A Structure-Preserving Divide-and-Conquer Method for Pseudosymmetric Matrices

TL;DR

This paper introduces a spectral divide-and-conquer method that preserves the pseudosymmetric structure of matrices, enabling efficient diagonalization especially for applications in quantum physics and chemistry.

Contribution

It presents a novel structure-preserving spectral divide-and-conquer algorithm tailored for pseudosymmetric matrices, with proven convergence and parallelizability.

Findings

Method preserves pseudosymmetric structure during spectral division.

Achieves full diagonalization through recursive application.

Converges rapidly using Zolotarev functions in quantum chemistry applications.

Abstract

We devise a spectral divide-and-conquer scheme for matrices that are self-adjoint with respect to a given indefinite scalar product (i.e. pseudosymmetic matrices). The pseudosymmetric structure of the matrix is preserved in the spectral division, such that the method can be applied recursively to achieve full diagonalization. The method is well-suited for structured matrices that come up in computational quantum physics and chemistry. In this application context, additional definiteness properties guarantee a convergence of the matrix sign function iteration within two steps when Zolotarev functions are used. The steps are easily parallelizable. Furthermore, it is shown that the matrix decouples into symmetric definite eigenvalue problems after just one step of spectral division.