DLASD1(3)

NAME
DLASD1 - 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) * ( 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:
   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)' 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