A decomposition method and acceleration techniques for a fixed point algorithm

Abstract

AbstractRecently, extensive studies have been carried out on the algorithms which constructively determine the solution of nonlinear equations. These algorithms are in general called fixed‐point algorithms, and their attraction lies in the global convergence. However, fixed‐point algorithms have been a problem in that the execution speed is decreased greatly with the increase of the size of the system of equations to consider. This paper proposes a decomposition method and acceleration techniques as a means to improve the computational efficiency of the fixedpoint algorithms. Kevorkian's method is used in this paper as decomposition. Since the method includes errors in the decomposed equations and function values, it is important to discuss the convergence of the solution algorithm and to improve the convergence speed taking the effect of those errors into consideration. This paper shows a solution algorithm in which the system of equations is decomposed by Kevorkian's method, and then Merrill's method, a typical fixedpoint algorithm, is applied. Accuracy of simplicial approximation and convergence properties of the algorithm are discussed, indicating that the algorithm exhibits local quadratic convergence under some suitable conditions. By numerical examples, it is verified that the algorithm exhibits quadratic convergence. the method is compared with the past methods, indicating the effectiveness of the proposed method.