Higher-order multiplicative derivative iterative scheme to solve the nonlinear problems

Grossman and Katz (ﬁve decade ago) suggested a new deﬁnition of diﬀerential and integral calculus which utilize the multiplicative and division operator as compared to the addition and subtraction. Multiplicative Calculus is a vital part of the applied Mathematics because of its application in the area of Biology, Science and Finance, Biomedical, Economic, etc. Therefore, we used a multiplicative calculus approach to develop a new fourth-order iterative scheme for multiple roots based on the well-known King’s method. In addition, we also propose the detailed convergence analysis of our scheme with the help of multiplicative calculus approach rather than the normal one. Diﬀerent kinds of numerical comparisons has been suggested and obtained results are very impressive as compared to the ordinary derivative methods. Finally, the convergence of our technique is also analyzed by basin of attractions that also support the theoretical aspects.


Introduction
In the 70s of the 20 th century multiplicative calculus introduced by Grossman and Katz [1].Many of scholars applied the multiplicative calculus in various branches.In 2008 Bashirov et al. [2] discussed the theoretical foundations as well as various applications of multiplicative calculus.Florack and Van Assen [3] used multiplicative calculus in Biomedical image analysis.Filip and Piatecki [4] used it to investigate economic growth.In addition, Mısırlı Gurefe [5], Riza et al. [6], and Özyapıcı and Mısırlı [7] used multiplicative calculus to develop multiplicative numerical methods.On the other hands, Bashirov et al. [8] applied it to develop multiplicative differential equations.Further, Bashirov and Riza [9] and Uzer [10] extended the multiplicative calculus to include complex valued functions of complex variables, which was previously applicable only to positive real valued functions of real variables.
From the above discussion, it is straightforward to say that multiplicative calculus approach is very important part of the applied Mathematics, Computational Engineering and Applied Sciences.In last few years, researchers used multiplicative derivatives for developing new iterative schemes for the solutions of the nonlinear equations.Özyzpici et al. [11] and Ali Özyzpici [12] adopted the multiplicative calculus approach (MCA) in order to construct an one-point and second order scheme.But, one-point methods have many problems regarding their order of convergence and efficiency index (See Traub [13] for more details).Obtaining the multiple root of a nonlinear equation is more complicated and challenging as compared to simple root.A few main reasons behind this are: the lengthy and complicated calculations and retaining or increasing order of convergence.
Keeping these things in our mind, we suggest a new multipoint iterative technique by adopting the MCA.Till today, we didn't have a single multipoint iterative method for multiple roots that utilize the multiplicative calculus approach.Our new scheme stands on the principles of MCA and the well-known fourth-order King's method [14].For a fair comparison of our methods with the existing methods, we choose the six different ways that are: (i) absolute error difference between two consecutive iterations (ii) order of convergence (iii) number of iterations (iv) CPU timing (v) the graphs of absolute errors and (vi) bar graphs.On the basis of six different ways of comparisons, we conclude that our new King's scheme perform much better in comparison of the existing methods.Finally, we study the basin of attraction which also support the numerical results.
The rest content of the paper are summarized as: Section 2 discussed the definition and basics terms of multiplicative calculus.The proposed method and its analysis of convergence is represented in Section 3. Section 4 depicts the numerical results.The basins of attraction of proposed method are discussed in Section 5. Finally, conclusion is represented in Section 6.

Basic terms of Multiplicative Calculus
Definition 2.1 Let g(x) be a real positive valued function in the open interval (a, b).Assume function g(x) be changes in x ∈ (a, b) s.t.g(x) changes to g(x + h).Then multiplicative forward operator [7] denoted as ∆ * defined as follows By considring the operator ∆ * (2.1), multiplicative derivative can be defined as below (2. 2) The function g * (x) is said to be multiplicative differentiable at x if the limit on R.H.S exists.If g is positive function and the derivative of g at x exist, then n th multiplicative derivatives of g exist and Theorem 2.3 (Multiplicative Newton-Raphson theorem) [22] Consider r be a simple root of nonlinear equation g(x) = 1 (or h(x) = g(x) − 1 = 0).According to the multiplicative analysis [19], the multiplicative Newton theorem can be expressed as follows For definite integrals, Equation (2.5) can be written using Newton Cotes quadrature of zeroth degree as In next section, we proposed the Multiplicative King's method scheme and its analysis of convergence.

The Proposed Method and Analysis of Convergence
The proposed King's iterative method in multiplicative derivative reprsented as Where q is iteration step, g * (x) is multiplicative derivative, and β is a free parameter.
For convergence analysis, we have proved the following theorem.
1 For an open interval I, let r ∈ I be a multiplicative zero of a sufficiently multiplicative differential function g : I ⊆ R → R + , then multiplicative King's method has fourth order of convergence with error Proof Let r be a simple root of equation g(x) = 1 and e q = x q −r be error at q th iteration.Using the multiplicative Taylor expansions (2.4) for function g(x), it can be written as If we take the natural logarithm on both sides, we get ln g(x q ) = ln g(r) + ln g * (r)e q + ln(g * ) 2 (r) where b j = 1 j! ln(g * ) j (r) ln g * (r) .
Now the first step of scheme (3.1) is obtained by using equation (3.5) By using the multiplicative Taylor expansion upon g(y q ) about r, we obtain g(y q ) = g(r)(g * (r)) eq ((g * ) 2 (r)) 2! ((g * ) 3 (r)) As a result of taking the natural logarithm from both sides, we get ln g(y q ) = ln g * (r)(e q + b 2 e 2 q + b 3 e 3 q + O(e 4 q )).
Numerical results indicate in the Tables 1-5 that the proposed method execute less number of iterations and reduce the computational time.
Remark: The meaning of expression m(±n) is m × 10 ±n and d represents that scheme is divergent in all the tables.In this model we evaluate the birth rate denoted as x, if in a specific local area has 1000 thousand people at first and 435 thousand move into the local area in the first year.Likewise, assume 1564 thousand individuals toward the finish of one year.The computed results towards the root x r = 0.1009979 . . .are displayed in Table 1.Clearly, the proposed methods M KM 1 , M KM 2 , M KM 3 shows better results in terms of consecutive error and number of iteration in comparison of existing ones.
Example 4.2 Next, we apply the proposed method on some of the following academic problems.In Table 2, 3 and 5 it is clearly seen that the proposed method shows more effective results as compared to others in terms of absolute error and consecutive error.In Table 4 the proposed method converges and giving the results while all other methods fails to converge.

Remark:
The Figure 1 represents the error analysis of numerical 4.1 to 4.2(d).It is clear from all sub figures of Figure 1 that proposed method error reduction is more faster than existing methods.Since, in the example 4.2(a) , (b) and (d), the methods KM 3 , and in the example 4.2(c) the methods KM 1 , KM 2 , KM 3 , CM, JM divergence so these are not shown in sub figures (a), (b), (c), and (d).In similar way, iteration comparisons of different existing methods with proposed methods is depicted in the Figure 2. Clearly, the proposed method converges to root in less number of iterations as compared with other schemes.Further, the Examples 4.2(a), (b), (c), (d) by the methods KM 3 , and KM 1 , KM 2 , KM 3 , CM, JM are not approaching to desired root so these are not shown in the Figure 2.

Basin of Attraction
The concept of basin of attraction confirms the convergence of all the possible roots of the nonlinear equation within specified rectangular region.So, here we presents the convergence of ordinary King's methods(KM 1 , KM 2 , KM 3 ), multiplicative King's methods(M KM 1 , M KM 2 , M KM 3 ), Chun method(CM ), and Jnawali method(JM ) on different initial values in the rectangular region [−2, 2] × [−2, 2] by dynamical planes explained in [25].The basin of attractions are shown in Figure 3 for the scaler equation z 3 − 1 and each image is plotted by an initial guess as an ordered pair of 256 complex points of abscissa and coordinate axis.If an initial point does not converge to the root then it is plotted with black color otherwise different colors are used to represent different roots 1, −i, i with a tolerance of 10 −3 .We observe that very less divergence area is depicted by proposed schemes M KM 1 , M KM 2 , M KM 3 .With the help of free disposable parameter β, we can obtain many new variants of fourth-order.In addition, we studied the convergence analysis of newly constructed scheme.We compare our methods with the existing techniques on the basis of absolute error difference between two consecutive iterations, order of convergence, number of iterations, CPU timing, the graphs of absolute errors and bar graphs.We found that our methods provide better approximations, which can be achieved with less computational time and complexity.Further, we also study the basin of attraction which also support the numerical results.In the future work, we will try to extend this idea for the system of nonlinear equations.In this way, this new approach of multiplicative calculus will open a new era of numerical techniques.

Example 4 . 1 − 1 )
Firstly, we consider the population growth model that formulate the following nonlinear equation g− 1.

Figure 3 :
Figure 3: Dynamical planes of new and existing methods for function z 3 − 1 = 0

Table 1 :
Results of population growth model with initial guess x 0 = 1