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

The famous Gelfand formula ρ(A)=limsupn→∞‖An‖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 ‖An‖1/n to ρ(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 ‖An‖1/n to ρ(A). The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.