A fourth order method for finding a simple root of univariate function

In this paper, we describe an iterative method for approximating a simple zero z of a real defined function. This method is a essentially based on the idea to extend Newton’s method to be the inverse quadratic interpolation. We prove that for a sufficiently smooth function f in a neighborhood of z the order of the convergence is quartic. Using Mathematica with its high precision compatibility, we present some numerical examples to confirm the theoretical results and to compare our method with the others given in the literature.


Introduction
Newton's method is one of the fundamental tools in numerical analysis, operations research, optimization and control.It has numerous applications in management science, industrial and financial research, data mining.Newton's method (also known as the Newton-Raphson method), is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.The Newton-Raphson method in one variable has a second order of convergence and is implemented as follows: Given a function f defined over the reals x, and its derivative f ′ , we begin with a first guess x 0 for a root of the function f .Provided the function is reasonably well-behaved a better approximation x 1 is Geometrically, (x 1 , 0) is the intersection with the x-axis of the line tangent to f at (x 0 , f (x 0 )).The process is repeated as until a sufficiently accurate value is reached.
An other manner to obtain the Newton method is that x n+1 can be computed by a simple osculatory linear interpolation of the inverse function of f at the point y n = f (x n ) using the known data: the value and the first derivative at y n (for more details on osculatory interpolation of the inverse function of f , one can see [21]).
There exist many iterative methods improving Newton's method for solving nonlinear equations.However, many of those iterative methods depend on the second or higher derivatives in computing process which make their practical application restricted strictly.As a result, Newton's method is frequently and alternatively used to solve nonlinear equations because of higher computational efficiency.There has been some progress on iterative methods improving Newton's method with cubic convergence that require the function and its first derivative evaluations for solving nonlinear equations, see [4,7,9,10,11,16,18,22] and the reference therein.
To further improve the order of convergence, some fourth-order iterative methods have been proposed and analyzed (see [1,5,13,14,21], for instance).These methods are free from second derivatives and require only three evaluations of both the function and its first derivative.
In this paper, we propose an efficient and rapid fourth order iterative method for approximating a simple zero of f , and use exactly the same type and number of data of the third order method given by Kasturiarachi [11] see also [8].Our method is essentially based on the idea to extend the Newton's method, since it can be obtained by the inverse linear interpolation, to be an inverse quadratic interpolation.To do this, firstly we compute an intermediate approximation t n of x n using the Newton's approximation, and secondly we compute x n+1 using the quadratic polynomial in y = f (x) which interpolates the inverse function of f at x n and t n and its first derivative at x n .The expression of our iterative method can be viewed as a simple modification of the Leap-frogging method's expression [11].
The paper is organized as follows: In Section 2, we give some tools needed to prove the convergence of the proposed iterative method.In Section 3, we propose the method for finding a simple zero of a real-valued function, starting with a reasonable initial guess and we state the main result concerning the fourth order of convergence and asymptotic error constant.In Section 4 we give some numerical example to illustrate the theoretical results and to compare our method to Leapfrogging method and some existing fourth order methods in the literature.

Preliminary
Let f : R −→ R be function having a simple zero z and sufficiently smooth in a neighborhood of this zero.An alternative sufficient method for computing a solution of the equation f (x) = 0 is given by rewriting it in the equivalent A fourth order method for finding a simple root of univariate function199 form g(x) − x = 0 where g is a certain real-valued function, defined and sufficiently smooth in a neighborhood of z.Upon such a transformation the problem of solving the equation f (x) = 0 is converted into one of finding the fixed point of g.In order to approximate this fixed point, we use the following theorem.
Theorem 2.1.There exists an interval J centered at z, such that for any x 0 ∈ J, the the sequence {x n } ∞ n=0 generated according to the following recurrence relation converges to z as n → ∞.

Proof:
Let r ∈ N with N be the set of natural numbers.Assume that g is of class C r in a neighborhood of z and ( Suppose also that x n belong to a sufficiently small neighborhood of z for n ≥ 0.
By applying Taylor's theorem we obtain where ξ ∈ (a, b) with a = min(z, x n ) and b = max(z, x n ).Since g(x) is continuous at z, there exists, for all given ε > 0, a number δ such that for every x n with |x n − z| < δ. r) is continuous on J, there exists a strictly positif real M such that |g (r) (x)| ≤ M, ∀x ∈ J.In the case r = 1, using Relation (2.2) we can take M < 1.
From Relation (2.4), we obtain where It is easy to see that the function g is a contraction on J for any r.Consequently g has a unique fixed point in J.Moreover, the sequence {x n } ∞ n=0 defined by Relation (2.1) converges to z as n → ∞ for any starting value x 0 in J (see [2,3,6,12,19,20], for instance).✷ From [6,19,21], we have the following definition Definition 2.2.Put e n = |x n − z|, then for the method (2.1) the order of the convergence is equal to r and the asymptotic error constant η is given by (2.6)

Fourth order modified Newton's method
In this Section, we propose a method for finding a simple zero of a real-valued function, starting with a reasonable initial guess.For this, let f be a real function having a simple zero z and six times differentiable in a neighborhood of z and x 0 be an initial guess.We compute t 0 using the Newton's approximation, i.e., t 0 can be obtained as the intersection of the line P 1 (y) interpolating the inverse function and its first derivative at f (x 0 ) with the x-axis.This interpolant is written in recursive Newton's interpolation form.Then if we denote by F the inverse function of f , it is easy to see that where P 0 (y) is the interpolant of degree 0 at f (x 0 ), i.e, P 0 (y) = x 0 , and . Now if we denote by P 2 (y) the quadratic polynomial which interpolates the inverse function of f at the points f (x 0 ) and f (t 0 ) and its first derivative at f (x 0 ), then P 2 (y) can be written as follows: Since we are looking for a zero of f , so we replace y = f (x) by 0 in this equation, then we can compute x 1 by the following relation (3.1) A fourth order method for finding a simple root of univariate function201 Repeating this process, we obtain a sequence of numbers x 1 , x 2 , . . ., x n , . . . the following iterative formula: To prove the convergence of the iterative method defined by Relation (3.2) we need some notations and lemmas.Put . Define the following functions: Using the Relations (3.5), (3.4) and (3.6), we rewrite the function g and we obtain The iterative method (3.2) can be written in a new form: Therefore, to approximate the zero z of f it suffice to approximate the fixed point of the function g.
Lemma 3.1.The value and the derivatives of t of order 1, . . ., 3 at x = z are given by and D. Sbibih, A. Serghini, A. Tijini and A. Zidna Using the fact that f (z) = 0, it is easy to see that t(z) = z, t ′ (z) = 0 and . By differentiating the function t ′′ and using the fact that f (z) = 0, we obtain easily that The value and derivatives of L of order 1, . . ., 4 at x = z are given by and Using Lemma 3.1, we get By differentiating the function L ′′′ and using the fact that The value and derivatives of S of order 1, . . ., 4 at x = z are given by and To compute the derivatives of R at x = z, we need the derivatives of order 1, . . ., 4 of U and V .
For the function U , we have and and U (4) (z) = 8f ′ (z)f ′′′ (z) + 6f ′′2 (z). (3.10) For the function V , we have and By differentiating the function V ′′′ and using the fact that t ′ (z) = 0, it is easy to see that Applying the Leibniz formula to the function R, we get then, using relations (3.9), (3.10), (3.11) and (3.12) we obtain we have also Then

.16) ✷
Now we state the main result.
Theorem 3.5.Let f be real valued function which has a simple real zero z and which is six times differentiable in a small neighborhood of z.Then the Modified Newton's method given by Relation 3.2 converges with order 4 and its asymptotic error constant is given by

17)
Proof: To prove this theorem, it suffice to prove that To do this, we begin by computing the derivatives of order 3 of both sides of Equation (3.7).Then, by the Leibnitz formula we obtain A fourth order method for finding a simple root of univariate function205 Using the fact that g(z) = z and Lemmas 3.1, 3.3 and 3.4, we get 6g ′ (z)f ′3 (z) = 0. Therefore, g ′ (z) = 0.By the same technique we compute the derivatives of order 4 of both sides of Equation (3.7).Then we get Consequently g ′′ (z) = 0. We continue with the Leibnitz formula and we differentiate Equation (3.7) to the order 5 and 6 at x = z.Then, for the order 5, we obtain For the order 6, we have Using Lemmas 3.1, 3.3 and 3.4 and by eliminating the terms −30t (4) (z)f ′3 (z) and and therefore By the results given in Section 2, it is easy to see that the iterative method (3.2) converge to the unique fixed point of g which is the simple zero of f .Furthermore, the order of convergence is 4 and the asymptotic error constant is given by Relation (3.17).✷ Remark 3.6.The method proposed in this paper can be viewed as a simple modification of the Leap-frogging Newton's method by taking the same type and number of data.We mention that our method is the fourth order of convergence.In the other side the Leap-frogging Newton's method is the third one.Furthermore, the efficiency index for our method is 4 1/3 ≈ 1.5874 which is better than 3 1/3 ≈ 1.4422 the one for the Leap-frogging Newton's method.

Numerical examples
In this section we give some numerical examples for testing the performance of the proposed method.To programming this method we used Mathematica which allows us to find the zero with higher precision.In the first part of this Section, we compare our results with the results given by the Leap-frogging Newton's method.For this, we consider two test functions and we compute |x n − z| the error between the zero obtained by our method and the exact zero for n = 0, . . ., 6. D. Sbibih, A. Serghini, A. Tijini and A. Zidna Exemple1: In this example we consider the function f defined by which has as a simple zero We choose as initial guess x 0 = 5.Exemple2: In this example we consider the function f defined by which has as a simple zero z = 2.We choose as initial guess x 0 = 2.5.
In Table 1 and Table 2, we give, respectively in each column, the errors obtained by the Leap-frogging Newton's method, the errors obtained by our method, the quantity e n+1 /e 4 n which gives the practical order of convergence and the value of the theoretical quantity asymptotic error constant η.These examples present the higher performance of our method comparing with the Leap-frogging Newton's method and illustrate the fourth order of convergence and confirm the value of the asymptotic error constant shown in the previous Section.
Table 1: Comparative rates of convergence between Leap-frogging Newton's method ( noted LFN) and our method for f (x) = x 3 − 3x In the second part of this Section we present some numerical experiments using our iterative method and we compare these results to well known fourth-order schemes.All computations were done using Mathematica programming using 128 digit floating point arithmetics.We accept an approximate solution rather than the exact root, depending on the precision ǫ of the computer.We use the following stopping criteria for computer programs: and so, when the stopping criterion is satisfied, x n+1 is taken as the exact zero z.
For numerical illustrations in this section we used the fixed stopping criterion ǫ < 10 −25 .We used the test functions given in [5,15,22].
The comparison is about the number of iterations to approximate the zero (IT), the number of functional evaluations (NFE) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative, the value f (z) and the distance δ of two consecutive approximations for the zero.The test results in Table 4 show that for most of the functions we tested, the iterative method introduced in this work have equal or better performance as compared to the other methods of the same order and use the same number and type of data.

Conclusion
In this paper, we have described an efficient iterative method for approximating a simple zero z of a real defined function.This method is a essentially based on the idea to extend Newton's method to be the inverse quadratic interpolation.We have proved that for a sufficiently smooth function f in a neighborhood of z, this method is a fourth order of convergence.We have used Mathematica programming with its high precision compatibility and we have presented some example tests to confirm the theoretical results.We also have showed the performance of our method comparing with the third order and fourth order existing methods in the literature.

Proof:Lemma 3 . 4 .
It suffices to use Leibniz formula and Lemma 3.2.✷ The value and derivatives of R of order 1, . . ., 4 at x = z are given by 2− 5.

Table 3 :
Test functions, zeros and initial guess.

Table 4 :
A Comparison of various existing fourth order methods and our method.