Abstract
We present iterative methods of convergence order three, five, and six for solving systems of nonlinear equations. Third-order method is composed of two steps, namely, Newton iteration as the first step and weighted-Newton iteration as the second step. Fifth and sixth-order methods are composed of three steps of which the first two steps are same as that of the third-order method whereas the third is again a weighted-Newton step. Computational efficiency in its general form is discussed and a comparison between the efficiencies of proposed techniques with existing ones is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples. It is shown that the present methods have an edge over similar existing methods, particularly when applied to large systems of equations.
Highlights
Solving the system F(x) = 0 of nonlinear equations is a common and important problem in various disciplines of science and engineering [1,2,3,4]
We present iterative methods of convergence order three, five, and six for solving systems of nonlinear equations
To compute F in any iterative method, we evaluate n scalar functions, whereas the number of scalar evaluations is n2 for any new derivative F
Summary
Solving the system F(x) = 0 of nonlinear equations is a common and important problem in various disciplines of science and engineering [1,2,3,4] This problem is precisely stated as follows: For a given nonlinear function F(x) : D ⊆. One of the basic procedures for solving systems of nonlinear equations is the classical Newton’s method [4, 5] which converges quadratically under the conditions that the function F is continuously differentiable and a good initial approximation x(0) is given. It is defined by x(k+1) = G2 (x(k)) = x(k) − F(x(k))−1F (x(k)) , (2). The methods such as those developed in [6,7,8] with second derivative are considered less efficient from a computational point of view
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