Algorithms for solving systems of nonlinear equations by Kevorkian's decomposition method and their quadratic convergence property

Abstract

AbstractKevorkian's decomposition method is known as one of the methods to decompose a large‐scale system of nonlinear equations into systems of equations with lower dimensions. However, his method suffers from the property that errors are included in the decomposed equations, function values, and the Jacobian matrix. Consequently, it is important to discuss the convergence of the solution algorithm and to improve the convergence speed, taking those errors into account. This paper presents an algorithm for the solution of systems of nonlinear equations using Kevorkian's decomposition method. the convergence property of the proposed method is discussed, indicating that the algorithm gives local quadratic convergence under some suitable conditions. A sufficient condition is given first for the existence and differentiability of the mapping G obtained by decomposition. Then a discussion is made of the errors in the equations and the function values by Kevorkian's method. A solution algorithm is proposed which retains the quadratic convergence under those errors, leading to a theorem for convergence. A numerical example is given, indicating that the quadratic convergence can be realized by the proposed method. A comparison is made with the case without using the proposed method, demonstrating its effectiveness.