# Stability of the Lanczos algorithm on matrices with regular spectral distributions

## TL;DR

This paper analyzes the stability of the Lanczos algorithm on matrices with spectral distributions close to well-behaved measures, showing it is stable for many large random matrices in finite precision arithmetic.

## Contribution

It provides a stability analysis of the Lanczos algorithm for matrices with specific spectral properties, demonstrating near stability in practical scenarios involving random matrices.

## Key findings

- Lanczos algorithm is forward stable on matrices with certain spectral distributions.
- Random matrices with regular spectral distributions exhibit near-deterministic behavior.
- Stability results depend on the spectral measure's properties, especially support and density.

## Abstract

We study the stability of the Lanczos algorithm run on problems whose eigenvector empirical spectral distribution is near to a reference measure with well-behaved orthogonal polynomials. We give a backwards stability result which can be upgraded to a forward stability result when the reference measure has a density supported on a single interval with square root behavior at the endpoints. Our analysis implies the Lanczos algorithm run on many large random matrix models is in fact forward stable, and hence nearly deterministic, even when computations are carried out in finite precision arithmetic. Since the Lanczos algorithm is not forward stable in general, this provides yet another example of the fact that random matrices are far from "any old matrix", and care must be taken when using them to test numerical algorithms.

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Source: https://tomesphere.com/paper/2302.14842