Variants of Newton’s method for functions of several variables☆
Introduction
Let us consider the problem of finding a real zero of a function , that is, a real solution , of the nonlinear equation system F(x) = 0, of n equations with n variables. This solution can be obtained as a fixed point of some function by means of the fixed point iteration methodwhere x(0) is the initial estimation. The best known fixed point method is the classical Newton’s method, given bywhere JF(x(k)) is the Jacobian matrix of the function F evaluated in x(k).
In the following, we remember the most common notions about the convergence of an iterative method. Definition 1 Let {x(k)}k⩾0 be a sequence in convergent to . Then, convergence is called linear, if there exists M, 0 < M < 1, and k0 such that quadratic, if there exists M, M > 0, and k0 such that cubic, if there exists M, M > 0, and k0 such thatAnd so on.
In practice, as the limit is unknown, we analyze for each p the behaviour of the quotientswhere p = 1, 2, 3, …, which are called convergence test or convergence rate.
If p = 1 and the convergence rate eventually tends to CL, 0 < CL < 1, the sequence {x(k)}k⩾0 is said to be linearly convergent to .
If p = 2 and there exists CC, CC > 0, such that the convergence rate eventually tends to CC, the sequence {x(k)}k⩾0 is said to be quadratically convergent to . In a similar way for p = 3 (cubic convergence CCu) and so on.
Some results about convergence and order of convergence of Newton’s method can be found in [1], [2].
Several modifications have been made on this classical method in order to accelerate the convergence or to reduce the number of operations and evaluations of functions in each step of the iterative process.
The variants on Newton’s method developed by Weerakoon and Fernando in [3] and the analysis of convergence made by Ford and Pennline in [4], for a nonlinear equation, suggest us their extension to functions of several variables. In Section 2 we develop some adjustments on Newton’s method to find solutions of systems of nonlinear equations, and in Section 3 quadratic convergence is proved for one of these methods. Last section is dedicated to the numeric results obtained by applying classical Newton’s method and its alternatives to several systems of nonlinear equations. From these results we compare the different methods and obtain some conclusions.
Section snippets
Description of the methods
Let be a sufficiently differentiable function and be a zero of the system of nonlinear equations F(x) = 0. The following result will be used to describe the Newton method and its modifications; its proof can be found in [1], [2]. Lemma 1 Let be continuously differentiable on a convex set D. Then, for any x, y ∈ D, F satisfies
Convergence of midpoint Newton’s method
Our aim in this section is to prove that, under certain conditions, midpoint Newton’s method converges quadratically. To do this, some previous results are needed and will be introduced in the following. The two first are technical lemmas whose proof can be found in [1], [2]. Lemma 2 Let be a differentiable function such thatfor any u, v ∈ D convex set. Then there exists γ > 0 such that for any x, y ∈ D, Lemma 3 Banach Let be nonsingular. If and ∥A−1∥ · ∥E
Numerical results
In this section we will check the effectiveness of the different numeric methods introduced in this work (including the classical Newton’s method), in order to find the zeros of the several nonlinear functions.
- (a)
,
- (b)
F(x1, x2) = ((x1 − 1)6 − x2, x2 − 1),
- (c)
,
- (d)
,
- (e)
,
- (f)
,
- (g)
,
- (h)
F(x) = (f1(x), f2(
References (4)
- et al.
A variant of Newton’s method with accelerated third-order convergence
Applied Mathematics Letters
(2000) Numerical Analysis. A Second Course
(1990)
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This research was supported by Ministerio de Ciencia y Tecnología MTM2004-03244 and by Generalitat Valenciana GRUPOS03/062.