slarre(3)

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

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) 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 ma

trix. Currently, the base representations are limited to being

positive or negative definite, and the eigenvalues of the defi

nite matrices are found by the dqds algorithm (subroutine

SLASQ2). As an added benefit, SLARRE also outputs the n Ger

schgorin
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 subdi

agonal elements 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 IS

PLIT(2), etc., and the NSPLIT-th consists of rows/columns IS

PLIT(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(3)