# Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations
## 1 Introduction  
Let \(T_{1}\), \(T_{2}\) stand for Banach spaces; \(\Omega\subset T_{1}\) be an open and convex set; and \(G:\Omega\longrightarrow T_{2}\) denote a differentiable operator according to Frechet [1; 2; 3; 4]. A plethora of applications from diverse areas of research such as Optimization and Computational Sciences are reduced using mathematical modeling [5; 6; 7; 8; 9; 10; 11; 12; 13; 14] to locate a solution \(x^{*}\) of a nonlinear equation like  
\[G(x)=0 \tag{1}\]  
The closed version of a solution \(x^{*}\in\Omega\) is possible only in special cases. Consequently, most solution approaches utilized by researchers and practitioners are iterative when the sequence is generated approximating the solution \(x^{*}\). The algorithm of successive substitutions or the algorithm of iteration or the Picard algorithm is a simple and important algorithm for solving linear as well as nonlinear equations. This algorithm originated in antiquity, appearing in the writings of [PERSON] of Alexandria in the second century B.C. in relation to root extraction. Later, [PERSON], as well as [PERSON], employed this algorithm to assure the existence of solutions of differential equations. It is defined for \(P:T_{1}\longrightarrow T_{1}\) and each \(n=0,1,2,\ldots\) by\[z_{0}\in\Omega,\ z_{n+1}=P(z_{n}).\]  
[PERSON] inaugurated the abstract formulation of this algorithm followed by [PERSON] and [PERSON]. We refer the readers to the reference by [PERSON] [15], [PERSON] [16], and [PERSON] et al. [3] for further information on the convergence conditions (see also [17; 18; 19; 20; 21; 22; 23]). Concerning the convergence order of this algorithm, it is only linear. Thus, there is a need for introducing algorithms of convergence order higher than one. [PERSON]'s algorithm is without a doubt the most well-known algorithm of convergence order two for solving transcendental as well as scalar equations. It is written for each \(n=0,1,2,\ldots\) as  
\[x_{0}\in\Omega,\ x_{n+1}=x_{n}-G^{\prime}(x_{n})^{-1}G(x_{n}),\]  
where \(G^{\prime}\) is the notation for the derivative of the operator \(G\) according to [PERSON] [8; 24; 25]. The construction of [PERSON]'s algorithm is based on linearization. Let \(x_{0}\in\Omega\) be an initial point. If the operator \(G\) is Frechet-differentiable, one can write the Ostrowski [21] representation  
\[G^{\prime}(x)=G(x_{0})+G^{\prime}(x_{0})(x-x_{0})+d(x,x_{0}),\]  
where \(\|d(x,x_{0})\|=O(\|x-x_{0}\|)\) as \(\|x-x_{0}\|\longrightarrow 0\). If \(x^{*}\in\Omega\) is a solution of the Equation (1), it follows by the preceding representation that  
\[G(x_{0})+G^{\prime}(x_{0})(x^{*}-x_{0})=-d(x^{*},x_{0}).\]  
If \(x^{*}\) is near to \(x_{0}\), then one can neglect the supposedly small quantity \(d(x^{*},x_{0})\), leading to a linear equation  
\[G(x_{0})+G^{\prime}(x_{0})(x-x_{0})=0.\]  
It is said that this equation is obtained from Equation (1) by the technique of linearization or the tangent algorithm since this is the equation of the tangent line to the curve \(y=G(x)\) at the point \((x_{0},G(x_{0}))\) provided that \(G\) is a real function. It follows by this linearization that \(x_{1}\) is a unique solution if \(G^{\prime}(x_{0})^{-1}\) exists. In this case, we can write  
\[x_{1}=x_{0}-G^{\prime}(x_{0})^{-1}G(x_{0}).\]  
If \(x_{0}\) is close to \(x^{*}\), then maybe \(x_{1}\) is even closer. Thus, this process can be repeated with \(x_{1}\) replacing \(x_{0}\) leading to [PERSON]'s algorithm. A main drawback with the implementation of [PERSON]'s algorithm is the inversion of the linear operator \(G^{\prime}(x_{n})\) at each step of the iteration. We address this issue in this paper. Our methodology applies to all single-step, multistep methods, and quasi-Newton [5; 10; 20; 26; 27] using inverses along the same lines. We shall demonstrate our methodology to a large class of algorithms involving inverses including [PERSON]'s algorithm as a special case.