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

Abstract

The famous Gelfand formula $\rho (A)={\mathrm{limsup}}_{n\to \infty }\Vert A^n{\Vert }^{1/n}$ 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 $\Vert A^n{\Vert }^{1/n}$ 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 $\Vert A^n{\Vert }^{1/n}$ to $\rho (A)$. The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.

arXiv commentCorrected typos, improved estimate for the kay constant in the paper, added references

References in corpus (3)

• Approximation of the joint spectral radius using sum of squares
• Inequalities for numerical invariants of sets of matrices
• On explicit a priori estimates of the joint spectral radius by the generalized Gelfand formula

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