Stability of a functional equation coming from the characterization of the absolute value of additive functions

In the present paper, we prove the stability of the functional equationmax{f((x ^ y)^ y), f(x)} = f(x ^ y) + f(y)for real valued functions defined on a square-symmetric groupoid with a leftunit element. As a consequence, we obtain the known result about the stabilityof the equationmax{f(x + y), f(x ? y)} = f(x) + f(y)for real valued functions defined on an abelian grou

Validated computation tool for the Perron-Frobenius eigenvalues

A matrix with non-negative entries has a special eigenvalue, the so called Perron-Frobenius eigenvalue, which plays an important role in several fields of science [1]. In this paper we present a numerical tool to compute rigorous upper and lower bounds for the Perron-Frobenius eigenvalue of non-negative matrices. The idea is to express a non-negative matrix in terms of a directed graph, and make use of R. Tarjan’s algorithm [5] which finds all strongly connected components of a directed graph very efficiently. This enables us to decompose the original matrix into irreducible components (possibly of small size), and then to enclose the aimed Perron-Frobenius eigenvalue. We also show a numerical example which demonstrates the efficiency of our tool

Validated computation tool for the Perron-Frobenius eigenvalues

A matrix with non-negative entries has a special eigenvalue, the so called Perron-Frobenius eigenvalue, which plays an important role in several fields of science [1]. In this paper we present a numerical tool to compute rigorous upper and lower bounds for the Perron-Frobenius eigenvalue of non-negative matrices. The idea is to express a non-negative matrix in terms of a directed graph, and make use of R. Tarjan’s algorithm [5] which finds all strongly connected components of a directed graph very efficiently. This enables us to decompose the original matrix into irreducible components (possibly of small size), and then to enclose the aimed Perron-Frobenius eigenvalue. We also show a numerical example which demonstrates the efficiency of our tool