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

The famous Gelfand formula $\rho (A)={\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}\mathrm{\parallel }{A}^n{\mathrm{\parallel }}^{1\mathrm{/}n}$ 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 $\mathrm{\parallel }{A}^n{\mathrm{\parallel }}^{1\mathrm{/}n}$ 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 $\mathrm{\parallel }{A}^n{\mathrm{\parallel }}^{1\mathrm{/}n}$ to $\rho (A)$. The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.