dlasd1 (l)  Linux Manuals
dlasd1: computes the SVD of an upper bidiagonal NbyM matrix B,
NAME
DLASD1  computes the SVD of an upper bidiagonal NbyM 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 NbyM 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: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.
secular equation problem is reduced by one.
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
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.
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 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 (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 offdiagonal 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 ith argument had an illegal value.
> 0: if INFO = 1, an singular value did not converge
FURTHER DETAILS
Based on contributions byMing Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA