slaed7(3F)

NAME
     SLAED7 - compute the updated eigensystem of a diagonal matrix after
     modification by a rank-one symmetric matrix

SYNOPSIS
     SUBROUTINE SLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ,
                        INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM,
                        GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, INFO )

         INTEGER        CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N, QSIZ,
                        TLVLS

         REAL           RHO

         INTEGER        GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), IWORK( * ),
                        PERM( * ), PRMPTR( * ), QPTR( * )

         REAL           D( * ), GIVNUM( 2, * ), Q( LDQ, * ), QSTORE( * ),
                        WORK( * )

PURPOSE
     SLAED7 computes the updated eigensystem of a diagonal matrix after
     modification by a rank-one symmetric matrix. This routine is used only
     for the eigenproblem which requires all eigenvalues and optionally
     eigenvectors of a dense symmetric matrix that has been reduced to
     tridiagonal form.  SLAED1 handles the case in which all eigenvalues and
     eigenvectors of a symmetric tridiagonal matrix are desired.

       T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)

        where Z = Q'u, u is a vector of length N with ones in the
        CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

        The eigenvectors of the original matrix are stored in Q, and the
        eigenvalues are in D.  The algorithm consists of three stages:

           The first stage consists of deflating the size of the problem
           when there are multiple eigenvalues or if there is a zero in
           the Z vector.  For each such occurence the dimension of the
           secular equation problem is reduced by one.  This stage is
           performed by the routine SLAED8.

           The second stage consists of calculating the updated
           eigenvalues. This is done by finding the roots of the secular
           equation via the routine SLAED4 (as called by SLAED9).
           This routine also calculates the eigenvectors of the current
           problem.

           The final stage consists of computing the updated eigenvectors
           directly using the updated eigenvalues.  The eigenvectors for
           the current problem are multiplied with the eigenvectors from
           the overall problem.
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.

     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.

     TLVLS  (input) INTEGER
            The total number of merging levels in the overall divide and
            conquer tree.

            CURLVL (input) INTEGER The current level in the overall merge
            routine, 0 <= CURLVL <= TLVLS.

            CURPBM (input) INTEGER The current problem in the current level in
            the overall merge routine (counting from upper left to lower
            right).

     D      (input/output) REAL array, dimension (N)
            On entry, the eigenvalues of the rank-1-perturbed matrix.  On
            exit, the eigenvalues of the repaired matrix.

     Q      (input/output) REAL array, dimension (LDQ, N)
            On entry, the eigenvectors of the rank-1-perturbed matrix.  On
            exit, the eigenvectors of the repaired tridiagonal matrix.

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

     INDXQ  (output) INTEGER array, dimension (N)
            The permutation which will reintegrate the subproblem just solved
            back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will be in
            ascending order.

     RHO    (input) REAL
            The subdiagonal element used to create the rank-1 modification.

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

            QSTORE (input/output) REAL array, dimension (N**2+1) Stores
            eigenvectors of submatrices encountered during divide and conquer,
            packed together. QPTR points to beginning of the submatrices.

     QPTR   (input/output) INTEGER array, dimension (N+2)
            List of indices pointing to beginning of submatrices stored in
            QSTORE. The submatrices are numbered starting at the bottom left
            of the divide and conquer tree, from left to right and bottom to
            top.

            PRMPTR (input) INTEGER array, dimension (N lg N) Contains a list
            of pointers which indicate where in PERM a level's permutation is
            stored.  PRMPTR(i+1) - PRMPTR(i) indicates the size of the
            permutation and also the size of the full, non-deflated problem.

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

            GIVPTR (input) INTEGER array, dimension (N lg N) Contains a list
            of pointers which indicate where in GIVCOL a level's Givens
            rotations are stored.  GIVPTR(i+1) - GIVPTR(i) indicates the
            number of Givens rotations.

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

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

     WORK   (workspace) REAL array, dimension (3*N+QSIZ*N)

     IWORK  (workspace) INTEGER array, dimension (4*N)

     INFO   (output) INTEGER
            = 0:  successful exit.
            < 0:  if INFO = -i, the i-th argument had an illegal value.
            > 0:  if INFO = 1, an eigenvalue did not converge