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NAME
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(
* )
#include <sunperf.h>
void slaed7(int icompq, int n, int qsiz, int tlvls, int
curlvl, int curpbm, float *d, float *q, int ldq,
int *indxq, float srho, int cutpnt, float *qstore,
int *qptr, int *prmptr, int *perm, int *givptr,
int *givcol, float *givnum, int *info);
PURPOSE
SLAED7 computes the updated eigensystem of a diagonal matrix
after modification by a rank-one symmetric matrix. This rou-
tine 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 prob-
lem 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 eigen-
values. 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 eigenvec-
tors directly using the updated eigenvalues. The eigenvec-
tors 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
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 sub-
problem 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 subma-
trices.
QPTR (input/output) INTEGER array, dimension (N+2)
List of indices pointing to beginning of subma-
trices stored in QSTORE. The submatrices are num-
bered 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 ille-
gal value.
> 0: if INFO = 1, an eigenvalue did not converge
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