CXML
        	    
LAPACK version 3.0
dlaed7(3)

PURPOSE
  DLAED7 - compute the updated eigensystem of a diagonal matrix after
  modification by a rank-one symmetric matrix

SYNTAX
  SUBROUTINE DLAED7( 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

      DOUBLE	     PRECISION RHO

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

      DOUBLE	     PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ), QSTORE( *
		     ), WORK( * )

DESCRIPTION
  DLAED7 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.
  DLAED1 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 DLAED8.

	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 DLAED9).
	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) 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	 (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) DOUBLE PRECISION
	 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (2, N lg N) Each
	 number indicates the S value to be used in the corresponding Givens
	 rotation.

  WORK	 (workspace) DOUBLE PRECISION 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

FURTHER DETAILS
  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA