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slarre

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 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
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 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 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