SLARRE - the tridiagonal matrix T, SLARRE 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 SLARRE( N, D, E, TOL, NSPLIT, ISPLIT, M, W,
WOFF, GERSCH, WORK, INFO )
INTEGER INFO, M, N, NSPLIT
REAL TOL
INTEGER ISPLIT( * )
REAL D( * ), E( * ), GERSCH( * ), W( * ),
WOFF( * ), WORK( * )
PURPOSE
Given the tridiagonal matrix T, SLARRE 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 SSTEGR 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
SLASQ2). As an added benefit, SLARRE 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) REAL 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) REAL 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.
TOL (input) REAL
The threshold for splitting. If on input |E(i)| <
TOL, then the matrix T is split into smaller
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) REAL 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) REAL array, dimension (N)
The NSPLIT base points sigma_i.
GERSCH (output) REAL array, dimension (2*N)
The n Gerschgorin intervals.
WORK (input) REAL array, dimension (4*N???)
Workspace.
INFO (output) INTEGER
Output error code from SLASQ2
FURTHER DETAILS
Based on contributions by
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Man(1) output converted with
man2html