A new sixth-order Jarratt-type iterative method for systems of nonlinear equations

Many real-life problems using mathematical modeling can be reduced to scalar and system of nonlinear equations. In this paper, we develop a family of three-step sixth-order method for solving nonlinear equations by employing weight functions in the second and third step of the scheme. Furthermore, we extend this family to the multidimensional case preserving the same order of convergence. Moreover, we have made numerical comparisons with the efficient methods of this domain to verify the suitability of our method.


Development of a sixth-order scheme for nonlinear scalar equation
We introduce a new sixth-order method for solving nonlinear equations.

Derivation of the scheme
For the development of our scheme, we use weight function approach. Our method is defined by the following three steps: where P : C → C and Q : C → C are weight functions that are analytic in the neighborhood of 1 and f (x n ) . Theorem 2.1 demonstrates that the order of convergence reaches at six using particular conditions on these weight functions.
Using the conditions on P and its derivatives as becomes With the help of (3) and (5), Let us consider Taylor's expansion for the weight function Q about v n = 1 up to fifth-order terms as Thus Therefore, the final step takes the form where From (8), it is clear that for the following conditions on Q and on its derivatives: our proposed scheme has the following error equation: The error equation shows that the proposed scheme (1) approaches the sixth-order of convergence.

Particular cases of weight functions
Here are some particular cases of weight functions written as Case 1, Case 2, and Case 3.

Case 2.2
If we take the weight functions P (u) and Q (v) of the following form: Then, for we get a new sixth-order scheme, called as F S1 Case 2.3 When the weight functions P (u) and Q (v) are the rational function of the following form: Then, a new sixth-order scheme is given namely as F S2 Case 2.4 Next, we consider weight functions P (u) and Q (v) of the following form: Then, another sixth-order new scheme, namely F S3, is obtained as

Numerical results
Now, we want to verify the numerical results of our new schemes that are presented in the previous section.
To demonstrate the suitability of our suggested schemes, we have considered some examples and compared the results of our schemes, namely, F S1, F S2, and F S3, with respect to the number of iterations n, absolute residual error of the corresponding function | f (x n ) |, error in two consecutive iterations |x n − x n−1 |, and computational order of convergence Example 2. 5 We choose a function from [4], which is The function has two real and four complex roots. We take the real root γ = 2 and an initial guess x 0 = 2.5.

Example 2.6
Consider the function from [11]. The desired root for the function is γ = 2.759 + 6.585i. We take an initial guess x 0 = 3 + 7.4i.

Extension of sixth-order method to the system of nonlinear equations
Now, we give an extension of our method to the system of nonlinear equations by preserving the order of convergence as in the case of scalar equations.

Derivation of the scheme
We use the weight function approach in the development of our scheme. Our method consists of three steps, which are given below. For the multidimensional case, the scheme (1) named as FS can be rewritten as for the multivariate vector-valued function F : D ⊆ C n → C n with n ∈ N Theorem 3.1 Let us suppose that F : D ⊆ C n → C n with n ∈ N be a sufficiently Frechet differentiable function in D containing simple root ϒ. In addition, that convergence is guaranteed if we consider that initial guess X (0) is close to the root ϒ. Then, the numerical scheme (14)  Proof Let us consider that E n = X (n) − ϒ be the error in the n th iteration. The Taylor's series expansion of the function F(X (n) ) and F (X (n) ) with the assumption | F (ϒ) | = 0 leads us to where for i = 2, 3, ... and Now, for the first substep Applying Taylor's series to (17), we get Also, F (Y (n) ) is given by where Next, for the Taylor's series expansion of the function U (n) = (F (Y (n) )) −1 F (X (n) ) Moreover, P(U (n) ) is given by ..C 6 , P(I ), P (I ), P (1), P (1), P iv (I )), 3 i 5.
Let us now consider the second substep as where becomes Similarly, F(Z (n) ) is given as where J i = J i (C 2 , C 3 , ...C 6 , P (I ), P iv (I ), P v (I )), 5 i 6.
It is apparent that taking the following conditions on the weight function Q: we obtain the following error equation from (21): This asymptotic error constant reveals that the proposed scheme (14) reaches at sixth-order convergence. It completes the proof. Next, we take some special cases of our proposed scheme (14), which are as follows: Case 1 When the weight functions P(U ) and Q(V ) are polynomial functions of the following form: Then, we get a sixth-order scheme, named as F S4 which is given below Case 2 If we take the weight functions P(U ) and Q(V ) of the following form: Then, we obtain the following sixth-order scheme called as F S5: Case 3 If we take weight functions P(U ) and Q(V ) of the following form: Then, we obtain the sixth-order scheme called as F S6 Table 3 Comparison of sixth-order methods for F 1 (X )

Efficiency of the methods
Consider the efficiency index [12] (EI), E I = p 1 d , where p represents the order and d represents the total number of functional evaluations. Moreover, the computational efficiency index (CE) [4] is characterized as C E = p 1 (d+op) where op is the operations cost per cycle. We have made comparisons of our scheme FS for E I and C E with the sixth-order methods given in Sect. 3, namely, B A, K C, L K , and B S.

Conclusion
We developed a new sixth-order scheme for the univariate as well as for the multidimensional case. The numerical results of our scheme compared with those of existing families of Jarratt-type methods show that our scheme performs better than the existing ones.
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