Inflationary dynamics for matrix eigenvalue problems

TL;DR

This paper introduces a novel approach to solving extremal eigenvalue problems by transforming them into a nonlinear mechanical system with a special constraint that causes the solutions to grow exponentially, aiding their computation.

Contribution

It presents a new method that converts eigenvalue problems into a nonlinear mechanical system with inflationary dynamics to efficiently find extremal eigenpairs.

Findings

Demonstrates exponential growth of solutions under the proposed method

Provides a new perspective on eigenvalue problems through classical mechanics analogy

Offers potential improvements over traditional algorithms for extremal eigenvalues

Abstract

Many fields of science and engineering require finding eigenvalues and eigenvectors of large matrices. The solutions can represent oscillatory modes of a bridge, a violin, the disposition of electrons around an atom or molecule, the acoustic modes of a concert hall, or hundreds of other physical quantities. Often only the few eigenpairs with the lowest or highest frequency (extremal solutions) are needed. Methods that have been developed over the past 60 years to solve such problems include the Lanczos [1,2] algorithm, Jacobi-Davidson techniques [3], and the conjugate gradient method [4]. Here we present a way to solve the extremal eigenvalue/eigenvector problem, turning it into a nonlinear classical mechanical system with a modified Lagrangian constraint. The constraint induces exponential inflationary growth of the desired extremal solutions.