CLAED8 - merge the two sets of eigenvalues together into a
single sorted set
SYNOPSIS
SUBROUTINE CLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z,
DLAMDA, Q2, LDQ2, W, INDXP, INDX,
INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM,
INFO )
INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N,
QSIZ
REAL RHO
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
INDXQ( * ), PERM( * )
REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ), W(
* ), Z( * )
COMPLEX Q( LDQ, * ), Q2( LDQ2, * )
PURPOSE
CLAED8 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
K (output) INTEGER
Contains the number of non-deflated eigenvalues.
This is 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 unitary matrix used to reduce
the dense or band matrix to tridiagonal form. QSIZ
>= N if ICOMPQ = 1.
Q (input/output) COMPLEX array, dimension (LDQ,N)
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.
The leading dimension of the array Q. LDQ >= max(
1, N ).
D (input/output) REAL array, dimension (N)
On entry, D contains the eigenvalues of the two
submatrices to be combined. On exit, D contains
the trailing (N-K) updated eigenvalues (those which
were deflated) sorted into increasing order.
RHO (input/output) REAL
Contains the off diagonal element associated with
the rank-1 cut which originally split the two sub
matrices which are now being recombined. RHO is
modified during the computation to the value
required by SLAED3.
CUTPNT (input) INTEGER Contains the location of the
last eigenvalue in the leading sub-matrix.
MIN(1,N) <= CUTPNT <= N.
Z (input) REAL array, dimension (N)
On input this vector contains the updating vector
(the last row of the first sub-eigenvector matrix
and the first row of the second sub-eigenvector
matrix). The contents of Z are destroyed during
the updating process.
DLAMDA (output) REAL array, dimension (N) Contains
a copy of the first K eigenvalues which will be
used by SLAED3 to form the secular equation.
Q2 (output) COMPLEX array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
Contains 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)
This will hold the first k values of the final
deflation-altered z-vector and will be passed to
SLAED3.
INDXP (workspace) INTEGER array, dimension (N)
This will contain the permutation used to place
deflated values of D at the end of the array. On
output INDXP(1:K)
points to the nondeflated D-values and INDXP(K+1:N)
points to the deflated eigenvalues.
This will contain the permutation used to sort the
contents of D into ascending order.
INDXQ (input) INTEGER array, dimension (N)
This contains 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.
PERM (output) INTEGER array, dimension (N)
Contains the permutations (from deflation and sort
ing) to be applied to each eigenblock.
GIVPTR (output) INTEGER Contains the number of
Givens rotations which took place in this subprob
lem.
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.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille
gal value.
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