DLASD7 merges the two sets of singular values together into a single


 sorted set. Then it tries to deflate the size of the problem. There


 are two ways in which deflation can occur:  when two or more singular


 values are close together or if there is a tiny entry in the Z


 vector. For each such occurrence the order of the related


 secular equation problem is reduced by one.


 DLASD7 is called from DLASD6.

ICOMPQ

          ICOMPQ is INTEGER


          Specifies whether singular vectors are to be computed


          in compact form, as follows:


          = 0: Compute singular values only.


          = 1: Compute singular vectors of upper


               bidiagonal matrix in compact form.

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


         N = NL + NR + 1 rows and


         M = N + SQRE >= N columns.

K

          K is INTEGER


         Contains the dimension of the non-deflated matrix, this is


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

D

          D is DOUBLE PRECISION array, dimension ( N )


         On entry D contains the singular values of the two submatrices


         to be combined. On exit D contains the trailing (N-K) updated


         singular values (those which were deflated) sorted into


         increasing order.

Z

          Z is DOUBLE PRECISION array, dimension ( M )


         On exit Z contains the updating row vector in the secular


         equation.

ZW

          ZW is DOUBLE PRECISION array, dimension ( M )


         Workspace for Z.

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.

VFW

          VFW is DOUBLE PRECISION array, dimension ( M )


         Workspace for VF.

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.

VLW

          VLW is DOUBLE PRECISION array, dimension ( M )


         Workspace for VL.

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.

DSIGMA

          DSIGMA is DOUBLE PRECISION array, dimension ( N )


         Contains a copy of the diagonal elements (K-1 singular values


         and one zero) in the secular equation.

IDX

          IDX is INTEGER array, dimension ( N )


         This will contain the permutation used to sort the contents of


         D into ascending order.

IDXP

          IDXP is INTEGER array, dimension ( N )


         This will contain the permutation used to place deflated


         values of D at the end of the array. On output IDXP(2:K)


         points to the nondeflated D-values and IDXP(K+1:N)


         points to the deflated singular values.

IDXQ

          IDXQ is INTEGER array, dimension ( N )


         This contains the permutation which separately sorts the two


         sub-problems in D into ascending order.  Note that entries in


         the first half of this permutation must first be moved one


         position backward; and entries in the second half


         must first have NL+1 added to their values.

PERM

          PERM is INTEGER array, dimension ( N )


         The permutations (from deflation and sorting) to be applied


         to each singular 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


         The 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, must be at least 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.

INFO

          INFO is INTEGER


         = 0:  successful exit.


         < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

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