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( * )
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 prob
lem 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
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 partially
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 ele
ments 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 >=
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 cor
responding 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 eigenval
ues.
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.
FURTHER DETAILS
Based on contributions by
Jeff Rutter, Computer Science Division, University of
California
at Berkeley, USA
Man(1) output converted with
man2html