SLASD2 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.


 SLASD2 is called from SLASD1.

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 REAL 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 REAL array, dimension (N)


         On exit Z contains the updating row vector in the secular


         equation.

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 contains the left singular vectors of two


         submatrices in the two square blocks with corners at (1,1),


         (NL, NL), and (NL+2, NL+2), (N,N).


         On exit U contains the trailing (N-K) updated left singular


         vectors (those which were deflated) in its last N-K columns.

LDU

          LDU is INTEGER


         The leading dimension of the array U.  LDU >= N.

VT

          VT is REAL array, dimension (LDVT,M)


         On entry VT**T contains the right singular vectors of two


         submatrices in the two square blocks with corners at (1,1),


         (NL+1, NL+1), and (NL+2, NL+2), (M,M).


         On exit VT**T contains the trailing (N-K) updated right singular


         vectors (those which were deflated) in its last N-K columns.


         In case SQRE =1, the last row of VT spans the right null


         space.

LDVT

          LDVT is INTEGER


         The leading dimension of the array VT.  LDVT >= M.

DSIGMA

          DSIGMA is REAL array, dimension (N)


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


         and one zero) in the secular equation.

U2

          U2 is REAL array, dimension (LDU2,N)


         Contains a copy of the first K-1 left singular vectors which


         will be used by SLASD3 in a matrix multiply (SGEMM) to solve


         for the new left singular vectors. U2 is arranged into four


         blocks. The first block contains a column with 1 at NL+1 and


         zero everywhere else; the second block contains non-zero


         entries only at and above NL; the third contains non-zero


         entries only below NL+1; and the fourth is dense.

LDU2

          LDU2 is INTEGER


         The leading dimension of the array U2.  LDU2 >= N.

VT2

          VT2 is REAL array, dimension (LDVT2,N)


         VT2**T contains a copy of the first K right singular vectors


         which will be used by SLASD3 in a matrix multiply (SGEMM) to


         solve for the new right singular vectors. VT2 is arranged into


         three blocks. The first block contains a row that corresponds


         to the special 0 diagonal element in SIGMA; the second block


         contains non-zeros only at and before NL +1; the third block


         contains non-zeros only at and after  NL +2.

LDVT2

          LDVT2 is INTEGER


         The leading dimension of the array VT2.  LDVT2 >= M.

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.

IDX

          IDX is INTEGER array, dimension (N)


         This will contain the permutation used to sort the contents of


         D into ascending order.

IDXC

          IDXC is INTEGER array, dimension (N)


         This will contain the permutation used to arrange the columns


         of the deflated U matrix into three groups:  the first group


         contains non-zero entries only at and above NL, the second


         contains non-zero entries only below NL+2, and the third is


         dense.

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 hlaf 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.

COLTYP

          COLTYP is INTEGER array, dimension (N)


         As workspace, this will contain a label which will indicate


         which of the following types a column in the U2 matrix or a


         row in the VT2 matrix is:


         1 : non-zero in the upper half only


         2 : non-zero in the lower half only


         3 : dense


         4 : deflated


         On exit, it is an array of dimension 4, with COLTYP(I) being


         the dimension of the I-th type columns.

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