Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices

TL;DR

This paper introduces a highly accurate, efficient algorithm for eigenvalue problems involving rank-one modifications of diagonal matrices, suitable for parallel computation and extendable to complex Hermitian matrices.

Contribution

The paper presents a novel shift-and-invert based algorithm that computes eigenvalues and eigenvectors of rank-one modified diagonal matrices with high accuracy in linear time.

Findings

Computes eigenvalues and eigenvectors with high relative accuracy.

Operates in O(n) complexity, suitable for large matrices.

Extends to complex Hermitian matrices.

Abstract

We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with high relative accuracy in $O(n)$ operations. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to the complex Hermitian case. The algorithm is similar to the algorithm for solving the eigenvalue problem for real symmetric arrowhead matrices from: N. Jakov\v{c}evi\'{c}~Stor, I. Slapni\v{c}ar and J. L. Barlow, {Accurate eigenvalue decomposition of real symmetric arrowhead matrices and applications}, Lin. Alg. Appl., 464 (2015).