Dissecting the FEAST algorithm for generalized eigenproblems

TL;DR

This paper provides an in-depth analysis of the FEAST algorithm for large sparse eigenproblems, identifying key factors affecting its performance and suggesting potential improvements.

Contribution

It establishes the connection between FEAST and Rayleigh-Ritz, and analyzes critical issues influencing FEAST's convergence and accuracy.

Findings

The choice of initial vector space significantly impacts convergence.

Stopping criteria and inner linear system solutions affect solution quality.

Numerical examples illustrate the identified issues and potential improvements.

Abstract

We analyze the FEAST method for computing selected eigenvalues and eigenvectors of large sparse matrix pencils. After establishing the close connection between FEAST and the well-known Rayleigh-Ritz method, we identify several critical issues that influence convergence and accuracy of the solver: the choice of the starting vector space, the stopping criterion, how the inner linear systems impact the quality of the solution, and the use of FEAST for computing eigenpairs from multiple intervals. We complement the study with numerical examples, and hint at possible improvements to overcome the existing problems.