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NAME
slaed8 - merge the two sets of eigenvalues together into a
single sorted set
SYNOPSIS
SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ,
RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
GIVCOL, GIVNUM, INDXP, INDX, INFO )
INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, QSIZ
REAL RHO
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), INDXQ( * ),
PERM( * )
REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ), Q( LDQ, * ), Q2(
LDQ2, * ), W( * ), Z( * )
#include <sunperf.h>
void slaed8(int icompq, int *k, int n, int qsiz, float *d,
float *q, int ldq, int *indxq, float *srho, int
cutpnt, float *sz, float *dlamda, float *q2, int
ldq2, float *w, int *perm, int *givptr, int
*givcol, float *givnum, int *indxp, int *indx, int
*info);
PURPOSE
SLAED8 merges the two sets of eigenvalues together into a
single sorted set. Then it tries to deflate the size of the
problem. There are two ways in which deflation can occur:
when two or more eigenvalues are close together or if there
is a tiny element in the Z vector. For each such occurrence
the order of the related secular equation problem is reduced
by one.
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.
K (output) INTEGER
The number of non-deflated eigenvalues, and the
order of the related secular equation.
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.
D (input/output) REAL array, dimension (N)
On entry, the eigenvalues of the two submatrices
to be combined. On exit, the trailing (N-K)
updated eigenvalues (those which were deflated)
sorted into increasing order.
Q (input/output) REAL array, dimension (LDQ,N)
If ICOMPQ = 0, Q is not referenced. Otherwise, on
entry, Q contains the eigenvectors of the par-
tially solved system which has been previously
updated in matrix multiplies with other partially
solved eigensystems. On exit, Q contains the
trailing (N-K) updated eigenvectors (those which
were deflated) in its last N-K columns.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >=
max(1,N).
INDXQ (input) INTEGER array, dimension (N)
The permutation which separately sorts the two
sub-problems in D into ascending order. Note that
elements in the second half of this permutation
must first have CUTPNT added to their values in
order to be accurate.
RHO (input/output) REAL
On entry, the off-diagonal element associated with
the rank-1 cut which originally split the two sub-
matrices which are now being recombined. On exit,
RHO has been modified to the value required by
SLAED3.
CUTPNT (input) INTEGER The location of the last
eigenvalue in the leading sub-matrix. min(1,N) <=
CUTPNT <= N.
Z (input) REAL array, dimension (N)
On entry, Z contains the updating vector (the last
row of the first sub-eigenvector matrix and the
first row of the second sub-eigenvector matrix).
On exit, the contents of Z are destroyed by the
updating process.
DLAMDA (output) REAL array, dimension (N) A copy
of the first K eigenvalues which will be used by
SLAED3 to form the secular equation.
Q2 (output) REAL array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise, a
copy of the first K eigenvectors which will be
used by SLAED7 in a matrix multiply (SGEMM) to
update the new eigenvectors.
LDQ2 (input) INTEGER
The leading dimension of the array Q2. LDQ2 >=
max(1,N).
W (output) REAL array, dimension (N)
The first k values of the final deflation-altered
z-vector and will be passed to SLAED3.
PERM (output) INTEGER array, dimension (N)
The permutations (from deflation and sorting) to
be applied to each eigenblock.
GIVPTR (output) INTEGER The number of Givens rota-
tions which took place in this subproblem.
GIVCOL (output) INTEGER array, dimension (2, N)
Each pair of numbers indicates a pair of columns
to take place in a Givens rotation.
GIVNUM (output) REAL array, dimension (2, N) Each
number indicates the S value to be used in the
corresponding Givens rotation.
INDXP (workspace) INTEGER array, dimension (N)
The permutation used to place deflated values of D
at the end of the array. INDXP(1:K) points to the
nondeflated D-values
and INDXP(K+1:N) points to the deflated eigen-
values.
INDX (workspace) INTEGER array, dimension (N)
The permutation used to sort the contents of D
into ascending order.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille-
gal value.
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