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.TH SLARRE l "15 June 2000" "LAPACK version 3.0" ")"
.SH 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 representations and (iii) eigenvalues of each L_i D_i L_i^T
.SH SYNOPSIS
.TP 19
SUBROUTINE SLARRE(
N, D, E, TOL, NSPLIT, ISPLIT, M, W, WOFF,
GERSCH, WORK, INFO )
.TP 19
.ti +4
INTEGER
INFO, M, N, NSPLIT
.TP 19
.ti +4
REAL
TOL
.TP 19
.ti +4
INTEGER
ISPLIT( * )
.TP 19
.ti +4
REAL
D( * ), E( * ), GERSCH( * ), W( * ), WOFF( * ),
WORK( * )
.SH 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) eigenvalues 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
.br
intervals for each L_i D_i L_i^T.
.br
.SH ARGUMENTS
.TP 8
N (input) INTEGER
The order of the matrix.
.TP 8
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.
.TP 8
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 bidiagonal
matrices L_i.
.TP 8
TOL (input) REAL
The threshold for splitting. If on input |E(i)| < TOL, then
the matrix T is split into smaller blocks.
.TP 8
NSPLIT (input) INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.
.TP 8
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.
.TP 8
M (output) INTEGER
The total number of eigenvalues (of all the L_i D_i L_i^T)
found.
.TP 8
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.
.TP 8
WOFF (output) REAL array, dimension (N)
The NSPLIT base points sigma_i.
.TP 8
GERSCH (output) REAL array, dimension (2*N)
The n Gerschgorin intervals.
.TP 8
WORK (input) REAL array, dimension (4*N???)
Workspace.
.TP 8
INFO (output) INTEGER
Output error code from SLASQ2
.SH FURTHER DETAILS
Based on contributions by
.br
Inderjit Dhillon, IBM Almaden, USA
.br
Osni Marques, LBNL/NERSC, USA
.br
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