When does the Lanczos algorithm compute exactly?

TL;DR

This paper identifies specific matrix and vector structures that ensure the Lanczos algorithm and related methods produce exact results in finite precision arithmetic, extending classical theory.

Contribution

It introduces conditions on matrices and vectors that guarantee exact Lanczos computations in floating point arithmetic, including variants and related algorithms.

Findings

Exact Lanczos computations occur under certain structured matrices and vectors.

Results extend to Arnoldi, nonsymmetric Lanczos, Golub-Kahan bidiagonalization, and block-Lanczos algorithms.

Applicable to solving linear systems with guaranteed precision under IEEE 754 standard.

Abstract

In theory, the Lanczos algorithm generates an orthogonal basis of the corresponding Krylov subspace. However, in finite precision arithmetic, the orthogonality and linear independence of the computed Lanczos vectors is usually lost quickly. In this paper we study a class of matrices and starting vectors having a special nonzero structure that guarantees exact computations of the Lanczos algorithm whenever floating point arithmetic satisfying the IEEE 754 standard is used. Analogous results are formulated also for a variant of the conjugate gradient method that produces then almost exact results. The results are extended to the Arnoldi algorithm, the nonsymmetric Lanczos algorithm, the Golub-Kahan bidiagonalization, the block-Lanczos algorithm and their counterparts for solving linear systems.