In order to find the zeros of nonlinear equations, in this paper, we propose a family of thirdorder and optimal fourthorder iterative methods. We have also obtained some particular cases of these methods. These methods are constructed through weight function concept. The multivariate case of these methods has also been discussed. The numerical results show that the proposed methods are more efficient than some existing third and fourthorder methods.
Newton’s iterative method is one of the eminent methods for finding roots of a nonlinear equation:
This paper is organized as follows. In Section
First we give some definitions which we will use later.
Let
Let
To improve the order of convergence of Newton’s method, some modified methods are given by GrauSánchez and DíazBarrero in [
Let
Suppose
Hence, from (
If we take
If we take
If we take
If we take
If we take
By taking different values of
The order of convergence of the methods obtained in the previous subsection is three with three function evaluations (one function and two derivatives) per step. Hence its efficiency index is
Let
Using (
If we take
If we take
If we take
If we take
If we take
If we take
By taking different values of
In this section, we extend some third and fourthorder methods from our proposed methods to solve the nonlinear systems. Similarly we can extend other methods also. The multivariate case of our thirdorder method (
Let
For the convenience of calculation, we replace
where
By virtue of (
The multivariate case of (
Let
For the convenience of calculation we replace
In this section, ten different test functions have been considered in Table
Functions and their roots.























Comparison of absolute value of the functions by different methods after fourth iteration (TNFE12).

Guess  HN3  M3  SL4  JM4  M4 





































































































































































































































































































































1.0 









1.2 





2.0 






1.5 






1.3 






1.8 





An effective way to compare the efficiency of methods is CPU time utilized in the execution of the programme. In present work, the CPU time has been computed using the command “
Comparison of CPU time (in seconds) between some existing methods and our proposed methods.
Function  CPU time  

Guess  HN3  M3  SL4  JM4  M4  

0.3  0.2867  0.2644  0.3060  0.2449  0.2449 

1.5  0.2943  0.2510  0.3049  0.2682  0.3043 

2.3  0.3019  0.3658  0.3457  0.3562  0.3483 

0.3  0.3091  0.2850  0.2832  0.2399  0.2428 

1.35  0.3399  0.3694  0.3938  0.4149  0.3940 

0.7  0.2896  0.2708  0.2388  0.2613  0.2550 

0.65  0.2517  0.2356  0.2938  0.2644  0.2880 

−1.00  0.2697  0.2279  0.2739  0.2934  0.2900 
Further, six nonlinear systems (Examples
Norm of the functions by different methods after first, second, third, and fourth iteration.
Example  Guess  Method 





Example 
(5.1, 6.1)  NR1 




NR2 





MM3 





BB4 





SH4 





MM4 







Example 
(1, 0.5, 1.5)  NR1 




NR2 





MM3 





BB4 





SH4 





MM4 







Example 
(0.5, 0.5, 0.5, −0.2)  NR1 




NR2 





MM3 





BB4 





SH4 





MM4 







Example 
(1.0, 2.0) 
NR1 




NR2 





MM3 





BB4 





SH4 





MM4 







Example 
(−0.8, 1.1, 1.1)  NR1 




NR2 





MM3 





BB4 





SH4 





MM4 







Example 
(0.5, 1.5)  NR1 




NR2 





MM3 





BB4 





SH4 





MM4 




Consider
Consider
Consider
Consider
Consider
Consider
In the present work, we have provided a family of third and optimal fourthorder iterative methods which yield some existing as well as many new thirdorder and fourthorder iterative methods. The multivariate case of these methods has also been considered. The efficiency of our methods is supported by Table
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to express their sincerest thanks to the editor and reviewer for their constructive suggestions, which significantly improved the quality of this paper. The authors would also like to record their sincere thanks to Dr. F. Soleymani for providing his efficient cooperation.