A NEW FAMILY OF SECANT-LIKE METHOD WITH SUPER-LINEAR CONVERGENCE

Another new family of Secant-like method is proposed in this paper. Analysis of convergence shows that the method has a super-linear convergence as Secant method. Numerical experiments show that the efficiency of the method is depended on the value of its parameter. AMS Subject Classification: 65D99, 65H05


Introduction
In this study, we are concerned of finding an approximated solution of a nonlinear equation of the form f (x) = 0.
Chen [4] rearranges (8) as Equation ( 11) can be expressed as By approximating x 2 in the right hand side of ( 12) with Secant method and simplifying, Chen obtains a super linearly convergent iterative method In this article we discuss another variation of Secant method and its convergent analysis in Section 2.Then, in Section 3 we compare the proposed method with the methods given in (10) and (13) using some test functions.

Proposed Method
Dividing both side equation (8) with f (x 1 )−f (x 0 ) Rearranging the terms on equation ( 14), we obtain By approximating x 2 in the right-hand side equation (15) with Secant method yields From ( 16), we propose the iterative formula where If we take α = 0, then equation ( 17) becomes Secant method.
Theorem 1. Assume that the function f : and second derivatives in the interval D, then the order of convergence of the method defined by (17 Proof.Let e n = x n − x * , then Taylor expansion of f (x n ) about x n = x * is given by where c j = f (j) (x * ) j!f ′ (x * ) .Similarly, we obtain Subtracting ( 21) from (20) yields Since e n is small enough, then substituting ( 22) into (19) we obtain Using geometry series, (23) can be written as By ignoring the terms containing e j n e k n , with k + j > 2 yields Then, using (21), (24), dan (25) we have On substituting (26) into (17), and recalling e n+1 = x n+1 − x * , we get Equation ( 27) is the same as equation ( 4), so the order of convergence the iterative method ( 17) is (1 + √ 5)/2.

Numerical Experiments
In this section we compare the proposed method (17) with the methods introduced by Kanwar (10) and Chen (13) using the following test functions x * = 2.094551481542327, x * = 0.5671432904097839.The results are shown in Table 1.In this table, NC indicates that the methods are not convergent or exceed the number iterations allowed or blow up.
From computational experiments, we can see that the proposed method (PM) is comparable with Kanwar's method (KM) and Chen's method (CM), especially for small enough α.Hence, we can conclude that the proposed method can be used as an alternative method for Secant-like methods.

Table 1 :
The number iterations needed to obtain the solution of f (x) = 0 by varying parameter α