Steffensentype methods are practical in solving nonlinear equations. Since, such schemes do not need derivative evaluation per iteration. Hence, this work contributes two new multistep classes of Steffensentype methods for finding the solution of the nonlinear equation
Finding rapidly and accurately the zeros of nonlinear functions is a common problem in various areas of numerical analysis. This problem has fascinated mathematicians for centuries, and the literature is full of ingenious methods, and discussions of their merits and shortcomings [
Over the last decades, there exist a large number of different methods for finding the solution of nonlinear equations either iteratively or simultaneously. Thus far, some better modified methods for finding roots of nonlinear equations cover mainly the pioneering work of Kung and Traub [
In this paper, we consider the problem of finding a numerical procedure to solve a simple root
Let the sequence
The efficiency of a method [
The socalled method of Newton is basically taken into account to solve (
This paper tries to overcome on the matter of derivative evaluation for nonlinear solvers by giving two general classes of threestep eighthorder convergence methods, which have the optimal efficiency index, optimal order, accurate performance in solving numerical examples as well as are totally free from derivative calculation per full cycle to proceed. Hence, after this short introduction in this section, we organize the rest of the paper as comes next. Section
In Mechanical Engineering, a trunnion (a cylindrical protrusion used as a mounting and/or pivoting point; in a cannon, the trunnions are two projections cast just forward of the center of mass of the cannon are fixed to a twowheeled movable gun carriage) has to be cooled before it is shrink fitted into a steel hub.
The equation that gives the temperature
Clearly (
For the first time, Steffensen in [
We here remind the wellwritten family of derivativefree threestep methods, which was given by Kung and Traub [
Another uniparametric threestep derivativefree iteration was presented recently by Zheng et al. in [
As usual to build highorder iterations, we must consider a multipoint cycle. Now in order to contribute, we pay heed to the follow up threestep scheme in which the first step is Steffensen, while the second and the third steps are Newton's iterations. This procedure is completely inefficient. Since it possesses
To annihilate the derivative evaluations of the structure (
Let us consider
Using Taylor's series and symbolic computations, we can determine the asymptotic error constant of the threestep uniparametric class (
A simple computational example from the class (
Although the structure (
The contributed class (
The scheme (
Let us consider
Applying Taylor series and symbolic computations, we can determine the asymptotic error constant of the threestep uniparametric family (
A very efficient example from our novel class (
Each method from the proposed derivativefree classes in this paper reaches the efficiency index
Some other examples from the class (
We check the effectiveness of the novel derivativefree classes of iterative methods (
The examples considered in this study.
Test functions  Simple zeros 

































The results of comparisons are given in Table
Results of convergence under fair circumstances for different derivativefree methods.

Guess  ( 
( 
( 
( 
( 
(  


IT  8  4  3  3  3  3  
TNE  16  12  12  12  12  12  
0.3 








−0.1 








0.5 








0.2 








 

IT  9  4  3  3  3  3  
TNE  18  12  12  12  12  12  
−0.3 








−0.6 








−0.8 








−0.4 








 

IT  9  4  3  3  3  3  
TNE  18  12  12  12  12  12  
1.9 




F. 



2 




F. 



2.1 

F. 


F. 



1.6 








 

IT  8  4  3  3  3  3  
TNE  16  12  12  12  12  12  
−1.5 








−1.2 








−1.6 




F. 



−1 








 

IT  8  4  3  3  3  3  
TNE  16  12  12  12  12  12  
0.3 








0.2 








0.1 








0.15 








 

IT  8  4  3  3  3  3  
TNE  16  12  12  12  12  12  
−3 








−2.5 








−3.2 








−2.1 








 

IT  8  4  3  3  3  3  
TNE  16  12  12  12  12  12  
3 








3.1 








1.4 








1.5 








 

IT  9  4  3  3  3  3  
TNE  18  12  12  12  12  12  
0.9 








1.2 








1 








0.3 








 

IT  8  4  3  3  3  3  
TNE  16  12  12  12  12  12  
0.9 








1 








0.4 








0.6 








 

IT  9  4  3  3  3  3  
TNE  18  12  12  12  12  12  
−0.3 








−0.6 








−0.9 








0.3 








 

IT  9  4  3  3  3  3  
TNE  18  12  12  12  12  12  
0.7 








0.5 








0.8 

F. 






0.4 








 

IT  9  4  3  3  3  3  
TNE  18  12  12  12  12  12  
2 

F. 






0.6 








1 








0.5 








 

IT  8  4  3  3  3  3  
TNE  16  12  12  12  12  12  
0.6 








0.55 








0.3 








0.4 








 

IT  9  4  3  3  3  3  
TNE  18  12  12  12  12  12  
11.5 








10 








8 








9.2 








 

IT  8  4  3  3  3  3  
TNE  16  12  12  12  12  12  
1.5 








1.2 








1.7 

F. 






0.9 








 

IT  9  4  3  3  3  3  
TNE  18  12  12  12  12  12  
−0.1 








−0.2 








−0.3 








0.2 







It can be observed from Table
We here remark that the eighthorder iterative methods such as (
Note that experimental results show that whatever the value of
A simple glance at Table
Multipoint methods without memory are methods that use new information at a number of points. Much literature on the multipoint Newtonlike methods for function of one variable and their convergence analysis can be found in Traub [
In this paper, two novel classes of iterations without memory were discussed fully. We have shown that each member of our contributions reach the optimal order of convergence eight by consuming only four function evaluations per full cycle. Thus, our classes support the optimality conjecture of Kung and Traub for building optimal without memory iterations. Our classes can be taken into account as the generalizations of the wellcited derivativefree method of Steffensen. We have given a lot of numerical examples in Section
The author cheerfully acknowledges the interesting comments of the reviewer, which have led to the improvement of this paper.