# Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations
## 1 Introduction
Let us consider the popular Newton-like algorithm defined for \(y_{0}\in\Omega\) and each \(n=0,1,2,...\) by  
\[y_{0}\in\Omega,\ \ y_{n+1}=y_{n}-L_{n}^{-1}G(y_{n}) \tag{2}\]  
Let \(L:\Omega\longrightarrow\mathcal{L}(T_{1},T_{2})\), the space of bounded linear operators mapping the space \(T_{1}\) into \(T_{2}\). Notice that for \(L_{n}=G^{\prime}(y_{0})\) or \(L_{n}=G^{\prime}(y_{n})\), we obtain the modified and [PERSON]'s algorithm, respectively. There are two types of convergence usually studied for iterative algorithms: the semi-local and the local analysis. The former uses the condition on the initial guess \(y_{0}\), and the operator \(F\) and the solution \(y^{*}\) are found in a neighborhood of \(x_{0}\). The latter differs from the former since the convergence conditions depend on \(y^{*}\) and demonstrate how difficult is to choose the initial guess \(y_{0}\). The main challenge of local analysis is that \(y^{*}\) is usually unknown. Numerous papers have been presented dealing with the semi-local as well as the local analysis of convergence for the Newton-like algorithm (2) [2; 24; 25; 26; 28]. The convergence conditions involve Lipchitz-Holder or generalized continuity conditions utilized to control the Frechet derivative \(G^{\prime}\) of the operator \(F\). Byin (2), we mean \(L_{n}=L(y_{n})\). The inversion of the linear operator at each step is computationally expensive or impossible in general. To essentially utilize the algorithm but without the inverse, we replace it with a finite sum of linear operators related to \(L\) as follows:  
Suppose:  
There exists \(\Delta\in\mathcal{L}(T_{1},T_{2})\) such that \(\Delta^{-1}\in\mathcal{L}(T_{2},T_{1})\) and \((I-\Delta^{-1}(\Delta-L(y))^{-1}\in\mathcal{L}(T_{2},T_{1})\). Then, the Newton-like algorithm (2) can be rewritten as  
\[y_{0}\in\Omega,\] \[y_{n+1} = y_{n}-[I-\Delta^{-1}(\Delta-L_{n})]^{-1}\Delta^{-1}G(y_{n}), \tag{3}\]  
where \(I\) denotes the identity operator. The iterates of algorithms (2) and (3) coincide, since  
\[z_{0}\in\Omega,\ z_{n+1}=P(z_{n}).\]0 \[z_{0}\in\Omega,\ z_{n+1}=P(z_{n}).\]1  
Even if we replace algorithm (2) with algorithm (3), we still need to invert the linear operator \((I-\Delta^{-1}(\Delta-L_{n}))\) at each step of the iteration. But we can avoid this inversion if we introduce for \(k\) a fixed natural number the operators \(\Gamma=\Gamma(x)=\Delta^{-1}(\Delta-L(x))\) and  
\[z_{0}\in\Omega,\ z_{n+1}=P(z_{n}).\]2  
Then, consider the replacement of algorithm (2) defined for \(x_{0}=y_{0}\in\Omega\) by  
\[z_{0}\in\Omega,\ z_{n+1}=P(z_{n}).\]3  
Algorithm (4) requires the inversion of the inversion of only the frozen linear operator \(\Delta\) at each step. Notice also that \(A\) is a linear operator. By letting \(k\longrightarrow+\infty\), and if \(\lim\limits_{k\rightarrow+\infty}A_{k}=L\) exists, then  
\[z_{0}\in\Omega,\ z_{n+1}=P(z_{n}).\]4  
The condition  
\[z_{0}\in\Omega,\ z_{n+1}=P(z_{n}).\]5  
guarantees the existence of this limit [5; 8; 16]. A possible choice for \(\Delta=I\). If \(T_{1}=T_{2}=\mathbb{R}^{i}\), \(i\) a natural number, and \(H\) denotes the Hessian of the operator \(G\), then we can choose \(\Delta=H(x_{0})\)(semi-local case) or \(\Delta=H(x^{*})\)(local case). The choice \(\Delta=H(\bar{x})\) has been considered in [29], where \(\bar{x}\in\Omega\) is an auxiliary point. In the more general setting of Banach space \(\Delta\in L(T_{1},T_{2})\) which is the space of bounded linear operator form \(T_{1}\) into \(T_{2}\). As a further example, if \(L_{n}=G^{\prime}(x_{n})\), then \(\Delta=G^{\prime}(x_{n})\) (semi-local case) or \(\Delta=G^{\prime}(x^{*})\) (local case) or \(\Delta=G^{\prime}(\bar{x})\). Other choices for \(\Delta\) are possible as long as they satisfy the convergence conditions (\(H_{4}\)) and (\(H_{5}\)) (semi-local case) and (\(C_{1}\)) and (\(C_{2}\)) (local case) (see Section 2 and Section 3, respectively).