Finding the root of a nonlinear equation

is a classical problem. It is well-known in scientific computation that Newton’s method (NM, see [1] ):

is widely used for root-finding, where is an initial guess of the root. However, when the derivative is unavailable or is expensive to be obtained, the derivative-free method is necessary. If the derivative is replaced by the divided difference in (2), Steffensen’s method (SM, see [1] ) is obtained as follows:

NM/SM converges quadratically and requires two function evaluations per iteration. The efficiency index of them is.

Besides H.T. Kung and J.F. Traub conjectured that an iterative method based on evaluations per iteration without memory would arrive at the optimal convergence of order (see [2] ), Traub proposed a self-acce- lerating two-point method of order 2.414 with memory (see [3] ):

where, and or, etc.

A lot of self-accelerating Steffensen-type methods were derived in the literature (see [1] -[7] ). Steffensen-type methods and their applications in the solution of nonlinear systems and nonlinear differential equations were discussed in [1] [4] [5] [8] . Recently, by a new self-accelerating technique based on the second-order Newtonian

interpolatory polynomial, J. Džunića

and M.S. Petkovića proposed a cubically convergent Steffensen-like method (see [7] ):

In this study, a one-step Steffensen-type method is proposed by doubly-self-accelerating in Section 2, its super-cubic convergence is proved in Section 3, and numerical examples are demonstrated in Section 4.

By the first-order Newtonian interpolatory polynomial and

we have

where

So, with some

should be better than to approximate.

Therefore, we suggest i.e., a two-parameter Steffensen’s method:

where, and are bounded constant sequences. The error equation of (7) is

. By defining and

recursively as the iteration proceeds without any new evaluation to

vanish the asymptotic convergence constant, we establish a self-accelerating Steffensen’s method with super quadratic convergence as follows:

Furthermore, we propose a one-step Steffensen-type method with super cubic convergence by doubly-self- accelerating as follows:

Lemma 3.1 where and

Proof. By Taylor formula, we have

So,

Then, the proof can be completed.

Theorem 3.2 Let be a sufficiently differentiable function with simple root, be an open set, be close enough to, then (9) achieve the convergence of order 3.383.

Proof. If converges to with order as:

and if converges to with order as:

Then

By Taylor formula and Lemma 3.1, we also have

So, comparing the exponents of in expressions of and for (9), we obtain the same system of two equations:

From its non-trivial solution and, we prove that the convergence of (9) is of order 3.383.

As the efficiency index is, without any additional function evaluations, the efficiency indices of (4), (5)

and (9) are and respectively.

Related one-step methods only using two function evaluations per iteration are showed in the following numeri- cal examples. The proposed method is a derivative-free two-point method with high computational efficiency.

Example 1. The numerical results of NM, SM, (4), (5) and (9) in Table 1 agree with the theoretical analysis. The computational order of convergence is defined by

Example 2. The numerical results of NM, SM, (4), (5) and (9) are in Table 2 for the following nonlinear functions:

Table 1..

Table 2. Numerical results for solving

By theoretical analysis and numerical experiments, we confirm that the proposed method which is a derivative- free two-point method has high computational efficiency. Its convergence order is 3.383 and its efficiency index is 1.839. We can see that the suggested method is suitable to solve nonlinear equations and can also be used for solving boundary-value problems of nonlinear ordinary differential equations.