DLARRE - the tridiagonal matrix T, DLARRE sets "small"
off-diagonal elements to zero, and for each unreduced
block T_i, it finds (i) the numbers sigma_i (ii) the base
T_i - sigma_i I = L_i D_i L_i^T representations and (iii)
eigenvalues of each L_i D_i L_i^T
SYNOPSIS
SUBROUTINE DLARRE( N, D, E, TOL, NSPLIT, ISPLIT, M, W,
WOFF, GERSCH, WORK, INFO )
INTEGER INFO, M, N, NSPLIT
DOUBLE PRECISION TOL
INTEGER ISPLIT( * )
DOUBLE PRECISION D( * ), E( * ), GERSCH( * ),
W( * ), WOFF( * ), WORK( * )
PURPOSE
Given the tridiagonal matrix T, DLARRE sets "small" off-
diagonal elements to zero, and for each unreduced block
T_i, it finds (i) the numbers sigma_i (ii) the base T_i -
sigma_i I = L_i D_i L_i^T representations and (iii) eigen
values of each L_i D_i L_i^T. The representations and
eigenvalues found are then used by DSTEGR to compute the
eigenvectors of a symmetric tridiagonal matrix. Currently,
the base representations are limited to being positive or
negative definite, and the eigenvalues of the definite
matrices are found by the dqds algorithm (subroutine
DLASQ2). As an added benefit, DLARRE also outputs the n
Gerschgorin
intervals for each L_i D_i L_i^T.
ARGUMENTS
N (input) INTEGER
The order of the matrix.
D (input/output) DOUBLE PRECISION array, dimension
(N)
On entry, the n diagonal elements of the tridiago
nal matrix T. On exit, the n diagonal elements of
the diagonal matrices D_i.
E (input/output) DOUBLE PRECISION array, dimension
(N)
On entry, the (n-1) subdiagonal elements of the
tridiagonal matrix T; E(N) need not be set. On
exit, the subdiagonal elements of the unit bidiag
onal matrices L_i.
The threshold for splitting. If on input |E(i)| <
TOL, then the matrix T is split into smaller
blocks.
NSPLIT (input) INTEGER
The number of blocks T splits into. 1 <= NSPLIT <=
N.
ISPLIT (output) INTEGER array, dimension (2*N)
The splitting points, at which T breaks up into
submatrices. The first submatrix consists of
rows/columns 1 to ISPLIT(1), the second of
rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
M (output) INTEGER
The total number of eigenvalues (of all the L_i
D_i L_i^T) found.
W (output) DOUBLE PRECISION array, dimension (N)
The first M elements contain the eigenvalues. The
eigenvalues of each of the blocks, L_i D_i L_i^T,
are sorted in ascending order.
WOFF (output) DOUBLE PRECISION array, dimension (N)
The NSPLIT base points sigma_i.
GERSCH (output) DOUBLE PRECISION array, dimension (2*N)
The n Gerschgorin intervals.
WORK (input) DOUBLE PRECISION array, dimension (4*N???)
Workspace.
INFO (output) INTEGER
Output error code from DLASQ2
FURTHER DETAILS
Based on contributions by
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Man(1) output converted with
man2html