On accuracy of approximation of the spectral radius by the Gelfand formula

TL;DR

This paper improves understanding of the Gelfand formula's accuracy by providing explicit convergence estimates for the spectral radius approximation, extending these results to joint spectral radius calculations.

Contribution

It introduces explicit, computable estimates for the convergence rate of the Gelfand formula using Bochi inequalities, enhancing its practical applicability.

Findings

Derived explicit convergence estimates for the Gelfand formula

Extended estimates to joint spectral radius of matrix sets

Improved bounds for spectral radius approximation

Abstract

The famous Gelfand formula $\rho(A)= \limsup_{n\to\infty}\|A^{n}\|^{1/n}$ for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities $\|A^{n}\|^{1/n}$ to $\rho(A)$. In the paper this deficiency is made up to some extent. By using the Bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities $\|A^{n}\|^{1/n}$ to $\rho(A)$. The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.