slaed8(3S)

NAME
     SLAED8 - merge the two sets of eigenvalues together into a single sorted
     set

SYNOPSIS
     SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z,
                        DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL, GIVNUM,
                        INDXP, INDX, INFO )

         INTEGER        CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, QSIZ

         REAL           RHO

         INTEGER        GIVCOL( 2, * ), INDX( * ), INDXP( * ), INDXQ( * ),
                        PERM( * )

         REAL           D( * ), DLAMDA( * ), GIVNUM( 2, * ), Q( LDQ, * ), Q2(
                        LDQ2, * ), W( * ), Z( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     SLAED8 merges the two sets of eigenvalues 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 eigenvalues are close
     together or if there is a tiny element in the Z vector.  For each such
     occurrence the order of the related secular equation problem is reduced
     by one.


ARGUMENTS
     ICOMPQ  (input) INTEGER
             = 0:  Compute eigenvalues only.
             = 1:  Compute eigenvectors of original dense symmetric matrix
             also.  On entry, Q contains the orthogonal matrix used to reduce
             the original matrix to tridiagonal form.

     K      (output) INTEGER
            The number of non-deflated eigenvalues, and the order of the
            related secular equation.

     N      (input) INTEGER
            The dimension of the symmetric tridiagonal matrix.  N >= 0.

     QSIZ   (input) INTEGER
            The dimension of the orthogonal matrix used to reduce the full
            matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

     D      (input/output) REAL array, dimension (N)
            On entry, the eigenvalues of the two submatrices to be combined.
            On exit, the trailing (N-K) updated eigenvalues (those which were
            deflated) sorted into increasing order.

     Q      (input/output) REAL array, dimension (LDQ,N)
            If ICOMPQ = 0, Q is not referenced.  Otherwise, on entry, Q
            contains the eigenvectors of the partially solved system which has
            been previously updated in matrix multiplies with other partially
            solved eigensystems.  On exit, Q contains the trailing (N-K)
            updated eigenvectors (those which were deflated) in its last N-K
            columns.

     LDQ    (input) INTEGER
            The leading dimension of the array Q.  LDQ >= max(1,N).

     INDXQ  (input) INTEGER array, dimension (N)
            The permutation which separately sorts the two sub-problems in D
            into ascending order.  Note that elements in the second half of
            this permutation must first have CUTPNT added to their values in
            order to be accurate.

     RHO    (input/output) REAL
            On entry, the off-diagonal element associated with the rank-1 cut
            which originally split the two submatrices which are now being
            recombined.  On exit, RHO has been modified to the value required
            by SLAED3.

            CUTPNT (input) INTEGER The location of the last eigenvalue in the
            leading sub-matrix.  min(1,N) <= CUTPNT <= N.

     Z      (input) REAL array, dimension (N)
            On entry, Z contains the updating vector (the last row of the
            first sub-eigenvector matrix and the first row of the second sub-
            eigenvector matrix).  On exit, the contents of Z are destroyed by
            the updating process.

            DLAMDA (output) REAL array, dimension (N) A copy of the first K
            eigenvalues which will be used by SLAED3 to form the secular
            equation.

     Q2     (output) REAL array, dimension (LDQ2,N)
            If ICOMPQ = 0, Q2 is not referenced.  Otherwise, a copy of the
            first K eigenvectors which will be used by SLAED7 in a matrix
            multiply (SGEMM) to update the new eigenvectors.
     LDQ2   (input) INTEGER
            The leading dimension of the array Q2.  LDQ2 >= max(1,N).

     W      (output) REAL array, dimension (N)
            The first k values of the final deflation-altered z-vector and
            will be passed to SLAED3.

     PERM   (output) INTEGER array, dimension (N)
            The permutations (from deflation and sorting) to be applied to
            each eigenblock.

            GIVPTR (output) INTEGER The number of Givens rotations which took
            place in this subproblem.

            GIVCOL (output) INTEGER array, dimension (2, N) Each pair of
            numbers indicates a pair of columns to take place in a Givens
            rotation.

            GIVNUM (output) REAL array, dimension (2, N) Each number indicates
            the S value to be used in the corresponding Givens rotation.

     INDXP  (workspace) INTEGER array, dimension (N)
            The permutation used to place deflated values of D at the end of
            the array.  INDXP(1:K) points to the nondeflated D-values
            and INDXP(K+1:N) points to the deflated eigenvalues.

     INDX   (workspace) INTEGER array, dimension (N)
            The permutation used to sort the contents of D into ascending
            order.

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

FURTHER DETAILS
     Based on contributions by
        Jeff Rutter, Computer Science Division, University of California
        at Berkeley, USA


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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