ZLAED8(3F)ZLAED8(3F)NAME
ZLAED8 - merge the two sets of eigenvalues together into a single sorted
set
SYNOPSIS
SUBROUTINE ZLAED8( 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
DOUBLE PRECISION RHO
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), INDXQ( * ),
PERM( * )
DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
Z( * )
COMPLEX*16 Q( LDQ, * ), Q2( LDQ2, * )
PURPOSE
ZLAED8 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
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*16 array, dimension (LDQ,N)
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 ).
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ZLAED8(3F)ZLAED8(3F)
D (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION
Contains the off diagonal element associated with the rank-1 cut
which originally split the two submatrices which are now being
recombined. RHO is modified during the computation to the value
required by DLAED3.
CUTPNT (input) INTEGER Contains the location of the last
eigenvalue in the leading sub-matrix. MIN(1,N) <= CUTPNT <= N.
Z (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N) Contains a
copy of the first K eigenvalues which will be used by DLAED3 to
form the secular equation.
Q2 (output) COMPLEX*16 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 DLAED7 in a
matrix multiply (DGEMM) to update the new eigenvectors.
LDQ2 (input) INTEGER
The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
W (output) DOUBLE PRECISION array, dimension (N)
This will hold the first k values of the final deflation-altered
z-vector and will be passed to DLAED3.
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.
INDX (workspace) INTEGER array, dimension (N)
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.
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ZLAED8(3F)ZLAED8(3F)
PERM (output) INTEGER array, dimension (N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.
GIVPTR (output) INTEGER Contains the number of Givens rotations
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) DOUBLE PRECISION array, dimension (2, N) Each
number indicates the S value to be used in the corresponding
Givens rotation.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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