DLAED1(l)								 )								 DLAED1(l)

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 DLAED3).
	     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/2.

       WORK   (workspace) DOUBLE PRECISION array, dimension (4*N + 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

FURTHER DETAILS

       Based on contributions by
	  Jeff Rutter, Computer Science Division, University of California
	  at Berkeley, USA
       Modified by Francoise Tisseur, University of Tennessee.

LAPACK version 3.0						   15 June 2000 							 DLAED1(l)