dlaed1(3F)

NAME
     DLAED1 - compute the updated eigensystem of a diagonal matrix after
     modification by a rank-one symmetric matrix

SYNOPSIS
     SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, INFO )

         INTEGER        CUTPNT, INFO, LDQ, N

         DOUBLE         PRECISION RHO

         INTEGER        INDXQ( * ), IWORK( * )

         DOUBLE         PRECISION D( * ), Q( LDQ, * ), WORK( * )

PURPOSE
     DLAED1 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 eigenvectors of a
     tridiagonal matrix.  DLAED7 handles the case in which eigenvalues only or
     eigenvalues and eigenvectors of a full symmetric matrix (which was
     reduced to tridiagonal form) 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 DLAED2.

           The second stage consists of calculating the updated
           eigenvalues. This is done by finding the roots of the secular
           equation via the routine DLAED4 (as called by SLAED3).
           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
     N      (input) INTEGER
            The dimension of the symmetric tridiagonal matrix.  N >= 0.

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

     Q      (input/output) DOUBLE PRECISION 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  (input/output) INTEGER array, dimension (N)
            On entry, the permutation which separately sorts the two
            subproblems in D into ascending order.  On exit, the permutation
            which will reintegrate the subproblems back into sorted order,
            i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

     RHO    (input) DOUBLE PRECISION
            The subdiagonal entry used to create the rank-1 modification.

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

     WORK   (workspace) DOUBLE PRECISION array, dimension (3*N+2*N**2)

     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