TY - JOUR

T1 - Algorithms for solving systems of nonlinear equations by Kevorkian's decomposition method and their quadratic convergence property

AU - Yamamura, Kiyotaka

AU - Oishi, Shin'Ichi

AU - Horiuchi, Kazuo

PY - 1983

Y1 - 1983

N2 - Kevorkian'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.

AB - Kevorkian'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.

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U2 - 10.1002/ecja.4400661106

DO - 10.1002/ecja.4400661106

M3 - Article

AN - SCOPUS:0020844359

SN - 8756-6621

VL - 66

SP - 33

EP - 41

JO - Electronics and Communications in Japan (Part I: Communications)

JF - Electronics and Communications in Japan (Part I: Communications)

IS - 11

ER -