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

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( * )

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 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 DLASQ2). As an added benefit, DLARRE also outputs the n Gerschgorin

intervals for each L_i D_i L_i^T.

N (input) INTEGER

The order of the matrix.

D (input/output) DOUBLE PRECISION 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) 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 subdiagonal 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 ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(N)=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

Based on contributions by

Inderjit Dhillon, IBM Almaden, USA

Osni Marques, LBNL/NERSC, USA