DLASD6 computes the SVD of an updated upper bidiagonal matrix B


 obtained by merging two smaller ones by appending a row. This


 routine is used only for the problem which requires all singular


 values and optionally singular vector matrices in factored form.


 B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.


 A related subroutine, DLASD1, handles the case in which all singular


 values and singular vectors of the bidiagonal matrix are desired.


 DLASD6 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 singular values of B can be computed using D1, D2, the first


 components of all the right singular vectors of the lower block, and


 the last components of all the right singular vectors of the upper


 block. These components are stored and updated in VF and VL,


 respectively, in DLASD6. Hence U and VT are not explicitly


 referenced.


 The singular values are stored in D. The algorithm consists of two


 stages:


       The first stage consists of deflating the size of the problem


       when there are multiple singular values or if there is a zero


       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 DLASD7.


       The second stage consists of calculating the updated


       singular values. This is done by finding the roots of the


       secular equation via the routine DLASD4 (as called by DLASD8).


       This routine also updates VF and VL and computes the distances


       between the updated singular values and the old singular


       values.


 DLASD6 is called from DLASDA.

ICOMPQ

          ICOMPQ is INTEGER


         Specifies whether singular vectors are to be computed in


         factored form:


         = 0: Compute singular values only.


         = 1: Compute singular vectors in factored form as well.

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 ( 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.

VF

          VF is DOUBLE PRECISION array, dimension ( M )


         On entry, VF(1:NL+1) contains the first components of all


         right singular vectors of the upper block; and VF(NL+2:M)


         contains the first components of all right singular vectors


         of the lower block. On exit, VF contains the first components


         of all right singular vectors of the bidiagonal matrix.

VL

          VL is DOUBLE PRECISION array, dimension ( M )


         On entry, VL(1:NL+1) contains the  last components of all


         right singular vectors of the upper block; and VL(NL+2:M)


         contains the last components of all right singular vectors of


         the lower block. On exit, VL contains the last components of


         all right singular vectors of the bidiagonal 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.

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.

PERM

          PERM is INTEGER array, dimension ( N )


         The permutations (from deflation and sorting) to be applied


         to each block. Not referenced if ICOMPQ = 0.

GIVPTR

          GIVPTR is INTEGER


         The number of Givens rotations which took place in this


         subproblem. Not referenced if ICOMPQ = 0.

GIVCOL

          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )


         Each pair of numbers indicates a pair of columns to take place


         in a Givens rotation. Not referenced if ICOMPQ = 0.

LDGCOL

          LDGCOL is INTEGER


         leading dimension of GIVCOL, must be at least N.

GIVNUM

          GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )


         Each number indicates the C or S value to be used in the


         corresponding Givens rotation. Not referenced if ICOMPQ = 0.

LDGNUM

          LDGNUM is INTEGER


         The leading dimension of GIVNUM and POLES, must be at least N.

POLES

          POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )


         On exit, POLES(1,*) is an array containing the new singular


         values obtained from solving the secular equation, and


         POLES(2,*) is an array containing the poles in the secular


         equation. Not referenced if ICOMPQ = 0.

DIFL

          DIFL is DOUBLE PRECISION array, dimension ( N )


         On exit, DIFL(I) is the distance between I-th updated


         (undeflated) singular value and the I-th (undeflated) old


         singular value.

DIFR

          DIFR is DOUBLE PRECISION array,


                  dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and


                  dimension ( N ) if ICOMPQ = 0.


         On exit, DIFR(I, 1) is the distance between I-th updated


         (undeflated) singular value and the I+1-th (undeflated) old


         singular value.


         If ICOMPQ = 1, DIFR(1:K,2) is an array containing the


         normalizing factors for the right singular vector matrix.


         See DLASD8 for details on DIFL and DIFR.

Z

          Z is DOUBLE PRECISION array, dimension ( M )


         The first elements of this array contain the components


         of the deflation-adjusted updating row vector.

K

          K is INTEGER


         Contains the dimension of the non-deflated matrix,


         This is the order of the related secular equation. 1 <= K <=N.

C

          C is DOUBLE PRECISION


         C contains garbage if SQRE =0 and the C-value of a Givens


         rotation related to the right null space if SQRE = 1.

S

          S is DOUBLE PRECISION


         S contains garbage if SQRE =0 and the S-value of a Givens


         rotation related to the right null space if SQRE = 1.

WORK

          WORK is DOUBLE PRECISION array, dimension ( 4 * M )

IWORK

          IWORK is INTEGER array, dimension ( 3 * N )

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.

September 2012

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