We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourthorder convergence. Each of the three methods uses three functional evaluations. Thus, according to KungTraub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourthorder convergence recently published.
One of the most important problems in numerical analysis is solving nonlinear equations. To solve these equations, we can use iterative methods such as Newton's method and its variants. Newton's classical method for a single nonlinear equation
Taking
In [
Chun presented a thirdorder iterative formula [
Li et al. presented a fifthorder iterative formula in [
The main goal and motivation in the development of new methods are to obtain a better computational efficiency. In other words, it is advantageous to obtain the highest possible convergence order with a fixed number of functional evaluations per iteration. In the case of multipoint methods without memory, this demand is closely connected with the optimal order considered in the KungTraub’s conjecture.
Multipoint methods which satisfy KungTraub's conjecture (still unproved) are usually called optimal methods; consequently,
The computational efficiency of an iterative method of order
On the case of multiple roots, the quadratically convergent modified Newton's method [
For this case, there are several methods recently presented to approximate the root of the function. For example, the cubically convergent Halley's method [
Another thirdorder method [
Recently, Chun and Neta [
All previous methods use the second derivative of the function to obtain a greater order of convergence. The objective of the new method is to avoid the use of the second derivative.
The new methods are based on a mixture of Lagrange's and Hermite's interpolations. That is to say not only Hermite’s interpolation. This is the novelty of the new methods. The interpolation process is a conventional tool for iterative methods; see [
The three new methods (for simple roots) in this paper use three functional evaluations and have fourthorder convergence; thus, they are optimal methods and their efficiency index is
In this paper, we consider iterative methods to find a simple root
Consider the polynomial
Consider the polynomial
Consider the polynomial
Consider the polynomial
Solving the system, we obtain
Let
Following an analogous procedure to find the error in Lagrange's and Hermite's interpolations, the polynomials (
Then, substituting
Let
The proof is based on the error of Lagrange's interpolation. Suppose that
Taking
Since
Thus,
Note that
In this section, we use numerical examples to compare the new methods introduced in this paper with Newton's classical method (NM) and recent methods of fourthorder convergence, such as Noor's method (NOM) with
All computations were done using MATLAB 2010. We accept an approximate solution rather than the exact root, depending on the precision
We used the functions in Tables
List of functions for a single root.






















Comparison of the methods for a single root.


IT  NOFE  

NM  NOM  CHM  CM  ZM  FAM1  FAM2  FAM3  NM  NOM  CHM  CM  ZM  FAM1  FAM2  FAM3  

1  6  4  5  4  4  3  3  3  12  12  15  12  12  9  9  9 

1  5  3  4  2  3  3  3  2  10  9  12  6  9  9  9  6 

2  5  2  4  3  3  2  3  2  10  6  12  9  9  6  9  6 

1.3  5  3  4  4  3  3  3  2  10  9  12  12  9  9  9  6 

1  6  4  5  4  4  4  4  3  12  12  15  12  12  12  12  9 

2  6  4  5  4  4  3  4  3  12  12  15  12  12  9  12  9 

3  7  4  5  4  4  4  3  3  14  12  15  12  12  12  9  9 

3.5  6  3  5  3  4  3  3  3  12  9  15  9  12  9  9  9 

1  6  4  6  4  4  3  3  3  12  12  18  12  12  9  9  9 

2  9  5  7  6  5  4  4  4  18  15  21  18  15  12  12  12 

0.5  5  4  4  4  3  3  3  3  10  12  12  12  9  9  9  9 
List of functions for a multiple root.














Comparison of the methods for a multiple root.


IT  NOFE  

NMM  HM  OM  ECM  CNM  FAM4  NMM  HM  OM  ECM  CNM  FAM4  
2  5  3  3  3  3  3  10  9  9  9  9  9  

1  5  3  4  3  3  3  10  9  12  9  9  9 
−0.3  54  5  5  5  5  4  108  15  15  15  15  12  
 
2.3  6  4  4  4  4  4  12  12  12  12  12  12  

2  6  4  4  4  4  4  12  12  12  12  12  12 
1.5  5  4  4  4  3  3  10  12  12  12  9  9  
 
0  4  3  3  3  3  2  8  9  9  9  9  6  

1  4  4  4  4  4  3  8  12  12  12  12  9 

5  4  4  4  4  3  10  12  12  12  12  9  
 
1.7  5  4  4  4  4  3  10  12  12  12  12  9  

1  4  3  3  3  3  3  8  9  9  9  9  9 
−3  99  8  8  8  8  7  198  24  24  24  24  21  
 
3  6  4  5  5  4  4  12  12  15  15  12  12  

−1  10  11  24  23  5  6  20  33  72  79  15  18 
5  8  9  15  15  5  5  16  27  45  45  15  15  
 

8  5  6  6  6  5  16  15  18  18  18  15  


5  3  4  3  3  5  10  9  12  9  9  15 

14  9  10  10  10  9  28  27  30  30  30  27  
 
1.7  5  3  4  3  4  4  10  9  12  9  12  12  

2  4  3  3  3  3  3  8  9  9  9  9  9 
3  5  3  3  3  3  3  10  9  9  9  9  9 
The computational results presented in Tables
In this paper, we introduce three new optimal fourthorder iterative methods to solve nonlinear equations. The analysis of convergence shows that the three new methods have fourthorder convergence; they use three functional evaluations, and thus, according to KungTraub's conjecture, they are optimal methods. In the case of multiple roots, the method developed here is cubically convergent and uses three functional evaluations without the use of second derivative. Numerical analysis shows that these methods have better performance as compared with Newton's classical method, Newton's modified method, and other recent methods of third (multiple roots) and fourthorder (simple roots) convergence.
The authors wishes to acknowledge the valuable participation of Professor Nicole Mercier and Professor Joelle Ann Labrecque in the proofreading of this paper. This paper is the result of the research project “Análisis Numérico de Métodos Iterativos Óptimos para la Solución de Ecuaciones No Lineales” developed at the Universidad del Istmo, Campus Tehuantepec, by ResearcherProfessor Gustavo FernándezTorres.