An unconstrained framework for eigenvalue problems

TL;DR

This paper introduces an unconstrained variational framework for solving eigenvalue problems in both discrete and continuous settings, enabling efficient computation of eigenvalues and eigenvectors without traditional constraints.

Contribution

It presents a novel unconstrained functional approach for eigenvalue problems, with convergence analysis and applications to differential operators, improving computational efficiency and theoretical understanding.

Findings

The framework effectively finds smallest eigenvalues and eigenvectors.

Algorithms converge reliably to critical points and local minimizers.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we propose an unconstrained framework for eigenvalue problems in both discrete and continuous settings. We begin our discussion to solve a generalized eigenvalue problem $A{\bf x} = \lambda B{\bf x}$ with two $N\times N$ real symmetric matrices $A, B$ via minimizing a proposed functional whose nonzero critical points ${\bf x}\in\mathbb{R}^N$ solve the eigenvalue problem and whose local minimizers are indeed global minimizers. Inspired by the properties of the proposed functional to be minimized, we provide analysis on convergence of various algorithms either to find critical points or local minimizers. Using the same framework, we will also present an eigenvalue problem for differential operators in the continuous setting. It will be interesting to see that this unconstrained framework is designed to find the smallest eigenvalue through matrix addition and multiplication and that a solution ${\bf x}\in\mathbb{R}^N$ and the matrix $B$ can compute the corresponding eigenvalue $\lambda$ without using $A$ in the case of $A{\bf x}=\lambda B{\bf x}$. At the end, we will present a few numerical experiments which will confirm our analysis.