# Estimating the Local Radius of Convergence for Picard Iteration
## 1 Introduction  
The well known [PERSON] theorem [1] gives a sufficient condition (the spectral radius of the Jacobian of the iteration mapping in the fixed point to be less than 1) for the local convergence of Picard iteration. \"However, no estimate for the size of an attraction ball is known\" [2] (2009). The problem of estimating the local radius of convergence for different iterative methods was considered by numerous authors and several results were obtained particularly for Newton method and its variants. Nevertheless \"... effective, computable estimates for convergence radii are rarely available\" [3] (1975). A similar remark was made in a more recent paper [4] (2015): \"The location of starting approximations, from which the iterative methods converge to a solution of the equation, is a difficult problem to solve\". It is worth noticing that the shape of the attraction basins is an unpredictable and sophisticated set, especially for high order methods, and therefore finding a good ball of convergence for these methods is indeed a difficult task. Among the oldest known results on this topic we could mention those given by [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON], [PERSON] and [PERSON], [PERSON] [3; 5; 6; 7; 8; 9]. Relatively recent results were communicated by [PERSON] [10; 11; 12], [PERSON] [13], [PERSON] and [PERSON] [4], [PERSON] [14], [PERSON] [15]. [PERSON] and [PERSON] [5] proposed the following estimation for [PERSON] method. Let \(F:D\subset X\to Y\) be a nonlinear mapping, where \(X,Y\) are Banach spaces. Suppose that \(F\) is Frechet differentiable on \(D\), that \(F^{\prime}(x)\) is invertible for each \(x\in D\), and that  
\[\|F^{\prime}(x)^{-1}(F^{\prime}(x+s(y-x))-F^{\prime}(x))(y-x)\|\leq s\omega\|y -x\|^{2},\]  
for all \(x,y\in D\), \(s\in[0,1]\). Under these conditions the equation \(F(x)=0\) has a solution \(y^{*}\in D\). Let \(y^{0}\in D\) such that  
\[\overline{B}(y^{*},\|y^{0}-y^{*}\|)=\{x:\|x-y^{*}\|\leq\|y^{0}-y^{*}\|\}\subset D\]  
and \(y^{0}\in B(y^{*},2/\omega)\). Then the Newton method remains in \(B(y^{*},\|y^{0}-y^{*}\|)\) and converges to \(y^{*}\). In [2] the authors propose a simple and elegant formula to estimate the radius of convergence for Picaed iteration and the algorithm presumptively gives a sharp value. More precisely, let \(G:D\subset X\)\(\mathcal{R}^{m}\to D\) be a nonlinear mapping and \(x^{*}\) a fixed point of \(G\). Suppose that \(G\) is differentiable on some ball centred in \(x^{*}\), \(B_{r_{1}}=\{x:\|x-x^{*}\|\leq r_{1}\}\), and the derivative of \(G\) satisfies  
\[\|G^{\prime}(x^{*})\|\leq q<1,\] \[\|G^{\prime}(x)-G^{\prime}(y)\|\leq k\|x-y\|^{p},\ \forall x\in B_{r_{1}}.\]  
Define  
\[r_{2}=\left(\frac{(1+p)(1-q)}{k}\right)^{\frac{1}{p}},\]  
then \(r=\min\{r_{1},r_{2}\}\) is an estimation of local convergence radius. [PERSON] and [PERSON] (2014) gave the following algorithm for a third-order multi-point method for solving nonlinear equations \(F(x)=0\) in Banach spaces [PERSON] and [PERSON] (2016). Suppose that \(p\) is a solution of the equation, there exists \(F^{\prime}(p)^{-1}\), \(\|F^{\prime}(p)^{-1}\|\leq\beta\) and \(F^{\prime}\) is k-Lipschitz continuous on some \(B_{r_{0}}=\{x:\|x-p\|\leq r_{0}\}\). Let \(\tilde{r}=\min\{r_{0},r\}\), where \(r=\zeta_{0}/[(1+\zeta_{0})\beta k]\) and \(\zeta_{0}\) is the positive real root of a certain algebraic equation of degree three. Then \(\tilde{r}\) is a local convergence radius. A particular method of this class, which will be investigated in the present study, obtained for a particular value of a numerical parameter of the general class, is the following one point method  
\[x_{n+1}=x_{n}-F^{\prime}(x_{n})^{-1}[F(x_{n})+F(x_{n}-F^{\prime}(x_{n})^{-1}F( x_{n}))]. \tag{1}\]  
In fact this is a modified Newton method in which the derivative is re-evaluated periodically after two steps. Note that, in this particular case, the equation giving the value of \(\zeta_{0}\), is \(x^{3}+4x^{2}-8=0\). We will call (1) the One point Ezquerro-Hernandez method. **Remark 1**.