DLAED1 (3lapack)

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