New Power Method for Solving Eigenvalue Problems

TL;DR

This paper introduces a novel power method that efficiently computes all eigenvalues of an operator or matrix without deflation, using a functional approach and parameter tuning, with demonstrated numerical accuracy.

Contribution

The paper presents a new power method capable of finding all eigenvalues simultaneously without deflation, utilizing a functional-based filtering technique.

Findings

Numerical results align well with analytical solutions.

The method can target specific eigenvalues by adjusting parameters.

It converges efficiently for various eigenvalue problems.

Abstract

We present a new power method to obtain solutions of eigenvalue problems. The method can determine not only the dominant or lowest eigenvalues but also all eigenvalues without the need for a deflation procedure. The method uses a functional of an operator (or a matrix) to select or filter an eigenvalue. The method can freely select a solution by varying a parameter associated to an estimate of the eigenvalue. The convergence of the method is highly dependent on how closely the parameter to the eigenvalues. In this paper, numerical results of the method are shown to be in excellent agreement with the analytical ones.