A numerical projection technique for large-scale eigenvalue problems

TL;DR

This paper introduces a generalized numerical projection method for large-scale eigenvalue problems, enabling convergence to exact eigenvalues and applicable across various scientific fields with matrices having dominant diagonal components.

Contribution

It extends the projection technique by performing both steps numerically, allowing convergence to exact eigenvalues in large-scale problems.

Findings

Method converges to exact eigenvalues

Applicable to matrices with dominant diagonal parts

Validated with two many-body models

Abstract

We present a new numerical technique to solve large-scale eigenvalue problems. It is based on the projection technique, used in strongly correlated quantum many-body systems, where first an effective approximate model of smaller complexity is constructed by projecting out high energy degrees of freedom and in turn solving the resulting model by some standard eigenvalue solver. Here we introduce a generalization of this idea, where both steps are performed numerically and which in contrast to the standard projection technique converges in principle to the exact eigenvalues. This approach is not just applicable to eigenvalue problems encountered in many-body systems but also in other areas of research that result in large scale eigenvalue problems for matrices which have, roughly speaking, mostly a pronounced dominant diagonal part. We will present detailed studies of the approach guided by two many-body models.