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.

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

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