1. Introduction

Finding the root of nonlinear equations is one of important problem in science and engineering [5]. In this paper, we present a new iterative method to find a simple root of the nonlinear equation , where for an open interval D is a scalar function. The multipoint root-solvers is of great practical importance since it overcomes theoretical limits of one-point methods concerning the convergence order and computational efficiency. In recent years, some modifications of the Newton-type methods for simple root have been proposed and analysed [5] and very little work has been done on the secant-type methods. Therefore, the purpose of this paper is to show further development of the secant-type methods. In addition, we will demonstrate the similarity between the Fernandez-Torres and Tiruneh et al. method. We have found that the efficiency index of new iterative method has a better efficiency index than the classical Newton method and equivalent to the Fernandez-Torres, Tiruneh et al. and McDougall et al. methods [1, 3, 6]. Furthermore, we demonstrate the performance of the new method is significantly better when compared with the established methods.

The two well-known iterative methods for finding simple root of nonlinear equations are namely, the classical secant method and the classical Newton method, given by

(1)

(2)

and their order of convergence is 1.618 and 2 respectively. For the purpose of this paper, we present a new secant-type method for finding simple root of nonlinear equations.

The structure of this paper is as follows: Some basic definitions relevant to the present work are presented in the section 2. In section 3 the new secant-type method and prove the order of convergence are described. In section 4 we demonstrate the equivalency of the two methods recently introduced. Finally, in section 5, numerical comparisons are made to demonstrate the performance of the presented method.

2. Review of Definitions

In order to establish the order of convergence of an iterative method, following definitions are used [1].

Definition 1 Let be a real-valued function with a root and let be a sequence of real numbers that converge towards . The order of convergence p is given by

(3)

where and is the asymptotic error constant [2, 5, 7].

Definition 2 Let be the error in the kth iteration, then the relation

(4)

is the error equation. If the error equation exists, then p is the order of convergence of the iterative method [2, 5, 7].

Definition 3 Let r be the number of function evaluations of the method. The efficiency of the method is measured by the concept of efficiency index and defined as

(5)

where p is the order of convergence of the method [4].

Definition 4 Suppose that and are three successive iterations closer to the root of (1). Then the computational order of convergence may be approximated by

(6)

further details may be found in [8].

3. The Method and Analysis of Convergence

In this section, we shall define the new iterative method, namely, the secant-type method. To obtain the solution of (1), the new method requires two evaluations of functions and set a particular initial point, ideally close to the simple root. Therefore the expression of the new iterative method for determining the simple root of (1) is given as

(7)

where , is the initial value and provided that the denominator of (7) is not equal to zero. It is essential to verify our finding and prove the order of convergence of the new iterative method.

Theorem

Assume that the function for an open interval D has a simple root . Let be sufficiently smooth in the interval D. If the initial points are sufficiently close to , then the asymptotic convergence order of the new method defined by (7) is .

Proof

Let be a simple root of, i.e. and , and the errors at iteration are expressed as and respectively.

Using Taylor expansion and taking into account that, we have

(8)

(9)

where

(10)

Furthermore, we have

(11)

(12)

By substituting the above expressions in the numerator of (7), we obtain

(13)

and similarly the denominator of (7) is given as

(14)

Dividing (14) by (13) and simplifying, we have

(15)

In order to prove the order of convergence of (15), we define positive real terms of as

(16)

(17)

Therefore, from (15) we obtain

(18)

In order to satisfy the asymptotic equation (18), the power of the error term shall approach zero, that is

(19)

It is obvious that quadratic equation (19) has two roots,

(20)

The order of convergence of the method (7) is determined by the positive root of (20). Hence, the new method defined by (7) has a convergence order of . This completes the proof.

The new method (7) requires 2 function evaluations and has the order of convergence and to determine the efficiency index of the new method, definition 3 shall be used. Hence, the efficiency index of the new iterative method given by (7) is

(21)

and the efficiency index of the Newton method and the secant method is given by

(22)

(23)

respectively. This indicates that the new method has a better efficiency index than the Newton method and is equivalent to the Fernandez-Torres, Tiruneh et al. and McDougall et al. methods.

4. The Established Methods

For the purpose of comparison, three well-known iterative methods are considered namely, the McDougall et al., the Fernandez-Torres method and the Tiruneh et al. methods. Since these methods are well established, the essential formulas are used to calculate the approximate solution of the given nonlinear equations and thus compare the effectiveness of the new method. First we will state the Fernandez-Torres method and the Tiruneh et al. method and then demonstrate the similarity between these methods.

In [1], Fernandez-Torres developed a super-quadratic convergence order method for finding simple root of nonlinear equations, the essential expression used in the method is given as,

(24)

The Tiruneh et al. method given in [7] is also a super-quadratic convergence order method for finding simple root of nonlinear equations and is expressed as

(25)

To demonstrate the equivalency of these methods we will first denote the following

(26)

Using (26) and simplifying the Fernandez-Torres method (24), we get

(27)

and similarly Tiruneh et al. method (25) is given as

(28)

To demonstrate the Fernandez-Torres and Tiruneh et al. methods produce identical approximation of the simple root of the nonlinear equation, hence, we simply equate the two methods.

Let we have

(29)

Cancelling common factors, we obtain

(30)

Simplifying, we get

(31)

Further simplification yields

(32)

Applying cross-multiplication rule to (32), we obtain

(33)

Expanding (33), we have

(34)

The equation (34) establishes that the Fernandez-Torres and Tiruneh et al. methods produce identical estimates of the simple root of the nonlinear equation. Furthermore, this is manifested in the numerical examples.

Another super-quadratic convergence order method for finding simple root of nonlinear equations was presented by McDougall et al. [3], we state the essential expressions used in the method.

The initial points are given as

(35)

(36)

Then the calculation of the simple root of the nonlinear equation is given by

(37)

where

(38)

5. Numerical Examples

The proposed iterative method given by (7) is employed to solve nonlinear equation with simple root. The difference between the simple root and the approximation for test function with initial guess is displayed in tables. Furthermore, the computational order of convergence approximations are displayed in tables and we observe that this perfectly coincides with the theoretical result. The numerical computations listed in the table was performed on an algebraic system called Maple and the errors displayed are of absolute value. The results in the tables were obtained by the following algorithm

Algorithm (Maple program)

sroots:=[fsolve(f(x),x)]:

root1:=sroots[1];

diff(f(x),x);

x[0]:=x0:

dfx[0]:= diff(f(x),x);

x[1]:=x[0]-fx[0]/dfx[0]:

for n from 1 to 6 do

fx[n]:=f(x);

dfx[n]:=diff(f(x),x);

x1[n]:=x[n]-x[n-1]:

dfxx[n]:=dfx[n]-dfx[n-1]:

x[n+1]:=evalf(x[n]-(2fx[n]dfx[n]x1[n])/

(2dfx[n]^2x1[n]- fx[n]dfxx[n]);

new[n]:=root1-x[n];

coc:=evalf(ln(new[n]/new[n-1])/(ln(new[n-1]/new[n-2]));

print(n,new[n],coc);

end:

Numerical example 1

We will demonstrate the convergence of the new iterative method for the following nonlinear equation

(39)

Having the exact value of the simple root of (39) is In Table 1 are the errors obtained by the methods described, based on the initial point (IP) . We observe that the new iterative method is converging to the expected order.

Table 1. Errors occurring in the estimates of the simple root of (39) by the methods described

Numerical example 2

We will demonstrate the convergence of the new iterative method for the following nonlinear equation

(40)

Having the exact value of the simple root of (40) is In Table 2 are the errors obtained by the methods described, based on the initial value

Table 2. Errors occurring in the estimates of the simple root of (40) by the methods described

Numerical example 3

We will demonstrate the convergence of the new method for the following nonlinear equation

(41)

having the exact value of the simple root of (41) is In Table 3 are the errors obtained by the methods described, based on the initial value .

Table 3. Errors occurring in the estimates of the simple root of (41) by the methods described

6. Conclusions

A new secant-type method for solving nonlinear equations with simple root has been presented. The effectiveness of the new method is examined by showing the accuracy of the simple root of several nonlinear equations. We have shown numerically and verified that the new iterative method has convergence of order 2.414. The major advantage of the new method is that it is very effective and produces high precision of approximation of the simple root. Finally, we conclude that the new method may be considered a very good alternative to the classical methods.