In recent studies, papers related to the multiplicative based numerical methods demonstrate applicability and efficiency of these methods. Numerical rootfinding methods are essential for nonlinear equations and have a wide range of applications in science and engineering. Therefore, the idea of rootfinding methods based on multiplicative and Volterra calculi is selfevident. NewtonRaphson, Halley, Broyden, and perturbed rootfinding methods are used in numerical analysis for approximating the roots of nonlinear equations. In this paper, NewtonRaphson methods and consequently perturbed rootfinding methods are developed in the frameworks of multiplicative and Volterra calculi. The efficiency of these proposed rootfinding methods is exposed by examples, and the results are compared with some ordinary methods. One of the striking results of the proposed method is that the rate of convergence for many problems are considerably larger than the original methods.
Ever since Grossman and Katz introduced multiplicative calculus in [
Let
If
Some properties and basic theorems of multiplicative derivative can be found in [
Let
Given
Let
In [
Furthermore, necessary concepts on Volterra calculus can easily be derived by using the above relations (
Taylor expansion for one variable cannot be obtained easily in Volterra calculus. Few factors of the Volterra type Taylor expansion are deduced, respectively, in [
As mentioned above, the zeros of functions (especially nonlinear functions) are very significant in real applications as well as mathematical applications such as critical points of nonlinear functions. Therefore, using numerical methods are essential in relation to these problems. NewtonRaphson, Chebyshev’s, Halley, Broyden, and perturbed methods are some of the important methods for approximating the zeros of functions. The papers [
It is also possible to create alternative methods based on multiplicative calculi which provide larger intervals for initial assumptions with more improved rate of convergence. Firstly, the multiplicative rootfinding methods have been discussed in master thesis [
In this section multiplicative rootfinding methods based on NewtonRaphson method are alternatively introduced. These methods will provide better performance for many problems.
The starting point of multiplicative Newton formulae will be the corresponding multiplicative Taylor theorems. In order to obtain Volterra taylor theorem, the relationships between multiplicative and Volterra derivatives, stated in Riza et al. [
Assume that
Since
The following theorem states the iteration of rootfinding algorithms with the convergent criteria in the framework of Volterra calculus.
Assume that
Insight can be easily given of the proof of multiplicative NewtonRaphson method.
It is very important to note that methods (
Alternatively the following theorem can be considered as a modification of Volterra NewtonRaphson method.
Assume that
Insight can be easily given of the proof of multiplicative NewtonRaphson method.
In this section, multiplicative perturbed rootfinding methods are derived based on corresponding Taylor theorems. The effort to derive these methods will provide better approximation with less computational time and complexity. The number of perturbed terms in corresponding ordinary perturbed methods (see [
We will start with the multiplicative Taylor theorem to derive corresponding multiplicative perturbed method. The first three factors in the multiplicative Taylor theorem can be given as
The solutions of the equations in (
We will derive Volterra perturbed method in the framework of Volterra calculus. The starting point is (
The selection of initial value is also very important for the convergence of the multiplicative rootfinding algorithms. In this section, the criterion of the convergence of the proposed methods will be given, which will also lead to a selection of the starting point for any given problem. Before giving the convergence of the multiplicative NewtonRaphson methods due to the initial value, the fixed point theorems should be derived in the multiplicative sense. Additionally, multiplicative Rolle and Mean Value theorems discussed in [
Suppose that
Let
By the assumption in multiplicative NewtonRaphson method,
Hence, the condition in (
Analogously, it is possible to derive similar condition for the corresponding Volterra method.
Assume that
It can easily be given by proof of multiplicative Rolle’s theorem.
Assume that
Let us consider the function
Then
Hence,
Suppose that
It can be easily shown by using Mean Value theorem in Volterra calculus.
Sufficient condition for initial value
Hence, condition in (
In this section, some examples will be considered to reveal the applicability of the introduced methods. Numerical results are reported which indicate that the proposed methods may allow a considerable saving in both the number of step sizes and reduction of computational cost. Besides, the examples consisting of different type of functions reveal the advantages of proposed methods compared to the ordinary methods.
It is important to note that finding a zero of the function
OP represents ordinary perturbation iterations, MP represents multiplicative perturbation iterations, and VP represents Volterra perturbation iterations.


OP  MP  VP 



NC 



NC 

 




NC 

NC 



 










 








NC 

 




NC 





 








NC 


NM represents ordinary NewtonRaphson iterations, MN represents multiplicative NewtonRaphson iterations, and AVN represents alternative Volterra NewtonRaphson iterations.


NM  MN  AVN 



NC 



NC 

 










 










 










 










 








NC 


Displayed in Table
According to the obtained results, multiplicative rootfinding algorithms can be used effectively and efficiently in real applications mentioned in section one. Moreover, these methods yield better approximations for the nonlinear equations especially when the equations involve exponential, logarithmic, and hyperbolic function. On the other hand, the ordinary methods can give more accurate results especially for polynomial equations. Thus, the method should be selected according to the functions appeared in the equations.
In this subsection, two examples will be considered to demonstrate possible impacts of the introduced methods to science and engineering.
The process in which chemicals interact to form new chemicals with different compositions is called chemical reactions. This process is the results of chemical properties of the element or compound causing changes in composition. These chemical changes are chemists’ main purpose. It is important to mention a remark from [
The number of function evaluations required such that


OP  MP  VP  NM  MN  AVN 


NC 


NC 

 

NC 





The Rayleigh function, which corresponds to Rayleigh distribution,
The number of function evaluations required such that


OP  MP  VP  NM  MN  AVN 


NC 

NC 


 







In this study, the multiplicative and Volterra based rootfinding methods are presented. These methods were tested for some nontrivial problems and compared with the original rootfinding method. The results show that in certain problems the multiplicative and/or Volterra methods give more accurate results compared to the original rootfinding methods. Especially, the examples showed that the nature of the underlying calculus plays an important role in approximating the zeros of the function.
The selection of the initial value is very important for the convergence of the iteration. Two theorems in Section
The authors declare that there is no conflict of interests regarding the publication of this paper.