SLAED7 (3lapack)

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

SYNOPSIS

  SUBROUTINE SLAED7( 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

      REAL           RHO

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

      REAL           D( * ), GIVNUM( 2, * ), Q( LDQ, * ), QSTORE( * ), WORK(
                     * )

PURPOSE

  SLAED7 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.
  SLAED1 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 SLAED8.

        The second stage consists of calculating the updated
        eigenvalues. This is done by finding the roots of the secular
        equation via the routine SLAED4 (as called by SLAED9).
        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) REAL array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.  On exit,
         the eigenvalues of the repaired matrix.

  Q      (input/output) REAL 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) REAL
         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) REAL 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) REAL array, dimension (2, N lg N) Each number
         indicates the S value to be used in the corresponding Givens
         rotation.

  WORK   (workspace) REAL 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