DLASD6(1) LAPACK auxiliary routine (version 3.2) DLASD6(1)NAME
DLASD6 - computes the SVD of an updated upper bidiagonal matrix B
obtained by merging two smaller ones by appending a row
SYNOPSIS
SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, IDXQ,
PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES,
DIFL, DIFR, Z, K, C, S, WORK, IWORK, INFO )
INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, NR,
SQRE
DOUBLE PRECISION ALPHA, BETA, C, S
INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), PERM( *
)
DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ), GIVNUM(
LDGNUM, * ), POLES( LDGNUM, * ), VF( * ), VL( * ),
WORK( * ), Z( * )
PURPOSE
DLASD6 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 vector matrices in factored form. B is an N-by-M
matrix with N = NL + NR + 1 and M = N + SQRE. A related subroutine,
DLASD1, handles the case in which all singular values and singular vec‐
tors of the bidiagonal matrix are desired. DLASD6 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 else‐
where; and the entry b is empty if SQRE = 0.
The singular values of B can be computed using D1, D2, the first compo‐
nents of all the right singular vectors of the lower block, and the
last components of all the right singular vectors of the upper block.
These components are stored and updated in VF and VL, respectively, in
DLASD6. Hence U and VT are not explicitly referenced.
The singular values are stored in D. The algorithm consists of two
stages:
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 dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine DLASD7.
The second stage consists of calculating the updated
singular values. This is done by finding the roots of the
secular equation via the routine DLASD4 (as called by DLASD8).
This routine also updates VF and VL and computes the distances
between the updated singular values and the old singular
values.
DLASD6 is called from DLASDA.
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) DOUBLE PRECISION 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.
VF (input/output) DOUBLE PRECISION array, dimension ( M )
On entry, VF(1:NL+1) contains the first components of all
right singular vectors of the upper block; and VF(NL+2:M) con‐
tains the first components of all right singular vectors of the
lower block. On exit, VF contains the first components of all
right singular vectors of the bidiagonal matrix.
VL (input/output) DOUBLE PRECISION 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) con‐
tains the last components of all right singular vectors of the
lower block. On exit, VL contains the last components of all
right singular vectors of the bidiagonal matrix.
ALPHA (input/output) DOUBLE PRECISION
Contains the diagonal element associated with the added row.
BETA (input/output) DOUBLE PRECISION
Contains the off-diagonal element associated with the added row.
IDXQ (output) INTEGER array, dimension ( N )
This contains the permutation which will reintegrate the sub‐
problem 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) INTE‐
GER The number of Givens rotations which took place in this sub‐
problem. Not referenced 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 refer‐
enced if ICOMPQ = 0. LDGCOL (input) INTEGER leading dimension
of GIVCOL, must be at least N. GIVNUM (output) DOUBLE PRECISION
array, dimension ( LDGNUM, 2 ) Each number indicates the C or S
value to be used in the corresponding Givens rotation. Not ref‐
erenced if ICOMPQ = 0. LDGNUM (input) INTEGER The leading
dimension of GIVNUM and POLES, must be at least N.
POLES (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
On exit, POLES(1,*) is an array containing the new singular val‐
ues 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) DOUBLE PRECISION array, dimension ( N )
On exit, DIFL(I) is the distance between I-th updated (unde‐
flated) singular value and the I-th (undeflated) old singular
value.
DIFR (output) DOUBLE PRECISION 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 con‐
taining the normalizing factors for the right singular vector
matrix. See DLASD8 for details on DIFL and DIFR.
Z (output) DOUBLE PRECISION array, dimension ( M )
The first elements of this array contain the components of the
deflation-adjusted updating row vector.
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) DOUBLE PRECISION
C contains garbage if SQRE =0 and the C-value of a Givens rota‐
tion related to the right null space if SQRE = 1.
S (output) DOUBLE PRECISION
S contains garbage if SQRE =0 and the S-value of a Givens rota‐
tion related to the right null space if SQRE = 1.
WORK (workspace) DOUBLE PRECISION array, dimension ( 4 * M )
IWORK (workspace) INTEGER array, dimension ( 3 * 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 (versioNovember 2008 DLASD6(1)
