A New Variant of Chebyshev-Halley’s Method Without Second Derivative with Convergence of Optimal Order

INTRODUCTION

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The problems of determining the roots of a nonlinear equation constitutes one of the very important problem in numerical analysis. It is well-known that the following Newton’ss method, i.e:

(1)

constitutes as classical iterative method that can be used to find a simple root of a nonlinear equation f(x) = 0, where f : D ⊂ ℜ→ ℜ is a scalar function. As Ostrowski¹ is stated that this method is quadratically convergent with efficiency indexes equal to

In order to improve the local order of convergence, many modification of the method have been proposed. A family of iterative method with third order-convergence has been reported by Amat et al.² and Hernandez and Salanova³ as follows:

(2)

Where:

(3)

The Eq. 2 is known as Chebyshev-Halley’s method for some β. In particularly, the Eq. 2 becomes the Chebyshev’s method if β = 0, Halley’s method if β = 1/2 and super Halley’s method if β = 1, according to Gutierrez and Hernandez⁴.

Note that the Eq. 2 requires a second derivative of f. It is evident that the Eq. 2 can not be used in the cases in which the second derivative of f is not exist. Recently, some modification of Eq. 2 have been studied to avoid the second derivative by using several approximation such as Taylor’s series expansion^5-8, finite different quotient^9-11, cubic polynomial¹², quadratic function¹³, linear combination¹⁴ and hyperbola¹⁵.

Motivated by the recent study, in this paper is presented a new variant of the classical Chebyshev-Halley’s method that contain two real parameters using a new approximation to avoid the second derivative o f f in Eq. 2. It is shown that for θ = 1, this new method constitutes a generalization of several previous methods that be proposed in Ostrowski¹, Chun¹⁶, Potra and Ptak’s¹⁷ and Sharma¹⁸. In the end of this study, a numerical simulation is presented for comparing several methods.

NEW METHOD

To derive this method, let us consider the Chebyshev-Halley’s method in Eq. 2 in the following form:

(4)

The iteration scheme of Eq. 4 has the third order of convergence and contains a second derivative function. In order to avoid the second derivative, f (x_n) is approximated by using an equality of two methods.

To derive an approximation for f (x_n) in Eq. 4, firstly use a circle of curvature that has a same tangent line at (x_n, y_n) of the curve y = f(x) that given by:

(5)

The circle of curvature in Eq. 5 that throughout at intersection at point (x_n+1, 0) is:

(6)

Let f(x_n) = y_n , f '(x_n) = y_n and f ''(x_n) = y"_n, then (6) can be written as:

(7)

Where:

(8)

By substituting (8) into (7), one have:

(9)

Furthermore, consider the following Potra-Ptak’ss method¹⁷:

(10)

where y_n is a Newton’ss method that defined by (8):

Based on Eq. 9 and 10, one get a new expression of f''(x_n) that given by:

(11)

By subtituting Eq. 11 into Eq. 4, one obtained a new two-parameters family of Chebyshev-Halley’s method which free of second derivative, that is:

(12)

The Eq. 12 is a variant of Chebyshev-Halley’s method with two parameters θ and β that requires three evaluation of functions f(x_n), f '(x_n) and f(y_n).

One can see that for θ = 1 and β∈ℜ, the family of Eq. 12 constitutes a generalization of Chebyshev-Halley’s method. For β →±∞, one get the Newton’ss method as defined by Eq. 1. For β = 0, one get the Potra-Ptak’ss method¹⁷:

For β = ½, one get the Newton’s-Steffensen’s method¹⁸:

For β =1, one get the Ostrowski’s method¹:

For β = -1/2, one get the Chun’s method¹⁶:

Furthermore, it will be shown that this new method has the fourth-order convergence.

Theorem: Let α ∈ D be a simple root of a differentiable function f : D ⊂ ℜ → ℜ. If the initial value x₀ is sufficiently close

to α, then the method defined by Eq. 12 has fourth order convergence for θ = 1 and β =1 with error:

(13)

Proof: Let α is the root of nonlinear equation f(x) = 0. If e_n = x_n-α and , then the expansion of Taylor’s series for f(x_n) and f '(x_n) around α is given by:

(14)

and:

(15)

because:

(16)

Using the Eq. 14 and 15, one get as follows:

(17)

Furthermore, using the Eq. 17 and x_n = α + e_n, one get:

(18)

and using the expansion of Taylor’s series around α, f(y_n) can be written as:

(19)

By using the Eq. 16 and 19, one get:

(20)

(21)

and:

(22)

Furthermore, by using the Eq. 20-22 and e_n = x_n-α, the Eq. 12 becomes:

(23)

Where:

A0	=	f(α)2
A1	=	1+f(α)2
A2	=	4 f(α)2-1
A3	=	f(α)6+22f(α)4+f(α)2+6
A4	=	-f(α)6-4f(α)4+7f(α)2+2
A5	=	f(α)2 (7f(α)4+27f(α)2+4)
A6	=	f(α)4 (5f(α)4+15f(α)2+2)
A7	=	f(α)2 (2f(α)4+7f(α)2-2)
A8	=	7f(α)4+24f(α)2+5

Using the Eq. 23 and by taking θ = 1, one get:

(24)

Because the Eq. 12 requires three evaluation of functions, it becomes an optimal iterative method when it has fourth order of convergence¹⁹. So, based on the Eq. 24, one can see that the order of convergence of Eq. 24 will increase quartically by taking β = 1 and it can be written as:

(25)

The Eq. 25 has fourth-order convergence and it requires three evaluation of functions with an efficiency index equal to ≈ 1.5874.

NUMERICAL EXAMPLE

In this section is presented a numerical example to illustrate efficiency of the proposed method by using several test functions. The zeros approximation α of the test functions was displayed 20th decimal places.

It is compared the performance of Eq. 12 both of β ≠ 1 (VCH3) and β = 1 (VCH4) with Newton’ss method (N2)²⁰, classical Chebyshev-Halley’s method with β = 1/2 (CH3)^3,4, Potra-Ptak’ss method (PP3)¹⁷. All computations are performed by using Maple 13.0 with 850 digits floating point arithmetics for the following several test functions:

•	f1(x)	=xe‾x -0.1, α = 0.11183255915896296483
•	f2(x)	=ex-4x2, α = 4.30658472822069929833
•	f3(x)	=cos(x)-1, α = 0,73908513321516064165
•	f4(x)	=(x-1)3-1, α = 2.00000000000000000000
•	f5(x)	=x3+4x2-10, α = 1.36523003414096845760
•	f6(x)	=-cos(x+1)+x3+1, α = -1.000000000 00000000000
•	f7(x)	=-x , α = 1.00000000000000000000

Table 1 shows the number of iteration (IT) that satisfies stopping criteria:

(26)

where, e = 10^-95 and all computation order of convergence (COC) in the parentheses by using the following formula:

(27)

Based on the Table 1, one can see that order of convergence of the proposed method is three for β ≠1 and four for β = 1.

The accuracy of the new method and several other methods by using the same total number of functional evaluation as comparison are presented at Table 2. Based on the Table 2, one can see that accuracy of the proposed method for β = 1 is better than other methods.

Table 1:	Number of iteration (IT) and COC

Table 2:	Absolute value of function |f (xn+1)| under same total number of functional evaluation (TNFE) with TNFE = 12