# On a fast Arnoldi method for BML matrices

## TL;DR

This paper introduces a fast Arnoldi method tailored for BML matrices, leveraging GMRES residuals and low-rank structures to efficiently generate Krylov bases and analyze Hessenberg matrices.

## Contribution

It presents a novel short recurrence Arnoldi algorithm for BML matrices using GMRES residuals and explores the low-rank structure of the Hessenberg matrix.

## Key findings

- Efficient Krylov basis generation for BML matrices
- Short recurrence relation based on GMRES residuals
- Low-rank structure of the Hessenberg matrix

## Abstract

Matrices whose adjoint is a low rank perturbation of a rational function of the matrix naturally arise when trying to extend the well known Faber-Manteuffel theorem, which provides necessary and sufficient conditions for the existence of a short Arnoldi recurrence. We show that an orthonormal Krylov basis for this class of matrices can be generated by a short recurrence relation based on GMRES residual vectors. These residual vectors are computed by means of an updating formula. Furthermore, the underlying Hessenberg matrix has an accompanying low rank structure, which we will investigate closely.