Fast and Inverse-Free Algorithms for Deflating Subspaces

TL;DR

This paper introduces inverse-free algorithms for computing spectral projectors of matrix pencils, combining rational approximation and structured iterative methods to improve efficiency and applicability in numerical linear algebra.

Contribution

It presents a high-level inverse-free framework for spectral projector computation, unifying existing methods and enabling new efficient algorithms for structured problems.

Findings

Framework captures existing inverse-free methods

Demonstrates effectiveness with Implicit Repeated Squaring

Adapts randomized eigensolvers for fast generalized Schur form

Abstract

This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil $(A,B)$, in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on spectral projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to computing these projectors, which combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function. In an appendix, we demonstrate that recent, randomized divide-and-conquer eigensolvers -- which are built on fast methods for individual projectors -- can be adapted to produce the generalized Schur form of any matrix pencil in nearly matrix multiplication time.