Research Article, Res Rep Math Vol: 2 Issue: 1

Enlarging the Radius of Convergence for the Halley Method to Solve Equations with Solutions of Multiplicity under Weak Conditions

Ioannis K Argyros^1* and Santhosh George²

¹Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

²Department of Mathematical and Computational Sciences, NIT Karnataka, India

*Corresponding Author : Ioannis K Argyros
Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
Tel: (580) 581-2200
E-mail: iargyros@cameron.edu

Received: August 18, 2017 Accepted: January 15, 2018 Published: February 10, 2018

Citation: Argyros IK, George S (2018) Enlarging the Radius of Convergence for the Halley Method to Solve Equations with Solutions of Multiplicity under Weak Conditions. Res Rep Math 2:1

Abstract

Keywords: Halley’s method; Solutions of multiplicity; Ball convergence; Derivative; Divided difference

Introduction

Many problems in applied sciences and also in engineering can be written in the form like

f (x) = 0, (1.1)

Using mathematical modeling, where is sufficiently many times differentiable and D is a convex subset in . In the present study, we pay attention to the case of a solution p of multiplicity m>1; namely and

The determination of solutions of multiplicity m is of great interest. In the study of electron trajectories, when the electron reaches a plate of zero speed, the function distance from the electron to the plate has a solution of multiplicity two. Multiplicity of solution appears in connection to Van Der Waals equation of state and other phenomena. The convergence order of iterative methods decreases if the equation has solutions of multiplicity m. Modifications in the iterative function are made to improve the order of convergence. The modified Newton’s method (MN) defined for each n=0,1,2,..

(1.2)

Where x₀∈D is an initial point is an alternative to Newton’s method in the case of solutions with multiplicity m that converges with second order of convergence.

A method with third order of convergence is defined by modified Halley method (MH) [4]

(1.3)

Method (1.3) is an extension of the classical Halley’s method of the third order. Other iterative methods of high convergence order can be found in [1-15] and the references therein.

Let denote an open ball and denote its closure. It is said that is a convergence ball for an iterative method, if the sequence generated by this iterative method converges to p; provided that the initial point But how close x₀ should be to x* so that convergence can take place. Extending the ball of convergence is very important, since it shows the difficulty; we confront to pick initial points. It is desirable to be able to compute the largest convergence ball. This is usually depending on the iterative method and the conditions imposed on the function f and its derivatives. We can unify these conditions by expressing them as:

(1.4)

(1.5)

for all x, y ∈ D; where are continuous and nondecreasing functions satisfying and

Then, we obtain the conditions under which the preceding methods were studied [1-17]. However, there are ceases where even (1.6) does not hold (see Example 4.1). Moreover, the smaller functions Ï�?₀, Ï�? are chosen, the larger the radius of convergence becomes. The technique, we present next can be used for all preceding methods as well as in methods where m=1: However, in the present study, we only use it for MH. This way, in particular, we extend the results in [4,5,12,13,16,17].

The rest of the paper is structured as follows. Section 2 contains some auxiliary results on divided differences and derivatives. The ball convergence of MH is given in Section 3. The numerical examples in the concluding Section 4.

Auxiliary Results

In order to make the paper as self-contained as possible, we restate some standard definitions and properties for divided differences [4,13,16,17].

Definition: The divided differences on k+1 distinct points y₀, y₁,…,y_k of a function f(x) are defined by

(2.1)

If the function f is sufficiently differentiable, then its divided differences can be defined if some of the arguments y_i coincide. for instance, if f(x) has k-th derivative at y₀; then it makes sense to define

Lemma: The divided differences f[y₀, y₁,…, y_k] are symmetric functions of their arguments,i.e., they are invariant to permutations of the y₀, y₁,…, y_k.

Lemma: If the function f has k-th derivative, and f ^(k)(x) is continuous on the interval then

(2.3)

Where

Lemma: If the function f has (k +1)-th derivative, then for every argument x; the following formulae holds

Where

(2.5)

Lemma: Assume the function f has continuous (m + 1)-th derivative, and x* is a zero of multiplicity m; we define functions g₀, g and g₁ as

(2.6)

Then,

(2.7)

Lemma: If the function f has an (m+1)-th derivative, and x* is a zero of multiplicity m, then for every argument x, the following formulae hold

(2.8)

(2.9)

And

where g₀(x); g(x) and g₁(x) are defined previously.

Local Convergence

It is convenient for the local convergence analysis that follows to define some real functions and parameters. Define the function ��?0 on + U{0} by

We have and Suppose

positive number of + ∞

for sufficiently large t. It then follows from the intermediate value theorem that function ψ₀ has zeros in the interval (0, + ∞): Denote by ρ₀ the smallest such zero. Define functions on the interval by

And

we get that and as

Denote by r₀ the smallest zero of function g0 in the interval (0, ρ₀): Moreover, we get that and

Denote by r the smallest zero of function ð�?�? on the interval (0,r₀ ): Then, we have that for each t ∈ [0, r)

The local convergence analysis is based on conditions (A):

(A₁) Function times differentiable and x* is a zero of multiplicity m.

(A₂) Conditions (1.4) and (1.5) hold

where the radius of convergence r is defined previously.

(A₄) Condition (3.1) holds

Theorem Suppose that the (A) conditions hold. Then, sequence generated for by MH is well defined in U(x*,r), remains in U(x*,r) for each n = 0, 1, 2…and converges to x*

Proof. We base the proof on mathematical induction. Set δ_n=x_n- x* and choose initial point x₀∈U(x*,r)- {x*}. Using (1.2), (2.8), (2.9) and (2.10), we have in turn that

(3.3)

so

(3.4)

or

(3.5)

Where

(3.6)

By (2.2) and (2.6), we can get

(3.7)

and by the condition (1.4) and

we obtain,

(3.8)

where y₀ is a point between x0 and x*, so g(x₀)≠0

(3.9)

Hence, we get

(3.10)

Using (2.2), (2.6), conditions (1.4), (1.5) and Lemma 2.3, we have

(3.11)

And

(3.12)

In view of (3.10), (3.11) and (3.12), we obtain

we get

(3.14)

We get by (3.8), (3.13) and (3.14)

(3.15)

Where

(3.16)

By simply replacing x₀, x₁ by x_k, x_k+1 in the preceding estimates, we get

(3.17)

so

Next, we present a uniqueness result for the solution x*

Proposition Suppose that the conditions (A) hold. Then, the limit point x* is the only solution of equation

Proof Let x** be a solution of equation f(x)=0 in D₁: We can write by (2.8) that

(3.18)

Using (1.4) and the properties of divided differences, we get in turn that

(3.19)

for some point between x** and x* It follows from (3.18) and (3.19) x**= x

Numerical Examples

We present a numerical example in this section.

Example Let D=[0; 1]; m=2; p=0 and define function f on D by

we have

function f ′′ cannot satisfy (1.5) with ð�?�? given by (1.6). Hence, the results in [4,5,12,13,16,17] cannot apply. However, the new results apply for and Moreover, the convergence radius is r=0.8.

Example Let D=[-1, 1], m=2; p=0 and define function f on D by

We get The convergence radius is r=1:4142; so choose r=1.

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