
DLASD6(l) ) DLASD6(l)
NAME
DLASD6  compute 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, GIV
COL, 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
byM matrix with N = NL + NR + 1 and M = N + SQRE. A related subroutine, DLASD1, handles
the case in which all singular values and singular vectors 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 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 singu
lar 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 NRbyNR square matrix.
= 1: the lower block is an NRby(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) contains the first compo
nents 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) contains the last compo
nents 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) DOUBLE PRECISION
Contains the diagonal element associated with the added row.
BETA (input) DOUBLE PRECISION
Contains the offdiagonal element associated with the added row.
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.
PERM (output) INTEGER array, dimension ( N )
The permutations (from deflation and sorting) to be applied to each block. Not ref
erenced if ICOMPQ = 0.
GIVPTR (output) INTEGER The number of Givens rotations which took place in this
subproblem. Not referenced if ICOMPQ = 0.
GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indi
cates 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) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) Each number indi
cates the C or S value to be used in the corresponding Givens rotation. Not refer
enced 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 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) DOUBLE PRECISION array, dimension ( N )
On exit, DIFL(I) is the distance between Ith updated (undeflated) singular value
and the Ith (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 Ith updated (undeflated) singular value and the
I+1th (undeflated) old singular value.
If ICOMPQ = 1, DIFR(1:K,2) is an array containing 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 deflationadjusted
updating row vector.
K (output) INTEGER
Contains the dimension of the nondeflated 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 Cvalue of a Givens rotation related to the
right null space if SQRE = 1.
S (output) DOUBLE PRECISION
S contains garbage if SQRE =0 and the Svalue of a Givens rotation 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 ith 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 version 3.0 15 June 2000 DLASD6(l) 
