On finite precision block Lanczos computations

TL;DR

This paper extends Greenbaum's finite precision analysis of the Lanczos algorithm to the block Lanczos method, providing a mathematical model, conditions for small perturbations, and empirical validation.

Contribution

It generalizes the finite precision analysis to block Lanczos, introduces a continuation process, and explores conditions for small perturbations in the model.

Findings

Block Lanczos results can be interpreted via a larger model matrix.

Sufficient conditions for small perturbations are derived.

Numerical experiments validate the practical implementation.

Abstract

In her seminal 1989 work, Greenbaum demonstrated that the results produced by the finite precision Lanczos algorithm after $k$ iterations can be interpreted as exact Lanczos results applied to a larger matrix, whose eigenvalues lie in small intervals around those of the original matrix. This establishes a mathematical model for finite precision Lanczos computations. In this paper, we extend these ideas to the block Lanczos algorithm. We generalize the continuation process and show that it can be completed in a finite number of iterations using carefully constructed perturbations. The block tridiagonal matrices produced after $k$ iterations can then be interpreted as arising from the exact block Lanczos algorithm applied to a larger model matrix. We derive sufficient conditions under which the required perturbations remain small, ensuring that the eigenvalues of the model matrix stay close to those of the original matrix. While in the single-vector case these conditions are always satisfiable, as shown by Greenbaum based on results by Paige, the question of whether they can always be satisfied in the block case remains open. Finally, we present numerical experiments demonstrating a practical implementation of the continuation process and empirically assess the validity of the sufficient conditions and the size of the perturbations.