claed8(3S)

NAME
     CLAED8 - merge the two sets of eigenvalues together into a single sorted
     set

SYNOPSIS
     SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2,
                        LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL,
                        GIVNUM, INFO )

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

         REAL           RHO

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

         REAL           D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ), Z( * )

         COMPLEX        Q( LDQ, * ), Q2( LDQ2, * )

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
     CLAED8 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
     K      (output) INTEGER
            Contains the number of non-deflated eigenvalues.  This is 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 unitary matrix used to reduce the dense or
            band matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

     Q      (input/output) COMPLEX array, dimension (LDQ,N)
            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 ).

     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.

     RHO    (input/output) REAL
            Contains the off diagonal element associated with the rank-1 cut
            which originally split the two submatrices which are now being
            recombined. RHO is modified during the computation to the value
            required by SLAED3.

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

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

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

     Q2     (output) COMPLEX array, dimension (LDQ2,N)
            If ICOMPQ = 0, Q2 is not referenced.  Otherwise, Contains 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)
            This will hold the first k values of the final deflation-altered
            z-vector and will be passed to SLAED3.
     INDXP  (workspace) INTEGER array, dimension (N)
            This will contain the permutation used to place deflated values of
            D at the end of the array. On output 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)
            This will contain the permutation used to sort the contents of D
            into ascending order.

     INDXQ  (input) INTEGER array, dimension (N)
            This contains 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.

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

            GIVPTR (output) INTEGER Contains 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.

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

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

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