slasd1.f3

!>
!> 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.
!> 

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
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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