# Long Time Simulation Analysis of Geometry Dynamics Model under Iteration
## 3 Newton Iterative  
Traditionally, the dynamics equation which is ODE type is solved by the Runge-Kutta method. However, the error accumulation of this method may lead to the distortion of long simulations. The Lie group method and the Hamel method can avoid this problem. One reason is that modeling the dynamics of the geometry method can maintain the geometry characteristics of the system, and the other is that both of these methods solve the nonlinear equation, which is the coal character of these two methods. The distinguishing feature between the Lie group method and the Hamel method is that the Lie group method transforms the structure preserving part to be a nonlinear equation, and the Hamel method transforms the ODE to be a nonlinear equation directly. The nonlinear equation is often solved by Newtonian iteration. Thus, the error of each step can be reduced to be the lowest level and avoids error accumulation, which can guarantee the reliability of long simulations. Suppose that the nonlinear equation is as Equation (25). \[\left\{ \begin{array}{cccc}f_{1}\big{(}\begin{array}{cccc}x_{1},&x_{2},& \ldots,&x_{n}\end{array}\end{array} \right)=0\\ f_{2}\big{(} \begin{array}{cccc}x_{1},&x_{2},&\ldots,&x_{n}\end{array} \end{array} \right)=0\\ f_{n}\big{(} \begin{array}{cccc}x_{1},&x_{2},&\ldots,&x_{n}\end{array} \end{array} \right)=0 \tag{25}\]  
Using \(x=(x_{1},x_{2},\ldots,x_{n})\), \(F(x)=(f_{1}(x),f_{2}(x),\ldots,f_{n}(x))^{T}\), Equation (25) can be written as a vector as in Equation (26). \[F(x)=0 \tag{26}\]  
Suppose that \(\Delta x=x-x^{(k)}\). \(x^{(k)}\) is the initial number of iterations. Substitute it into Equation (26). Then the linear equation system is Equation (28). \[F\Big{(}x^{(k)}\Big{)}+F^{\prime}\Big{(}x^{(k)}\Big{)}\Delta x^{(k)}=0 \tag{27}\]  
Expanding Equation (28), we get the expression of \(x\). \[x=x^{(k)}-\Big{[}F^{\prime}\Big{(}x^{(k)}\Big{)}\Big{]}^{-1}F\Big{(}x^{(k)} \Big{)} \tag{28}\]  
In Equation (29), \(F^{\prime}\Big{(}x^{(k)}\Big{)}\) is the Jacobi matrix of \(F\Big{(}x^{(k)}\Big{)}\), which is written as Equation (30). \[F^{\prime}\Big{(}x^{(k)}\Big{)}=\left[ \begin{array}{cccc}\frac{\partial f_{ 1}}{\partial x_{1}}&\frac{\partial f_{1}}{\partial x_{2}}&\ldots&\frac{ \partial f_{1}}{\partial x_{n}}\\ \vdots&\vdots&\vdots&\vdots\\ \frac{\partial f_{n}}{\partial x_{1}}&\frac{\partial f_{n}}{\partial x_{2}}& \ldots&\frac{\partial f_{n}}{\partial x_{n}}\end{array} \right]_{x=x^{(k)}} \tag{29}\]  
According to continuous iteration under the given initial value, the exact solution of the system can be obtained which can satisfy the limiting condition. Then, the stop criterion of iteration is  
\[\left\|x^{(k+1)}-x^{(k)}\right\|<\varepsilon \tag{30}\]For example, Equation (18) can be discrete with Lie group or Lie algebra. Using the discrete model of a discrete Lie group variational integrator, the discrete rotation matrix can be written as Equation (32). \[R_{k+\frac{1}{2}}=\frac{R_{k+1}+R_{k}}{2} \tag{31}\]  
According to \(\Gamma_{i}=R^{T}e_{i}\), the discrete pattern on vector domain is shown in Equation (33). \[\Gamma_{ik+\frac{1}{2}}=\frac{\Gamma_{ik+1}+\Gamma_{ik}}{2} \tag{32}\]  
\(\Gamma_{ik+1}\) can be obtained by the given \(\Gamma_{ik}\). The calculation can also use the difference as the unknown quantity, and the discrete schemes can be written as  
\[\Gamma_{ik+\frac{1}{2}}=\Gamma_{ik}+\frac{\Delta\Gamma_{ik}}{2} \tag{33}\]  
Based on the given values of \(\Gamma_{ik}\) and \(\Delta\Gamma_{ik}\), the accuracy value of \(\Delta\Gamma_{ik}\) can be obtained according to the iterations, along with the accuracy value of \(\Gamma_{ik+1}\). Suppose that there is a transfer matrix between \(R_{k+1}\) and \(R_{k}\). The discrete value of the rotation matrix is  
\[R_{k+1}=R_{k}F_{k} \tag{34}\]  
Substitute Equation (35) into Equation (32) and get the expression of Equation (36).