Man page - lasd1(3)

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lasd1

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine dlasd1 (integer nl, integer nr, integer sqre, double precision,dimension( * ) d, double precision alpha, double precision beta, doubleprecision, dimension( ldu, * ) u, integer ldu, double precision,dimension( ldvt, * ) vt, integer ldvt, integer, dimension( * ) idxq,integer, dimension( * ) iwork, double precision, dimension( * ) work,integer info)
subroutine slasd1 (integer nl, integer nr, integer sqre, real, dimension( *) d, real alpha, real beta, real, dimension( ldu, * ) u, integer ldu,real, dimension( ldvt, * ) vt, integer ldvt, integer, dimension( * )idxq, integer, dimension( * ) iwork, real, dimension( * ) work, integerinfo)
Author

NAME

lasd1 - lasd1: D&C step: merge subproblems

SYNOPSIS

Functions

subroutine dlasd1 (nl, nr, sqre, d, alpha, beta, u, ldu, vt, ldvt, idxq, iwork, work, info)
DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
subroutine slasd1 (nl, nr, sqre, d, alpha, beta, u, ldu, vt, ldvt, idxq, iwork, work, info)
SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.

Detailed Description

Function Documentation

subroutine dlasd1 (integer nl, integer nr, integer sqre, double precision,dimension( * ) d, double precision alpha, double precision beta, doubleprecision, dimension( ldu, * ) u, integer ldu, double precision,dimension( ldvt, * ) vt, integer ldvt, integer, dimension( * ) idxq,integer, dimension( * ) iwork, double precision, dimension( * ) work,integer info)

DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.

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**T a Z2**T b ) * VT(in)
( 0 0 D2(in) 0 )

= U(out) * ( D(out) 0) * VT(out)

where Z**T = (Z1**T a Z2**T b) = u**T VT**T, 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:

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 occurrence 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.

Parameters

NL

NL is INTEGER
The row dimension of the upper block. NL >= 1.

NR

NR is INTEGER
The row dimension of the lower block. NR >= 1.

SQRE

SQRE is 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

D is 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

ALPHA is DOUBLE PRECISION
Contains the diagonal element associated with the added row.

BETA

BETA is DOUBLE PRECISION
Contains the off-diagonal element associated with the added
row.

U

U is 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.

LDU

LDU is INTEGER
The leading dimension of the array U. LDU >= max( 1, N ).

VT

VT is DOUBLE PRECISION array, dimension(LDVT,M)
where M = N + SQRE.
On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
the right singular vectors of the lower block. On exit
VT**T contains the right singular vectors of the
bidiagonal matrix.

LDVT

LDVT is INTEGER
The leading dimension of the array VT. LDVT >= max( 1, M ).

IDXQ

IDXQ is 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

IWORK is INTEGER array, dimension( 4 * N )

WORK

WORK is DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, a singular value did not converge

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

subroutine slasd1 (integer nl, integer nr, integer sqre, real, dimension( *) d, real alpha, real beta, real, dimension( ldu, * ) u, integer ldu,real, dimension( ldvt, * ) vt, integer ldvt, integer, dimension( * )idxq, integer, dimension( * ) iwork, real, dimension( * ) work, integerinfo)

SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.

Purpose:

SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0.

A related subroutine SLASD7 handles the case in which the singular
values (and the singular vectors in factored form) are desired.

SLASD1 computes the SVD as follows:

( D1(in) 0 0 0 )
B = U(in) * ( Z1**T a Z2**T b ) * VT(in)
( 0 0 D2(in) 0 )

= U(out) * ( D(out) 0) * VT(out)

where Z**T = (Z1**T a Z2**T b) = u**T VT**T, 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:

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 occurrence the dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine SLASD2.

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 SLASD4 (as called
by SLASD3). 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.

Parameters

NL

NL is INTEGER
The row dimension of the upper block. NL >= 1.

NR

NR is INTEGER
The row dimension of the lower block. NR >= 1.

SQRE

SQRE is 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

D is REAL array, dimension (NL+NR+1).
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

ALPHA is REAL
Contains the diagonal element associated with the added row.

BETA

BETA is REAL
Contains the off-diagonal element associated with the added
row.

U

U is REAL 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.

LDU

LDU is INTEGER
The leading dimension of the array U. LDU >= max( 1, N ).

VT

VT is REAL array, dimension (LDVT,M)
where M = N + SQRE.
On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
the right singular vectors of the lower block. On exit
VT**T contains the right singular vectors of the
bidiagonal matrix.

LDVT

LDVT is INTEGER
The leading dimension of the array VT. LDVT >= max( 1, M ).

IDXQ

IDXQ is 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

IWORK is INTEGER array, dimension (4*N)

WORK

WORK is REAL array, dimension (3*M**2+2*M)

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, a singular value did not converge

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Author

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