The Lanczos algorithm for matrix functions: a handbook for scientists

TL;DR

This handbook provides an accessible introduction to Lanczos-based methods for matrix functions, clarifying their behavior in finite precision arithmetic and unifying knowledge across communities.

Contribution

It offers a comprehensive, conceptual overview of Lanczos algorithms for matrix functions, addressing misconceptions and serving as a starting point for new researchers.

Findings

Clarifies the behavior of Lanczos methods in finite precision

Unifies knowledge from different research communities

Provides practical guidance for scientists and students

Abstract

Lanczos-based methods have become standard tools for tasks involving matrix functions. Progress on these algorithms has been driven by several largely disjoint communities, resulting many innovative and important advancements which would not have been possible otherwise. However, this also has resulted in a somewhat fragmented state of knowledge and the propagation of a number of incorrect beliefs about the behavior of Lanczos-based methods in finite precision arithmetic. This monograph aims to provide an accessible introduction to Lanczos-based methods for matrix functions. The intended audience is scientists outside of numerical analysis, graduate students, and researchers wishing to begin work in this area. Our emphasis is on conceptual understanding, with the goal of providing a starting point to learn more about the remarkable behavior of the Lanczos algorithm. Hopefully readers will come away from this text with a better understanding of how to think about Lanczos for modern problems involving matrix functions, particularly in the context of finite precision arithmetic.