# Local Comparison between Two Ninth Convergence Order Algorithms for Equations
## 1 Introduction  
In this study, we consider the problem of finding a solution \(x_{*}\) of the nonlinear equation  
\[F(x)=0, \tag{1}\]  
where \(F:D\subset B_{1}\to B_{2}\) is a continuously differentiable nonlinear operator acting between the Banach spaces \(B_{1}\) and \(B_{2}\), and \(D\) stands for an open non empty convex subset of \(B_{1}\). One would like to obtain a solution \(x_{*}\) of (1) in closed form. However, this can rarely be done. So most researchers and practitioners develop iterative algorithms which converge to \(x_{*}\). It is worth noticing that a plethora of problems from diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, physics, engineering and scientific computing reduce to solving an equation like (1) [1; 2; 3; 4]. Therefore, the study of these algorithms in the general setting of a Banach space is important. At this generality we cannot use these algorithms to find solutions of multiplicity greater than one, since we assume the invertibility of \(F^{\prime}(x)\). There is an extensive literature on algorithms for solving systems [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. Our technique can be used to look at the local convergence of these algorithms along the same lines. Algorithms (2) and (3) when (i.e, when \(B_{1}=B_{2}=\mathbb{R}^{j}\)) cannot be used to solve undetermined systems in this form. However, if these derivatives are replaced by Moore-Penrose inverses (as in the case of [PERSON]'s and other algorithms [1; 3; 4]), then these modified algorithms can be used to solve undetermined systems too. A similar local convergence analysis can be carried out. However we do not pursue this task here. We cannot discuss local versus global convergence in the setting of a Banach space. However, we refer the reader to subdivision solvers that are global and guarantee to find all solutions (when \(B_{1}=B_{2}=\mathbb{R}\)) [2; 3; 5; 6; 18; 20; 24]. Then, using these ideas we can make use of Algorithms (2) and (3) we do not pursue this here. In this paper we study efficient ninth order-algorithms studied in [23], defined for \(n=1,2,\ldots,\) by  
\[y_{n} = x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n})\] \[z_{n} = x_{n}-\frac{2}{3}(A_{n}^{-1}+F^{\prime}(x_{n})^{-1})F(x_{n})\] \[v_{n} = z_{n}-\frac{1}{2}(4A_{n}^{-1}+F^{\prime}(x_{n}))^{-1})F(z_{n})\] \[x_{n+1} = v_{n}-\frac{1}{3}(A_{n}^{-1}+F^{\prime}(x_{n})^{-1})F(v_{n}) \tag{2}\]  
and  
\[y_{n} = x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n})\] \[z_{n} = x_{n}-\frac{1}{2}(F^{\prime}(y_{n})^{-1}+F^{\prime}(x_{n})^{-1}) F(x_{n})\] \[v_{n} = z_{n}-\frac{1}{2}(B_{n}+F^{\prime}(x_{n}))^{-1})F(z_{n})\] \[x_{n+1} = v_{n}-\frac{1}{2}(B_{n}+F^{\prime}(x_{n})^{-1})F(v_{n}), \tag{3}\]  
where \(A_{n}=3F^{\prime}(y_{n})-F^{\prime}(x_{n}),\)\(B_{n}=F^{\prime}(y_{n})^{-1}F^{\prime}(x_{n})F^{\prime}(y_{n})^{-1}.\)  
The analysis in [23] uses assumptions on the 10 th order derivatives of \(F.\) However, the assumptions on higher order derivatives reduce the applicability of Algorithms (2) and (3). For example: Let \(B_{1}=B_{2}=\mathbb{R},\)\(D=[-\frac{1}{2},\frac{3}{2}].\) Define \(f\) on \(D\) by  
\[f(s)=\left\{ \begin{array}{cc}s^{3}\log s^{2}+s^{5}-s^{4},&s\
eq 0\\ 0,&s=0.\end{array} \right.\]  
Then, we get \(x_{*}=1,\)  
\[f^{\prime}(s)=3s^{2}\log s^{2}+5s^{4}-4s^{3}+2s^{2},\] \[f^{\prime\prime}(s)=6s\log s^{2}+20s^{3}-12s^{2}+10s,\] \[f^{\prime\prime\prime}(s)=6\log s^{2}+60s^{2}=24s+22,\]  
and \(s_{*}=1.\) Obviously \(f^{\prime\prime\prime}(s)\) is not bounded on \(D.\) Hence, the convergence of Algorithms (2) and (3) are not guaranteed by the analysis in [23]. We are looking for a ball centered at \(x_{*}\) and of a certain radius such that if one chooses a starter \(x_{0}\) from inside this ball, then the convergence of the method to \(x_{*}\) is guaranteed. That is we are interested in the ball convergence of these methods. Moreover, we also obtain upper bounds on \(\|x_{n}-x_{*}\|,\) radius of convergence and results on the uniqueness of \(x_{*}\) not provided in [23].