SLASD2(3)      LAPACK routine of NEC Numeric Library Collection      SLASD2(3)



NAME
       SLASD2

SYNOPSIS
       SUBROUTINE SLASD2 (NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,
           LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC, IDXQ, COLTYP,
           INFO)



PURPOSE
            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.




ARGUMENTS
           NL        (input)
                     NL is INTEGER
                    The row dimension of the upper block.  NL >= 1.

           NR        (input)
                     NR is INTEGER
                    The row dimension of the lower block.  NR >= 1.

           SQRE      (input)
                     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         (output)
                     K is INTEGER
                    Contains the dimension of the non-deflated matrix,
                    This is the order of the related secular equation. 1 <= K <=N.

           D         (input/output)
                     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         (output)
                     Z is REAL array, dimension (N)
                    On exit Z contains the updating row vector in the secular
                    equation.

           ALPHA     (input)
                     ALPHA is REAL
                    Contains the diagonal element associated with the added row.

           BETA      (input)
                     BETA is REAL
                    Contains the off-diagonal element associated with the added
                    row.

           U         (input/output)
                     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       (input)
                     LDU is INTEGER
                    The leading dimension of the array U.  LDU >= N.

           VT        (input/output)
                     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      (input)
                     LDVT is INTEGER
                    The leading dimension of the array VT.  LDVT >= M.

           DSIGMA    (output)
                     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        (output)
                     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      (input)
                     LDU2 is INTEGER
                    The leading dimension of the array U2.  LDU2 >= N.

           VT2       (output)
                     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     (input)
                     LDVT2 is INTEGER
                    The leading dimension of the array VT2.  LDVT2 >= M.

           IDXP      (output)
                     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       (output)
                     IDX is INTEGER array, dimension (N)
                    This will contain the permutation used to sort the contents of
                    D into ascending order.

           IDXC      (output)
                     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      (input/output)
                     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    (output)
                     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      (output)
                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.



LAPACK routine                  31 October 2017                      SLASD2(3)