Stability of the Lanczos Method for Matrix Function Approximation

TL;DR

This paper proves that the Lanczos method for matrix function approximation remains stable and effective in finite precision arithmetic, matching the ideal error bounds with logarithmic bits of precision, and explores special cases like matrix inversion.

Contribution

It extends the theoretical understanding of Lanczos stability in finite precision, providing bounds that match exact arithmetic guarantees and analyzing the inverse case with improved insights.

Findings

Lanczos error bounds hold in finite precision with logarithmic bits of precision

For matrix inversion, Lanczos converges faster than Greenbaum's bounds suggest

Finite precision Lanczos can achieve polylogarithmic iteration complexity for well-conditioned matrices

Abstract

The ubiquitous Lanczos method can approximate $f(A)x$ for any symmetric $n \times n$ matrix $A$, vector $x$, and function $f$. In exact arithmetic, the method's error after $k$ iterations is bounded by the error of the best degree-$k$ polynomial uniformly approximating $f(x)$ on the range $[\lambda_{min}(A), \lambda_{max}(A)]$. However, despite decades of work, it has been unclear if this powerful guarantee holds in finite precision. We resolve this problem, proving that when $\max_{x \in [\lambda_{min}, \lambda_{max}]}|f(x)| \le C$, Lanczos essentially matches the exact arithmetic guarantee if computations use roughly $\log(nC\|A\|)$ bits of precision. Our proof extends work of Druskin and Knizhnerman [DK91], leveraging the stability of the classic Chebyshev recurrence to bound the stability of any polynomial approximating $f(x)$. We also study the special case of $f(A) = A^{-1}$, where stronger guarantees hold. In exact arithmetic Lanczos performs as well as the best polynomial approximating $1/x$ at each of $A$'s eigenvalues, rather than on the full eigenvalue range. In seminal work, Greenbaum gives an approach to extending this bound to finite precision: she proves that finite precision Lanczos and the related CG method match any polynomial approximating $1/x$ in a tiny range around each eigenvalue [Gre89]. For $A^{-1}$, this bound appears stronger than ours. However, we exhibit matrices with condition number $\kappa$ where exact arithmetic Lanczos converges in $polylog(\kappa)$ iterations, but Greenbaum's bound predicts $\Omega(\kappa^{1/5})$ iterations. It thus cannot offer significant improvement over the $O(\kappa^{1/2})$ bound achievable via our result. Our analysis raises the question of if convergence in less than $poly(\kappa)$ iterations can be expected in finite precision, even for matrices with clustered, skewed, or otherwise favorable eigenvalue distributions.