# ART: adaptive residual--time restarting for Krylov subspace matrix exponential evaluations

## TL;DR

This paper introduces an adaptive residual-based restarting method for Krylov subspace matrix exponential computations, improving efficiency by adjusting restart length based on residual convergence, supported by numerical comparisons.

## Contribution

The paper presents a novel adaptive residual-based restarting technique for Krylov subspace methods, with convergence analysis and practical implementation details.

## Key findings

- The new method outperforms three existing restarting techniques in numerical tests.
- Adaptive restart length improves convergence efficiency.
- The algorithms are implemented in the expmARPACK MATLAB package.

## Abstract

In this paper a new restarting method for Krylov subspace matrix exponential evaluations is proposed. Since our restarting technique essentially employs the residual, some convergence results for the residual are given. We also discuss how the restart length can be adjusted after each restart cycle, which leads to an adaptive restarting procedure. Numerical tests are presented to compare our restarting with three other restarting methods. Some of the algorithms described in this paper are a part of the Octave/Matlab package expmARPACK available at http://team.kiam.ru/botchev/expm/.