slaed2(3S)

NAME
     SLAED2 - merge the two sets of eigenvalues together into a single sorted
     set

SYNOPSIS
     SUBROUTINE SLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W, Q2,
                        INDX, INDXC, INDXP, COLTYP, INFO )

         INTEGER        INFO, K, LDQ, N, N1

         REAL           RHO

         INTEGER        COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), INDXQ(
                        * )

         REAL           D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), 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
     SLAED2 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 entry in the Z vector.  For each such
     occurrence the order of the related secular equation problem is reduced
     by one.


ARGUMENTS
     K      (output) INTEGER
            The number of non-deflated eigenvalues, and the order of the
            related secular equation. 0 <= K <=N.

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

     N1     (input) INTEGER
            The location of the last eigenvalue in the leading sub-matrix.
            min(1,N) <= N1 <= N/2.

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

     Q      (input/output) REAL array, dimension (LDQ, N)
            On entry, Q contains the eigenvectors of two submatrices in the
            two square blocks with corners at (1,1), (N1,N1) and (N1+1, N1+1),
            (N,N).  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/output) 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 N1 added to their values.
            Destroyed on exit.

     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.

     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 have been
            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.

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

     Q2     (output) REAL array, dimension (N1**2+(N-N1)**2)
            A copy of the first K eigenvectors which will be used by SLAED3 in
            a matrix multiply (SGEMM) to solve for the new eigenvectors.

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

     INDXC  (output) INTEGER array, dimension (N)
            The permutation used to arrange the columns of the deflated Q
            matrix into three groups:  the first group contains non-zero
            elements only at and above N1, the second contains non-zero
            elements only below N1, and the third is dense.

     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.

            COLTYP (workspace/output) INTEGER array, dimension (N) During
            execution, a label which will indicate which of the following
            types a column in the Q2 matrix is:
            1 : non-zero in the upper half only;
            2 : dense;
            3 : non-zero in the lower half only;
            4 : deflated.  On exit, COLTYP(i) is the number of columns of type
            i, for i=1 to 4 only.

     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
     Modified by Francoise Tisseur, University of Tennessee.


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

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