SLAED7 - 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 tridiago
nal form. SLAED1 handles the case in which all eigenval
ues 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.
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 eigen
vectors for
the current problem are multiplied with the eigen
vectors from
the overall problem.
ARGUMENTS
ICOMPQ (input) INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense sym
metric 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 (count
ing 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
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 subprob
lem 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 encoun
tered 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 submatri
ces stored in QSTORE. The submatrices are numbered
starting at the bottom left of the divide and con
quer 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 prob
lem.
PERM (input) INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sort
ing) 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.
IWORK (workspace) INTEGER array, dimension (4*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille
gal 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
Man(1) output converted with
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