SLARRE(l) ) SLARRE(l)NAME
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 represen‐
tations 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 ele‐
ments to zero, and for each unreduced block T_i, it finds (i) the num‐
bers sigma_i (ii) the base T_i - sigma_i I = L_i D_i L_i^T representa‐
tions and (iii) eigenvalues of each L_i D_i L_i^T. The representations
and eigenvalues found are then used by SSTEGR to compute the eigenvec‐
tors of a symmetric tridiagonal matrix. Currently, the base representa‐
tions are limited to being positive or negative definite, and the ei‐
genvalues 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 tridiagonal 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 ele‐
ments of the unit bidiagonal matrices L_i.
TOL (input) REAL
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) 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
LAPACK version 3.0 15 June 2000 SLARRE(l)
