A generalization of the divide and conquer algorithm for the symmetric tridiagonal eigenproblem

TL;DR

This paper generalizes the divide-and-conquer algorithm for symmetric tridiagonal eigenproblems to handle rank-two modifications, introducing new deflation techniques and eigenvalue counting methods, resulting in faster computations.

Contribution

It extends Cuppen's algorithm to rank-two modifications with novel deflation and eigenvalue counting techniques, improving computational efficiency.

Findings

Eigenvalue computation is twice as fast as previous methods.

New deflation technique improves eigenvalue distribution analysis.

Eigenvectors are computed while preserving orthogonality.

Abstract

In this paper, we present a generalized Cuppen's divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We extend the Cuppen's work to the rank two modifications of the form $A =T +\beta_1\bw_1\bw_1^T + \beta_2\bw_2\bw_2^T$, where $T$ is a block tridiagonal matrix having three blocks. We introduce a new deflation technique and obtain a secular equation, for which the distribution of eigenvalues is nontrivial. We present a way to count the number of eigenvalues in each subinterval. It turns out that each subinterval contains either none, one or two eigenvalues. Furthermore, computing eigenvectors preserving the orthogonality are also suggested. Some numerical results, showing our algorithm can calculate the eigenvalue twice as fast as the Cuppen's divide-and-conquer algorithm, are included.