ArnouxRauzy

ArnouxRauzy refers to a method for finding a fixed point of a continuous function from a compact convex set to itself. It is a practical implementation of the Brouwer fixed-point theorem, which guarantees the existence of such a fixed point. The method was developed by Arnoux and Rauzy in the late 1980s.

The ArnouxRauzy algorithm is particularly well-suited for problems in computational geometry and game theory. It operates by iteratively refining a simplex within the given set, narrowing down the region that contains the fixed point. The core idea involves subdividing the simplex and discarding parts that are guaranteed not to contain the fixed point, based on the function's behavior. This process continues until a desired level of precision is reached.

While the theoretical foundation is Brouwer's theorem, the ArnouxRauzy algorithm provides a constructive and computationally feasible approach to locating the fixed point. Its efficiency can vary depending on the dimension of the problem and the specific function being analyzed. It has found applications in areas such as the analysis of economic models and the study of dynamical systems. The algorithm's development was a significant step in providing practical tools for problems that rely on fixed-point existence theorems.