DLASD1(3S)DLASD1(3S)NAME
DLASD1 - compute the SVD of an upper bidiagonal N-by-M matrix B,
SYNOPSIS
SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, IDXQ,
IWORK, WORK, INFO )
INTEGER INFO, LDU, LDVT, NL, NR, SQRE
DOUBLE PRECISION ALPHA, BETA
INTEGER IDXQ( * ), IWORK( * )
DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( *
)
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
PURPOSE
DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N =
NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.
A related subroutine DLASD7 handles the case in which the singular values
(and the singular vectors in factored form) are desired.
DLASD1 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 left singular vectors of the original matrix are stored in U, and the
transpose of the right singular vectors are stored in VT, and the
singular values are in D. The algorithm consists of three stages:
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DLASD1(3S)DLASD1(3S)
The first stage consists of deflating the size of the problem
when there are multiple singular values or when there are zeros 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 DLASD2.
The second stage consists of calculating the updated
singular values. This is done by finding the square roots of the
roots of the secular equation via the routine DLASD4 (as called
by DLASD3). This routine also calculates the singular vectors of
the current problem.
The final stage consists of computing the updated singular vectors
directly using the updated singular values. The singular vectors
for the current problem are multiplied with the singular vectors
from the overall problem.
ARGUMENTS
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 (N = 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.
ALPHA (input) DOUBLE PRECISION
Contains the diagonal element associated with the added row.
BETA (input) DOUBLE PRECISION
Contains the off-diagonal element associated with the added row.
U (input/output) DOUBLE PRECISION array, dimension(LDU,N)
On entry U(1:NL, 1:NL) contains the left singular vectors of
the upper block; U(NL+2:N, NL+2:N) contains the left singular
vectors of the lower block. On exit U contains the left singular
vectors of the bidiagonal matrix.
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DLASD1(3S)DLASD1(3S)
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max( 1, N ).
VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)' contains the
right singular
vectors of the upper block; VT(NL+2:M, NL+2:M)' contains the right
singular vectors of the lower block. On exit VT' contains the
right singular vectors of the bidiagonal matrix.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= max( 1, M ).
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.
IWORK (workspace) INTEGER array, dimension( 4 * N )
WORK (workspace) DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )
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
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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