CXML
        	    
LAPACK version 3.0
dlaed1(3)

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

SYNTAX
  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( * )

DESCRIPTION
  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.