# Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations
## 3 Local Analysis  
In Section 3, we exchange the role of \(x^{*}\) and the \"\(w\)\" functions with \(x_{0}\), and the \"\(\psi\)\" function, respectively. But the computations are similar. Suppose:  
1. There exists a solution \(x^{*}\in\Omega\) of the equation \(G(x)=0\), and an invertible operator \(\Delta\in\mathcal{L}(T_{1},T_{2})\) such that for each \(\tau\in[0,1]\), \(x\in\Omega\) \[\|\int_{0}^{1}\Delta^{-1}(G^{\prime}(x^{*}+\tau(x-x^{*}))-\Delta)d\tau\|\] \[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\] for some (FNDC) \(\psi:S\longrightarrow S\). 2. There exists \(\gamma\in[0,\frac{1}{2})\), such that \[\|\Delta^{-1}(A(x)-\Delta)\|\leq\gamma.\] Define the function \(g:S\longrightarrow S\) by \[g(t)=\frac{1-\gamma^{k+1}}{1-\gamma}\int_{0}^{1}\psi(\tau t)d\tau+\gamma\frac {1-\gamma^{k}}{1-\gamma}.\]
3. The equation \(g(t)-1=0\) has an SPS. Denote such a solution by \(\rho_{4}\). and
4. \(U[x^{*},\rho_{4}]\subset\Omega\). Theorem 3.1: _Suppose that the conditions \((C_{1})-(C_{4})\) hold. Then, the sequence \(\{x_{n}\}\) with initial guess \(x_{0}\in U(x^{*},\rho_{4})-\{x^{*}\}\) exists in \(U(x^{*},\rho_{4})\), stays in \(U(x^{*},\rho_{4})\), and converges to \(x^{*}\) such that_  
\[\|x_{n+1}-x^{*}\| \leq g(\|x_{n}-x^{*}\|)\|x_{n}-x^{*}\|\leq\|x_{n}-x^{*}\|\leq\rho_{4}. \tag{18}\]  
Proof: Assertion (18) is shown by mathematical induction. Notice that all the iterates \(x_{m}\) exist by algorithm (4). We can also write, in turn, that  
\[x_{m+1}-x^{*} = x_{m}-x^{*}-A\Delta^{-1}G(x_{m}) \tag{19}\] \[= x_{m}-x^{*}-A\Delta^{-1}\int_{0}^{1}G^{\prime}(x^{*}+\tau(x_{m} -x^{*}))d\tau(x_{m}-x^{*})\] \[= [I-A\Delta^{-1}(\int_{0}^{1}G^{\prime}(x^{*}+\tau(x_{m}-x^{*})) -\Delta+\Delta)d\tau](x_{m}-x^{*})\] \[= [(I-A-A\Delta^{-1}(\int_{0}^{1}G^{\prime}(x^{*}+\tau(x_{m}-x^{*}) )-\Delta)d\tau](x_{m}-x^{*}).\]  
But, we have as in the semi-local case  
\[\|A\|\leq\frac{1-\gamma^{k+1}}{1-\gamma},\]  
and  
\[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\]0  
Hence, (19) turns into \[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\]1  
Using the conditions (\(C_{1}\))-(\(C_{3}\)), we obtain  
\[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\]2 \[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\]3  
Thus, assertion (18) holds and the iterate \(x_{m+1}\in U(x^{*},\rho_{4})\). Then, by (22), we obtain  
\[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\]4  
where \(d=g(\|x_{0}-x^{*}\|)\in[0,1)\). Therefore, we deduce from (23) that \(\lim_{m\to+\infty}=x^{*}\), and the iterate \(x_{m+1}\in U(x^{*},\rho_{4})\). Next, the uniqueness of the \(x^{*}\) set is determined. Proposition 2: Suppose: There exists \(\rho_{5}>0\) such that the condition (\(C_{1}\)) holds in the ball \(U(x^{*},\rho_{5})\),  
\[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\]5  
for each \(u\in\Omega\) and some (FNDC) \(\psi_{0}:S\longrightarrow S\) and there exists \(\rho_{6}\geq\rho_{5}\) such that  
\[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\]6  
Define the set \(\Omega_{2}=U[x^{*},\rho_{6}]\cap\Omega\). Then, the only solution of the equation \(G(x)=0\) in the set \(\Omega_{2}\) is \(x^{*}\). Proof: Suppose that there exists \(y\in\Omega_{3}\) solving the equation \(G(x)=0\) and \(y_{0}\
eq x^{*}\). Define the linear operator \(E_{1}=\int_{0}^{1}G^{\prime}(x^{*}+\tau(y_{0}-x^{*}))d\tau\). By the condition (\(C_{1}\)), (24), and (25), we have in turn that  
\[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\]7 \[\leq\int_{0}^{1}\psi(\tau\|x-x^{*}\|)d\tau,\]8  
Hence, \(L_{1}\) is invertible. Finally, from the identity \(y_{0}-x^{*}=E^{-1}(G(y_{0})-G(x^{*}))=E^{-1}(0)=0\), we conclude that \(y_{0}=x^{*}\). Remark 2:
1. We can set in Proposition 2 \(y_{0}=x^{*}\) and \(\rho_{5}=\rho_{4}\). 2. The parameter \(\rho_{4}\) defined in the condition (\(C_{3}\)) is the radius of convergence for algorithm (4). 3. As in the semi-local analysis if \(k\longrightarrow+\infty\), we obtain the results for algorithm (3). Another choice for \(k=n\). However, we shall choose a small value of \(k\) to save computational cost.