dlasd1 (l) - Linux Manuals

dlasd1: computes the SVD of an upper bidiagonal N-by-M matrix B,

NAME

DLASD1 - computes 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( * )

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) * (   Z1aq   a   Z2aq    b ) * VT(in)

        (   0     0   D2(in) 0 )

 = U(out) * ( D(out) 0) * VT(out)
where Zaq = (Z1aq a Z2aq b) = uaq VTaq, 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 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/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.
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.
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)aq contains the right singular
vectors of the upper block; VT(NL+2:M, NL+2:M)aq contains the right singular vectors of the lower block. On exit VTaq 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

Pages related to dlasd1

  • dlasd1 (3)
  • dlasd0 (l) - a divide and conquer approach, DLASD0 computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE
  • dlasd2 (l) - merges the two sets of singular values together into a single sorted set
  • dlasd3 (l) - finds all the square roots of the roots of the secular equation, as defined by the values in D and Z
  • dlasd4 (l) - subroutine compute the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix whose entries are given as the squares of the corresponding entries in the array d, and that 0 <= D(i) < D(j) for i < j and that RHO > 0
  • dlasd5 (l) - subroutine compute the square root of the I-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) * diag( D ) + RHO The diagonal entries in the array D are assumed to satisfy 0 <= D(i) < D(j) for i < j
  • dlasd6 (l) - computes the SVD of an updated upper bidiagonal matrix B obtained by merging two smaller ones by appending a row
  • dlasd7 (l) - merges the two sets of singular values together into a single sorted set
  • dlasd8 (l) - finds the square roots of the roots of the secular equation,
  • dlasd9 (l) - find the square roots of the roots of the secular equation,
  • dlasda (l) - a divide and conquer approach, DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE