dlarre(3)

NAME

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) eigenvalues 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 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

DLASQ2). As an added benefit, DLARRE 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) 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 subdi

agonal elements of the unit bidiagonal matrices L_i.

TOL (input) DOUBLE PRECISION

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) 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

LAPACK version 3.0 15 June 2000

DLARRE(3)