# Extended High Order Algorithms for Equations under the Same Set of Conditions
## 2 Ball Convergence  
Our ball convergence analysis is requires the development of some scalar parameters and functions. Set \(M=[0,\infty)\). Suppose function:  
* \(\lambda_{0}(s)-1\) has a smallest root \(r_{0}\) in \(M\setminus\{0\}\) for some function \(\lambda_{0}:M\to M\) which is non-decreasing and continuous. Set \(M_{0}=[0,r_{0})\). * \(\mathcal{U}_{1}(s)-1\) has a smallest root \(\rho_{1}\) in \(M_{0}\setminus\{0\}\) for some functions \(\lambda:[0,2r_{0})\to M,\lambda_{1}:M_{0}\to M\) which are non-decreasing and continuous with \(\mathcal{U}_{1}:M_{0}\to M\) defined by \[\mathcal{U}_{1}(s)=\frac{\int_{0}^{1}\lambda((1-\theta)s)\;d\theta+|1-\alpha| \int_{0}^{1}\lambda_{1}(\theta s)\;d\theta}{1-\lambda_{0}(s)}.\]
* \(\lambda_{0}(\mathcal{U}_{1}(s)s)-1\) has a smallest root \(r_{1}\) in \(M_{0}\setminus\{0\}\). Set \(r_{2}=\min\{r_{0},r_{1}\}\) and \(M_{1}=[0,r_{2})\). * \(\mathcal{U}_{2}(s)-1\)has a smallest root \(\rho_{2}\) in \(M_{1}\setminus\{0\}\), where \[\mathcal{U}_{2}(s) =\frac{\int_{0}^{1}\lambda((1-\theta)s)\ d\theta}{1-\lambda_{0}(s)}\] \[+\frac{3}{8}\bigg{(}\frac{3(\lambda_{0}(s)+\lambda_{0}(\mathcal{U} _{1}(s)s))}{1-\lambda_{0}(\mathcal{U}_{1}(s)s)}+\frac{\lambda_{0}(s)+\lambda_{ 0}(\mathcal{U}_{1}(s)s)}{1-\lambda_{0}(s)}\bigg{)}\frac{1}{1-\lambda_{0}(s)}.\] 5. \(\lambda_{0}(\mathcal{U}_{2}(s)s)-1\) has a smallest root \(r_{3}\) in \(M_{1}\setminus\{0\}\). Set \(r=\min\{r_{2},r_{3}\}\) and \(M_{2}=[0,r)\). 6. \(\mathcal{U}_{3}(s)-1\) has a smallest root \(\rho_{3}\) in \(M_{2}\setminus\{0\}\), where \[\mathcal{U}_{3}(s) =\bigg{[}\frac{\int_{0}^{1}\lambda((1-\theta)\mathcal{U}_{2}(s)s )\ d\theta}{1-\lambda_{0}(\mathcal{U}_{2}(s)s))}\] \[+\frac{(\lambda_{0}(\mathcal{U}_{1}(s)s)+\lambda_{0}(\mathcal{U}_{2 }(s)s))\int_{0}^{1}\lambda_{1}(\theta\mathcal{U}_{2}(s)s)\ d\theta}{(1- \lambda_{0}(\mathcal{U}_{1}(s)s))(1-\lambda_{0}(\mathcal{U}_{2}(s)s))}\] \[+\frac{1}{8}\bigg{(}15\frac{(\lambda_{0}(s)+\lambda_{0}(\mathcal{U} _{1}(s)s))}{1-\lambda_{0}(\mathcal{U}_{1}(s)s)}+11\frac{(\lambda_{0}(s)+ \lambda_{0}(\mathcal{U}_{1}(s)s))}{1-\lambda_{0}(s)}+16\bigg{)}\] \[\times\frac{\int_{0}^{1}\lambda_{1}(\theta\mathcal{U}_{2}(s)s)\ d \theta}{1-\lambda_{0}(\mathcal{U}_{1}(s)s)}\bigg{]}\mathcal{U}_{2}(s).\] The scalar \(\rho\) given as \[\rho=\min\{\rho_{k}\},k=1,2,3\] (5) shall be shown to be a convergence radius for SM1. Set \(T=[0,\rho)\). It is implied by (5) that \[0\leq\lambda_{0}(s)<1,\] (6) \[0\leq\lambda_{0}(\mathcal{U}_{1}(s)s)<1,\] (7) \[\mathcal{U}_{2}(s) =\frac{\int_{0}^{1}\lambda((1-\theta)s)\ d\theta}{1-\lambda_{0}(s)}\]0 (8) and \[\mathcal{U}_{2}(s) =\frac{\int_{0}^{1}\lambda((1-\theta)s)\ d\theta}{1-\lambda_{0}(s)}\]1 (9) hold for each \(s\) in \(T\). The notation \(\overline{U}(v_{*},r)\) stands for the closure of a ball of radius \(r>0\) and center \(v_{*}\in\mathcal{U}_{1}\). We suppose from now on that \(v_{*}\) is a simple root of \(\mathcal{X}\), scalar functions \"\(\lambda\)\" are as given previously and \(\mathcal{X}:\mathcal{X}\rightarrow\mathcal{Y}_{2}\) is differentiable. Further, conditions \((C)\) hold: \((C_{1})\) For each \(v\in\mathcal{X}\) \[\mathcal{U}_{2}(s) =\frac{\int_{0}^{1}\lambda((1-\theta)s)\ d\theta}{1-\lambda_{0}(s)}\]2 Set \(\mathcal{Z}_{0}=\mathcal{Z}\cap U(v_{*},r_{0})\). \((C_{2})\) For each \(u,v\in\mathcal{Z}_{0}\) \[\mathcal{U}_{2}(s) =\frac{\int_{0}^{1}\lambda((1-\theta)s)\ d\theta}{1-\lambda_{0}(s)}\]3 and \[\mathcal{U}_{2}(s) =\frac{\int_{0}^{1}\lambda((1-\theta)s)\ d\theta}{1-\lambda_{0}(s)}\]4 \((C_{3})\)\(\overline{U}(v_{*},\tilde{\rho})\subset\mathcal{Z}\) for some \(\tilde{\rho}\) to be defined later. \((C_{4})\) There exists \(\rho_{*}\geq\tilde{\rho}\) satisfying \[\mathcal{U}_{2}(s) =\frac{\int_{0}^{1}\lambda((1-\theta)s)\ d\theta}{1-\lambda_{0}(s)}\]5Set \(\mathcal{Z}_{1}=\mathcal{Z}\cap\overline{U}(v_{*},\rho_{*})\). Next, the main convergence result for SM1 is developed utilizing conditions \((\mathbb{C})\). **Theorem 1**. : _Suppose that the conditions \((\mathbb{C})\) hold for \(\bar{\rho}=\rho\).