# Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations
## 5 Error Analysis  
The sequences \(\{y_{n}\}\) and \(\{x_{n}\}\) are generated by formulas (2) and (4), respectively. We select a portion of the standard semi-local convergence result for the Newton-like algorithm (2) [28]. **Theorem 7**. : _Let \(G:\Omega\longrightarrow Y\) be Frechet-differentiable and let \(L(x)\in L(T_{1},T_{2})\) be an approximation to the linear operator \(G^{\prime}(x)\). Suppose that there exist an open convex subset \(\hat{\Omega}\) of \(\Omega\), \(x_{0}\in\hat{\Omega}\), a bounded linear invertible operator \(K_{0}(=L(x_{0}))\), and constants \(\eta,K>0,K_{0},K_{1},\mu,l\geq 0\) such that for all \(x,y\in\hat{\Omega}\) the following conditions hold:_  
\[\|L_{0}^{-1}G(x_{0})\| \leq \eta,\] \[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\] \[\|K_{0}(F^{\prime}(x)-K_{1}(x)) \leq K_{0}\|x-x_{0}\|+\mu,\] \[\|K_{0}(A(x)-K_{0})\| \leq K_{1}\|x-x_{0}\|+l,\] \[l_{1} = \mu+l<1.\]  
_In addition, suppose that_  
\[h:=\sigma\eta\leq\frac{1}{2}(1-l_{1})^{2},\]_where \(\sigma=max\{K,K_{0}+K_{1}\}\), and_  
\[\bar{U}=U[x_{0},t^{*}]\subset\bar{\Omega}\]  
_where_  
\[t^{*}=(1-l_{1}-\sqrt{(1-l_{1})^{2}-2h)})/\sigma_{0}\]  
_Then, the following assertions hold_  
_The sequence \(\{y_{n}\}\) generated by algorithm (2) remains in the ball \(U(x_{0},t^{*})\) and converges to a solution \(x^{*}\in\bar{U}\) of the equation \(G(x)=0\) and_  
\[\|L_{n}^{-1}K_{0}\|\leq s_{1}:=\frac{1}{1-(K_{0}t^{*}+l)} \tag{26}\]  
_Next, the sequences \(\{y_{n}\}\) and \(\{x_{n}\}\) are related to each other._  
**Lemma 1**. : _Suppose that the conditions \((H_{1})-(H_{5})\) for \((\Delta=K_{0})\), and those of Theorem 7 hold. Then, the following error assertion holds for each \(n=0,1,2,...\)_  
\[\|x_{n+1}-y_{n+1}\|\leq e_{n}, \tag{27}\]  
_where_  
\[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]0 \[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]1  
Proof. : Under the conditions of Theorems 2 and 7, the iterates \(\{y_{n}\}\) and \(\{x_{n}\}\) are well defined by formulas (2), and (4), respectively. By subtracting (2) from (4) and pulling out \(L_{n}^{-1}\), we can write, in turn, that  
\[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]2 \[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]3 \[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]4 \[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]5 \[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]6  
We need, in turn, the following estimates obtained by the conditions of Theorems 2 and 7  
\[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]7 \[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]8 \[\|K_{0}^{-1}(F^{\prime}(y)-F^{\prime}(x))\| \leq K\|y-x\|,\]9  
\[\|K_{0}(F^{\prime}(x)-K_{1}(x)) \leq K_{0}\|x-x_{0}\|+\mu,\]0 \[\|K_{0}(F^{\prime}(x)-K_{1}(x)) \leq K_{0}\|x-x_{0}\|+\mu,\]1 (32) \[\|K_{0}(F^{\prime}(x)-K_{1}(x)) \leq K_{0}\|x-x_{0}\|+\mu,\]2 \[\|K_{0}(F^{\prime}(x)-K_{1}(x)) \leq K_{0}\|x-x_{0}\|+\mu,\]3  
and  
\[\|K_{0}(F^{\prime}(x)-K_{1}(x)) \leq K_{0}\|x-x_{0}\|+\mu,\]4  
By summing up (30)-(34) and using the triangle inequality in (34), we obtain (27). It is convenient for the next result to define the function \(\varphi:[0,+\infty)\longrightarrow R\) by  
\[\|K_{0}(F^{\prime}(x)-K_{1}(x)) \leq K_{0}\|x-x_{0}\|+\mu,\]5  
Proposition 3: _Let all the conditions of Lemma 1 hold. Suppose, in addition, that the equation \(\psi(t)=0\) has the smallest solution \(s^{*}>s_{2}\). Then, the following assertion holds for each \(n=0,1,2,\dots\)_  
\[\|K_{0}(F^{\prime}(x)-K_{1}(x)) \leq K_{0}\|x-x_{0}\|+\mu,\]6  
Proof: The estimate (27) for \(n=0\) implies \(\|x_{1}-y_{1}\|\leq s_{2}\leq s^{*}\), which is true by the choice of \(s^{*}\). Suppose that (35) holds for all integer values \(m\) smaller or equal to \(m\). Then, we have by the induction hypothesis, and (2) that  
\[\|K_{0}(F^{\prime}(x)-K_{1}(x)) \leq K_{0}\|x-x_{0}\|+\mu,\]7  
by the definition of \(s^{*}\).