Abstract
In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not require the evaluation of second or higher order Fréchet derivatives and still reaches fourth order convergence. The multi-step version converges with order 2r+4, where r is a positive integer and r ≥ 1. We have proved that the root α is a point of attraction for a general iterative function, whereas the proposed new schemes also satisfy this result. Numerical experiments including an application to 1-D Bratu problem are given to illustrate the efficiency of the new methods. Also, the new methods are compared with some existing methods.
Highlights
An often discussed problem in many applications of science and technology is to find a real zero of a system of nonlinear equations F (x) = 0, where F (x) = (f1 (x), f2 (x), ..., fn (x))T, x = (x1, x2, ..., xn )T, Algorithms 2015, 8 fi : Rn → R, ∀i = 1, 2, . . . , n and F : D ⊂ Rn → Rn is a smooth map and D is an open and convex T (0)set, where we assume that α = (α1, α2, ..., αn )T is a zero of the system and x(0) = x1, x2, ..., xn is an initial guess sufficiently close to α
The differential equations are reduced to system of nonlinear equations, which are in turn solved by the familiar Newton’s iteration method having convergence order two [1]
Homeier [2] has proposed a third order iterative method called Harmonic Mean Newton’s method for solving a single nonlinear equation. Analogous to this method [2], we consider the following extension to solve a system of nonlinear equation F (x) = 0, called as 3rd HM : x(k+1) = G3rd HM (x(k) ) = x(k) −
Summary
Homeier [2] has proposed a third order iterative method called Harmonic Mean Newton’s method for solving a single nonlinear equation. We note that [F 0 (x(k) )]−1 + [F 0 (x(k) − u(x(k) ))]−1 is the average of the inverses of two Jacobians Such third order methods free of second derivatives like Equation (2) can be used for solving system of nonlinear equations. An improved fourth order version from a third order method for solving a single nonlinear equation is found in [7]. In. Section 2, we present a new algorithm (optimal) that has fourth order convergence by using only three function evaluations and a multi-step version with order 2r + 4, where r is a positive integer and r ≥ 1 for solving systems of nonlinear equations.
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