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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 Argyros1* 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: [email protected]

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

The objective of this paper is to enlarge the ball of convergence and improve the error bounds of the Halley method for solving equations with solutions of multiplicity under weak conditions.

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

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,..

equation (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]

equation (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 equation denote an open ball and equation denote its closure. It is said thatequation is a convergence ball for an iterative method, if the sequence generated by this iterative method converges to p; provided that the initial point equation 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:

equation (1.4)

equation (1.5)

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

equation

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 ϕ0, ϕ 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 equation on k+1 distinct points y0, y1,…,yk of a function f(x) are defined by

equation

equation

equation (2.1)

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

equation

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

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

equation (2.3)

Where equation

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

equation

Where

equation (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

equation (2.6)

equation

Then,

equation (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

equation (2.8)

equation

equation (2.9)

equation

And

equation

equation

equation

equation

equation

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 𝜓0 on + U{0} by

equation

We have equation and equationSuppose

equation 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 equation on the interval equation by

equation

equation

equation

equation

And

equation

we get that equation and equation asequation

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

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

equation

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

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

(A2) Conditions (1.4) and (1.5) hold

equation where the radius of convergence r is defined previously.

(A4) Condition (3.1) holds

Theorem Suppose that the (A) conditions hold. Then, sequence equation generated for equation 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 δn=xn- 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

equation

equation (3.3)

so

equation

equation (3.4)

or

equation (3.5)

Where

equation

equation (3.6)

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

 (3.7)

and by the condition (1.4) and

equation we obtain,

equation

equation (3.8)

equation

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

equation (3.9)

Hence, we get

equation (3.10)

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

equation

equation

equation (3.11)

And

equation

equation

equation

equation (3.12)

equation

equation

equation

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

equation

equation

equation we get

equation

equation

equation

equation

equation

equation

equation

equation (3.14)

equation

equation

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

equation (3.15)

equation

equation

Where

equation (3.16)

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

equation (3.17)

so equation

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 equation

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

equation (3.18)

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

equation

equation (3.19)

equation

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

equation

we have equation

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

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

equation

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

References

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