# Extended High Order Algorithms for Equations under the Same Set of Conditions
## 1 Introduction  
We consider two Banach spaces \(\mathcal{Y}_{1}\) and \(\mathcal{Y}_{2}\) with an open and convex subset \(\mathcal{X}(\
eq\emptyset)\) of \(\mathcal{Y}_{1}\). Let us denote the set \(\{B_{L}:\mathcal{Y}_{1}\rightarrow\mathcal{Y}_{2}\text{ linear and bounded operators}\}\) by \(L(\mathcal{Y}_{1},\mathcal{Y}_{2})\). Suppose \(\mathcal{X}:\mathcal{Z}\subseteq\mathcal{Y}_{1}\rightarrow\mathcal{Y}_{2}\) is Frechet derivable. Equations of the kind  
\[\mathcal{X}(v)=0 \tag{1}\]  
are often utilized in science and other applied areas to solve several highly challenging problems. We should not ignore the fact that solving these equations is a difficult process, as the solution could only be discovered analytically on rare instances. This is why iterative processes are generally used for solving these equations. However, it is an arduous task to develop an effective iterative approach for addressing (1). The classical [PERSON]'s iterative strategy is most typically employed for this issue. In addition, a lot of studies on higher order modifications of conventional processes like [PERSON]'s, [PERSON]'s, [PERSON]'s, etc. are presented in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. [PERSON] et al. [16] presented a sixth order variant of [PERSON]'s algorithm, which adds the evaluation of the function at an additional point in the iteration procedure of [PERSON]'s method [12]. [PERSON] and [PERSON] [10] by applying Obreshkov-like techniques described two families of zero-finding iterative approaches. An efficient family of nonlinear system algorithms is suggested by [PERSON] et al. [8] using a reduced composition technique on [PERSON]'s and [PERSON]'s algorithm. [PERSON] et al. [17] composed two weighted-Newton steps to construct an efficient fourth order weighted-Newton method tosolve nonlinear systems. [PERSON] and [PERSON] [18] constructed iterative algorithms of fourth and sixth convergence order for solving nonlinear systems. Two bi-parametric fourth order families of predictor-corrector iterative algorithms are discussed in [9]. Newton-like iterative approaches of fifth and eighth rate of convergence are also designed by [PERSON] and [PERSON] [19]. Additional studies on other algorithms with their convergence and dynamics are available in [20; 21; 22; 23; 24; 25]. Notice that higher convergence order algorithms using Taylor expansions suffer from the following problems:  
1. Higher order derivatives (not on the algorithms) should exist although convergence may be possible without these conditions. 2. We do not know in advance how many iterations should be performed to reach a certain error tolerance. 3. The choice of initial points is limited, since we do not know a convergence ball. 4. No information is provided on the uniqueness of \(v_{*}\). 5. Results are limited on the multidimensional Euclidean space. Hence, there is a need to address these problems. The novelty of our article lies in the fact that we handle (1)-(5) as follows. 1. We only use the derivative that actually appears on these algorithms. The convergence order is recovered again, since we by pass Taylor series, (which require the higher order derivatives) and use instead the computational order of convergence (COC) given by \[COC=\frac{\ln\left(\frac{||v_{n+1}-v_{*}||}{||v_{n}-v_{*}||}\right)}{\ln\left( \frac{||v_{n}-v_{*}||}{||v_{n-1}-v_{*}||}\right)}\] and the approximate computational order of convergence (ACOC) given by \[ACOC=\frac{\ln\left(\frac{||v_{n+1}-v_{n}||}{||v_{n}-v_{n-1}||}\right)}{\ln \left(\frac{||v_{n}-v_{*}||}{||v_{n-1}-v_{n-2}||}\right)}.\] These formulae use the algorithms (which depend on the first derivative). In the case of ACOC no knowledge of \(v_{*}\) is needed. 2. We use generalized Lipschitz-type conditions which allow us to provide upper bounds on \(||v_{n}-v_{*}||\) which in turn can be used to determine the smallest number of iterations to reach the error tolerance. 3. Under our local convergence analysis a convergence ball is determined.