPHA, BETA, C, S
INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK(
* ), PERM( * )
REAL D( * ), DIFL( * ), DIFR( * ), GIVNUM(
LDGNUM, * ), POLES( LDGNUM, * ), VF( *
), VL( * ), WORK( * ), Z( * )
PURPOSE
SLASD6 computes the SVD of an updated upper bidiagonal
matrix B obtained by merging two smaller ones by appending
a row. This routine is used only for the problem which
requires all singular values and optionally singular vec
tor matrices in factored form. B is an N-by-M matrix with
N = NL + NR + 1 and M = N + SQRE. A related subroutine,
SLASD1, handles the case in which all singular values and
singular vectors of the bidiagonal matrix are desired.
SLASD6 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 singular values of B can be computed using D1, D2, the
first components of all the right singular vectors of the
lower block, and the last components of all the right sin
gular vectors of the upper block. These components are
stored and updated in VF and VL, respectively, in SLASD6.
Hence U and VT are not explicitly referenced.
The singular values are stored in D. The algorithm
The first stage consists of deflating the size of
the problem
when there are multiple singular values or if there
is a zero
in the Z vector. For each such occurence the dimen
sion of the
secular equation problem is reduced by one. This
stage is
performed by the routine SLASD7.
The second stage consists of calculating the updated
singular values. This is done by finding the roots
of the
secular equation via the routine SLASD4 (as called
by SLASD8).
This routine also updates VF and VL and computes the
distances
between the updated singular values and the old sin
gular
values.
SLASD6 is called from SLASDA.
ARGUMENTS
ICOMPQ (input) INTEGER Specifies whether singular vectors
are to be computed in factored form:
= 0: Compute singular values only.
= 1: Compute singular vectors in factored form as well.
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 ( 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
singular values of the modified matrix.
On entry, VF(1:NL+1) contains the first components
of all
right singular vectors of the upper block; and
VF(NL+2:M) contains the first components of all
right singular vectors of the lower block. On exit,
VF contains the first components of all right sin
gular vectors of the bidiagonal matrix.
VL (input/output) REAL array, dimension ( M )
On entry, VL(1:NL+1) contains the last components
of all
right singular vectors of the upper block; and
VL(NL+2:M) contains the last components of all
right singular vectors of the lower block. On exit,
VL contains the last components of all right singu
lar vectors of the bidiagonal 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.
IDXQ (output) INTEGER array, dimension ( N )
This contains the permutation which will reinte
grate the subproblem just solved back into sorted
order, i.e. D( IDXQ( I = 1, N ) ) will be in
ascending order.
PERM (output) INTEGER array, dimension ( N )
The permutations (from deflation and sorting) to be
applied to each block. Not referenced if ICOMPQ =
0.
GIVPTR (output) INTEGER The number of Givens rota
tions which took place in this subproblem. Not ref
erenced if ICOMPQ = 0.
GIVCOL (output) INTEGER array, dimension ( LDGCOL,
2 ) Each pair of numbers indicates a pair of
columns to take place in a Givens rotation. Not
referenced if ICOMPQ = 0.
LDGCOL (input) INTEGER leading dimension of GIVCOL,
must be at least N.
GIVNUM (output) REAL array, dimension ( LDGNUM, 2 )
Each number indicates the C or S value to be used
in the corresponding Givens rotation. Not refer
enced if ICOMPQ = 0.
GIVNUM and POLES, must be at least N.
POLES (output) REAL array, dimension ( LDGNUM, 2 )
On exit, POLES(1,*) is an array containing the new
singular values obtained from solving the secular
equation, and POLES(2,*) is an array containing the
poles in the secular equation. Not referenced if
ICOMPQ = 0.
DIFL (output) REAL array, dimension ( N )
On exit, DIFL(I) is the distance between I-th
updated (undeflated) singular value and the I-th
(undeflated) old singular value.
DIFR (output) REAL array,
dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and dimension
( N ) if ICOMPQ = 0. On exit, DIFR(I, 1) is the
distance between I-th updated (undeflated) singular
value and the I+1-th (undeflated) old singular
value.
If ICOMPQ = 1, DIFR(1:K,2) is an array containing
the normalizing factors for the right singular vec
tor matrix.
See SLASD8 for details on DIFL and DIFR.
Z (output) REAL array, dimension ( M )
The first elements of this array contain the compo
nents of the deflation-adjusted updating row vec
tor.
K (output) INTEGER
Contains the dimension of the non-deflated matrix,
This is the order of the related secular equation.
1 <= K <=N.
C (output) REAL
C contains garbage if SQRE =0 and the C-value of a
Givens rotation related to the right null space if
SQRE = 1.
S (output) REAL
S contains garbage if SQRE =0 and the S-value of a
Givens rotation related to the right null space if
SQRE = 1.
WORK (workspace) REAL array, dimension ( 4 * M )
IWORK (workspace) INTEGER array, dimension ( 3 * N )
INFO (output) INTEGER
= 0: successful exit.
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
Man(1) output converted with
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