Improved Ostrowski-Like Methods Based on Cubic Curve Interpolation

Janak Raj Sharma, Rangan Kumar Guha, Rajni Sharma Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, India Department of Applied Sciences, D.A.V. Institute of Engineering and Technology Kabirnagar, Jalandhar, India E-mail: jrshira@yahoo.co.in, rangankguha@yahoo.com, rajni_gandher@yahoo.co.in Received December15, 2010; revised May 10, 2011; accepted May 13, 2011


Introduction
Finding the root of a non-linear equation ( ) 0 f x  is a common and important problem in science and engineering.Analytic methods for solving such equations are almost non-existent and therefore, it is only possible to obtain approximate solutions by relying on numerical methods based on iteration procedures.Traub [1] has classified numerical methods into two categories viz. 1) one-point iteration methods with and without memory, and 2) multipoint iteration methods with and without memory.Two important aspects related to these classes of methods are order of convergence and computational efficiency.Order of convergence shows the speed with which a given sequence of iterates converges to the root while the computational efficiency concerns with the economy of the entire process.Investigation of onepoint iteration methods with and without memory, has demonstrated theoretical restrictions on the order and efficiency of these two classes of methods (see [1]).However, Kung and Traub [2] have conjectured that multipoint iteration methods without memory based on evaluations have optimal order .In particular, with three evaluations a method of fourth order can be constructed.The well-known Ostrowski's method [3] is an example of fourth order multipoint methods without memory which is defined as where 0,1, 2, i   and 0 x is the initial approximation sufficiently close to the required root.The method requires two function f and one derivative f  evaluations per step and is seen to be efficient than classical Newton's method.
Recently, based on Ostrowski's method (1) Grau and Díaz-Barrero [4] have developed a sixth order method requiring four evaluations, namely three f and one f  per iteration.Sharma and Guha [5] have shown that there exists a family of such sixth order methods with equal number of evaluations.
In the present paper, we derive two modified Ostrowski's-type methods which improve the local order of convergence from four for Ostrowski's method to eight for new methods.The important feature of these methods is that per step they require three evaluations of f and one evaluation of f  .Thus, the new methods support the conjecture of Kung and Traub for eighth order methods based on four evaluations.
The paper is organized in six sections.In Section 2, methods are developed and their eighth order convergence is established.In Section 3, computational effi-ciency of the methods is discussed.Section 4 contains the numerical experimentations and comparison with some well known methods.Concluding remarks are given in Section 5.In Section 6, references are given.

The Methods and Their Convergence
Method One Consider the Ostrowski scheme (1) now defined by In what follows, we construct the method to obtain the approximation 1 i x  to the root by considering the cubic curve interpolation.Let be an interpolatory polynomial of degree three such that and Our interest is to find the unknown parameters a, b, c and d introduced in the polynomial.In order to achieve that, we make use of the expressions (4) -( 7) in (3).From (3), ( 4) and (5), it is easy to show that   Substituting the values of a and b in (3) then using ( 6) and (7), we obtain after some simple calculations Solving these equations using Ostrowski iteration (2), we obtain where The tangent line to the curve of cubic polynomial (3) at the point Assuming that the root estimate x  is point of intersection of the tangent line (14) with x-axis, then Thus, from ( 7), ( 8) and ( 14), we obtain Now using the approximation (8) in (15), we can obtain the new improvement as given by where i is the Ostrowski point.It is quite obvious that formula (16) together with (2) requires five evaluations per iteration.However, we can reduce the number of evaluations to four by utilizing the approximation (8).Therefore, (3) and (8) yield Substituting the values of c and in (17), we obtain where Then the formula (16) in its final form is given by where is the Ostrowski iteration (2) and Thus, we derive a multipoint method based on the composition of two sub steps, Ostrowski sub step (2) followed by (19) obtained by tangential cubic interpolation.It is straight forward to see that per step the method util-izes four pieces of information namely- , Since we are using the approximation (8) for the derivative, therefore the error is given by (see [6]) x is sufficiently close to  then the method defined by ( 19) is of order eight.
 In order to show that the method is of order eight, we prove the following theorem: be errors in the ith iteration.Using Taylor's series expan- , where f

x e A e A A e A A A A e A A A A A A A e f x A A A A A A A A A
Expanding   i f w about  and using (24), we obtain Using Equations ( 21), ( 23) and (25), we obtain From second step of (2) it follows that

i i i i i e A A A e A A A A A A e   A A A A A A A A
Using ( 27) in (28), we get

i i i i i f z f A A A e A A A A A A e A A A A A A A A
Using the results of ( 21), ( 25) and (29) in and simplifying, we get From ( 22) and (30), we get Using ( 28) and (31) in we find the error equation as given by

e e e e e Ae A A A A A e O e A A A A A e O e A e A A A A A e e O e A A A A A A A A
.
Thus Equation (32) establishes the maximum order of convergence equal to eight for the iteration scheme defined by (19).This completes the proof of the theorem.
Remark 1.The error (20) is now given by .
This shows that the error in derivative approximation is of order four.
Remark 2. Upon using Taylor's expansion which is same as obtained in equation (31 This verifies the correctness of error (33) and calculation of are two iterative functions of order 1 and 2 , respectively, then the new composite iterative function has the order 1 2 In our case, the Ostrowski method (2) comprising the first two steps, is of order four.Thus to produce eighth order method the formula (19) should be of order two (neglecting how is obtained).From (34), it turns out that  Also, the Taylor's series expansion of the function  On substitution and simplifying, we see that the Newton-like method (15) and hence (19), has the order two, thus verifying the convergence theorem on composition of two iterative functions to produce eighth order iterative method.

Method Two
Here we consider the inverse interpolation.Let be an inverse interpolatory polynomial of degree three such that and From ( 36) and (37), we can calculate A and as given by Substituting the values of A and in (35), then using (38) and (39), we obtain The Equations ( 42) and (43) when solved, yield where The tangent line to the curve of cubic polynomial (35) at the point The approximation to the root 1 i x  is now obtained by intersecting this tangent line with x-axis.This yields where From ( 40), ( 44) and ( 45), we have where Hence, the iteration formula ( 46) is given by where is the Ostrowski iteration (2) and i z  is shown in (47).Thus, we obtain second modified Ostrowski-like method (48) developed by tangential inverse interpolation.In this method also, the number of evaluations required is same as in the first method.Error in the approximation (40), likewise the error (20), can be given by where Copyright © 2011 SciRes.AM In the following theorem we prove that the method is of order eight.

e Ae O e A A A A A e O e e e Ae A A A A A e O e A e A A A A A e e O e 8 A A A A A A A A
This result shows the eighth order convergence of method (48).
Remark 4. The error (49) upon using ( 21), ( 25) and ( 29) is given by  about  and using the fact that we can obtain the error as   This verifies calculations of rm (55) and error te therefore, similar to remark 3, the iterative formula (48) combined with the Ostrowski iteration (2) verifies the iterative functions to produce eig In order to obtain an assessment of the efficiency of our methods we shall make use of Traub's efficiency index ([1], Appendix C), according to which computational efficiency of an iterative method is given by

Computational Efficiency
where p is the order of the method and c is the cost per ction and derivative at is, iterative step of computing the fun required by the iterative formula, th , The value 0 j  simply gives the function f .Designating Ostrowski's method (1) as 4 , M sixth order method [4]

Numerical Illustrations
In this section, we ply the modified methods ap to solve some nonl ar equa ich not only illustrate the methods practically but also serve to check the validity of theoretical results we have derived.The performance is comp M .In mes order to compare the higher ord necessary that we use higher pre er m s it cision in computations.
pre-ethod beco Therefore, the calculations are performed with highcision arithmetic and terminated after three iterations.To check the theoretical order of convergence, we obtain the computational order of convergence (p) using the formula (see [7]) We consider the following test problems:

5.
the p int iterat q alua with theoretical analys ency di

Conclusions
In this work, we have obtained two multipoint methods of order eight using an additional evaluation of function at o ed by Ostrowski's method of order four for solving e uations.Thus, one requires three ev tions of the function f and one of its first-derivative f  per full step and therefore, the efficiency of the me s ki's method.The superiority of preorated by numerical resul displayed in the table 1.The computational order of con- thods i better than Ostrows sent methods is also corrob ts vergence (p) overwhelmingly supports the eighth order convergence of our methods.These methods also p vide the examples of eighth order methods requiring four evaluations for Kung and Traub conjecture.Finally, we conclude the paper with the remarks that such higher order methods are useful in the numerical applications requiring high precision in their computations.
composition of two hth order iterative method.

Table 1
shows the absolute difference