Newton-type iterative methods for finding zeros having higher multiplicity

Abstract: In this paper, using the idea of Gander, families of several iterative methods for solving non-linear equations f (x) = 0 having zeros of higher multiplicity are presented. The families of methods presented here include methods of Newton type, Steffensen type and their variant. We obtain families of methods of order 2 as well as 3. Some numerical examples are also presented in support of these methods.


Introduction
From the application point of view, solving non-linear equations has always been of great importance. Rarely, it becomes possible to find exact solutions of such equations. In that case, numerical methods are employed and among them one of the most widely used is the Newton-Raphson method, commonly known as Newton's method. It is known that under suitable conditions, if a function has a simple zero, Newton's method is of second order. Many people keep contributing towards the improvement of Newton's method by way of relaxing conditions or trying to increase the order of the method. If f (x) = 0 is a given non-linear equation, then the Newton method is given by: Gander (1985) used a different strategy to improve the Newton method. He produced a family of Newton-type methods by introducing a function G and proved the following: Theorem A (Gander, 1985) Let the functions f , G have a sufficient number of continuous derivatives in a neighbourhood of which is a simple zero of f. Then, the one point iteration method Gander made an observation that Theorem A can only be used if the zero of f is known in advance which is never the case. Thus, in order to use Theorem A effectively, he proved the following: Theorem B (Gander, 1985) Let f be a continuous function and H be any function satisfying H(0) = 1 and H � (0) = 1 2 ; then, the iterative method where F is defined by with is of order 3.
Though the Newton method (1.1) is very widely used, it has certain shortcomings. This method cannot be proceeded if during the iterative process, f ′ vanishes at some point. To overcome this problem, Wu (1999) made a modification as follows: where ≠ 0 is a real number chosen suitably. Wu (1999) proved that, in general, this method is of order 2. However, in Sharma (2007), it was proved that for = 1 2 f �� ( ) f � ( ) , the method is of order 3. As Gander obtained Theorems A and B for the method (1.1), similar results using the method (1.3) were obtained by Jain (2013). In all the above methods, the function f is assumed to have simple zero , i.e. f ( ) = 0 and f � ( ) ≠ 0.
It is known that for zeroes with multiplicity p > 1, the Newton method (1.1) is only linearly convergent. In order to retrieve second-order convergence, the following modified method is considered: In view of the fact that we do come across several non-linear equations having roots with higher multiplicity, a lot of iterative methods have appeared to deal with such situations. Some of them can be found in Farmer and Loizou (1985), Li, Liao andCheng (2009), Petković andMilošević (2012), Petković, Petković and Džunić (2014). The families of Newton-type methods given by Theorems A, B and also corresponding to method (1.3) are all pertaining to the equations having simple roots. There do not seem to exist these methods for roots with higher multiplicity. One of the aims of the present paper is to obtain all the results discussed above for the functions having zeros with multiplicity p > 1. The corresponding results are mentioned in Sections 2 and 3.
(1.2) x n+1 = F(x n ), , then the method becomes As the other aim of this paper, we obtain families of methods of orders 2 and 3 based on the last method for simple zeros as done in Theorems A and B. The case = 0 was discussed in Jain (2007).
These results are mentioned in Section 4.
In Section 5, we demonstrate the applications of our results. In particular, we show that the third order convergence of a method of Kasturiarachi (2002) can be obtained more easily by one of the results in the present paper. Finally, in Section 6, we provide examples in support of some of the methods obtained in this paper.
Throughout the paper, the functions used are real valued of one single variable. It is of interest if these results are studied for functions of several variables or even in Banach space setting.

The case of Newton method
Recall that if the function f has a simple zero , then the Newton method (1.1) is of order 2. For zero of multiplicity p > 1, this method is only linearly convergent. However, in such case, the two order convergence of the method can be retained by considering the method We prove the following: Proof If G( ) = p, then by taking the constant function G(x) = p, the method (2.2) is known to be of order 2.
Conversely, since is a p-order zero of f, we can write where h( ) ≠ 0 so that and we have (2.1) If we write then it can be checked by L'Hospital rule that u( ) = 0 and that Further, writing (2.3) as we find that Now, in order that the method (2.2) is of order 2, we must have F � ( ) = 0, which by (2.5) and (2.6) gives and the proof is complete. ✷ Remark 2.2 In order to apply Proposition 2.1, the zeros of f need to be known in advance but this is never the case in practice and so this proposition is of hardly any use in this form. However, following the idea of Gander (1985) is of order 2.
Note that the corresponding methods involve only first-order derivative of f. If we allow the higher derivatives as well, the order of the method can be increased to 3. In this regard, first we prove the following.
Proposition 2.6 Under the assumptions of Proposition 2.1, the iterative method Proof Following the proof of Proposition 2.1, if we again write then and with some calculations, it can be checked that

From (2.8), we have
and by applying L'Hospital rule, we can see that so that (2.9) gives Now, differentiating (2.6), we get so that But in order that the method (2.2) is of order 3, we must have which yields Remark 2.9 In Proposition 2.6 and Theorem 2.7, if p = 1, i.e., if f has a simple zero, then these results reduce, respectively, to Theorems A and B of Gander (1985).

The case of the method of Wu
Let us recall that Wu (1999) considered the method as a modification of Newton method in order to deal with the situation when f � (x n ) becomes zero during the iteration process. For simple zero, the method (3.1) is of order 2. Jain (2013) obtained a family of such methods.
For zeros of multiplicity p > 1, the method (3.1) can be modified as follows to retain the order of convergence 2: In this Section, we shall consider the case of multiple zeros and obtain families of methods of order 2 as well as of order 3. We follow the scheme of Section 2.  . Note that the corresponding methods involve only first derivative of f. If we allow the higher derivatives as well, the order of the method can be increased to 3. Precisely, we prove the following Proof Following the proof of Proposition 3.1, if we again write then we know that w( ) = 0, w � ( ) = 1 p and with some calculations, it can be shown that In the light of Remark 3.2, in order to make proper use of Proposition 3.5, we prove the following.

Theorem 3.6 Let f be a sufficiently differentiable function having a p-order zero, w be given by (3.8) and w be defined by
Let H be any function with H(0) = p and H � (0) = p; then, the iterative method is of order 3.
Proof Recall from the proof of Proposition 3.1 that the function w is such that w( ) = 0 and w � ( ) = 1 p . Also, it can be calculated that w( ) = 0 and So, if the function H is chosen so that

The case of simple zero for Steffensen-type method
It is very well known that the Steffensen method is the one which is obtained from Newton method . The method, therefore, read as This method is derivative free and known to be of order 2. The families of order 2 and 3 methods were obtained in Jain (2007). In this Section, we consider method (3.1) replacing again f � (x) by , i.e. the method and based on this method obtain families of second-order and third-order methods as was done in the previous sections. We consider the case when the function f has simple zero. We begin with the following: We find that (4.1) so that Using the L'Hospital rule, it can be calculated that Therefore, Since, for the method to be of second order, a necessary and sufficient condition is that F � ( ) = 0, it follows that and the assertion is proved. ✷ Next, we prove the following:

Theorem 4.2 Let f be a continuous function and H be any function with H(0) = 1. The iterative method
is of order 2, where the function C is defined by (4.4).
Proof Since C( ) = 0, the proof follows by Proposition 4.1 on taking G(x) = H(C(x)). ✷ Towards obtaining a family of order 3 methods, we prove the following: Proof If we consider the function C defined in (4.4), then we know that and with some tedious calculations, it can be shown that Now (4.5) gives so that (4.7) gives But in order that the method (4.2) is of order 3, a necessary and sufficient condition is that C( ) = 0 and C � ( ) = 1. Remark 4.5 The function D involves second derivative and consequently the family of the methods (4.8) involves second derivative. We can modify it so as to involve only first derivative. In fact, in D, , and denote this new function by D (x), i.e.

Application
In this Section, we shall apply the tools developed and discussed in the previous sections on some concrete methods. Kasturiarachi (2002) studied the method where In fact, this is an amalgamation of the secant method and Newton method. Kasturiarachi proved that the method (5.1) is of order 3. We shall prove that the order 3 of the method (5.1) can be obtained more easily using Theorem A.

Examples
Example 6.1 We consider the equation  which is of order 3. Now, we consider two equations and on which we apply methods (3.2), (6.1) and (6.2) and tabulate the iterations in Tables 2 and 3, respectively. (6.2)