Biorthogonal Extended Krylov Subspace Methods

TL;DR

This paper introduces a unified framework for oblique projections of nonhermitian matrices onto rational Krylov subspaces, generalizing classical methods and enabling efficient computations through structured pencils.

Contribution

It develops a general structured pencil framework for oblique projections, including a biorthogonal tridiagonal pencil, extending classical Krylov subspace methods.

Findings

Structured pencils can be efficiently used for oblique projections.

The framework generalizes classical biorthogonal Krylov methods.

Numerical experiments confirm the approach's validity.

Abstract

A general framework for oblique projections of nonhermitian matrices onto rational Krylov subspaces is developed. To obtain this framework we revisit the classical rational Krylov subspace algorithm and prove that the projected matrix can be written efficiently as a structured pencil, where the structure can take several forms, such as Hessenberg or inverse Hessenberg. One specific instance of the structures appearing in this framework for oblique projections is a tridiagonal pencil. This is a direct generalization of the classical biorthogonal Krylov subspace method where the projection becomes a single nonhermitian tridiagonal matrix and of the Hessenberg pencil representation for rational Krylov subspaces. Based on the compact storage of this tridiagonal pencil in the biorthogonal setting, we can develop short recurrences. Numerical experiments confirm the validity of the approach.