英语翻译If the method is convergent,L2 norm of the difference ve

英语翻译
If the method is convergent,L2 norm of the difference vector,Δp,and the residual vector,A(p),converge to zero [see 12].We are reporting convergence of both of
these vectors.For better understanding the error reducing property of these methods,we report variation of ‖A(pk)‖L2/‖A(p0)‖L2 and ‖Δ(pk)‖L2/‖Δ(p0)‖L2
with iterations (k).
We performed three experiments with different initialization in the algorithms (7) and (9).In the first test,let the initial vector be zero for both the algorithms.Figure 2 reports the result.Figure 2(a) presents convergence of the residual vector while the Figure 2(b) presents convergence of the difference vector.In these figures,NIM stands for Newton Iterative Method while AJNIM stands for Alterted Newton Iterative Method.These figures show that both the methods converges at the same rate (quadratically),but still the Altered Jacobian Newton Iterative Method is better in reducing the error.
In the second case,let us select initial vector whose elements 10.Figure 3 presents comparison of the two methods for an initial vector whose elements are 10.It can be seen in the Figures 3(a) and 3(b) that the Altered Jacobian
Newton Iterative Method converges faster than the Newton Iterative Method.Let us finally take an initial vector with elements equal to 100.Figure 4 presents comparison
of the two methods for an initial vector whose elements are 100.The Figures 4(a) and 4(b) show that the Newton Iterative Method is not converging while the Altered
Jacobian Newton Iterative Method still converges.The table 1 presents error after 10 iterations of the two methods.These experiments does show the independence of the convergence of the Altered Jacobian Newton Iterative Method with respect to initialization.We saw that the Newton Iterative Method converges quadratically for the first case (initial guess is zero vector) but its convergence rate decreases as we selected other initial guesses.On the other hand,for all the initial guesses the Altered Jacobian Newton Iterative Method converges quadratically.
We have developed a nonlinear algorithm named Altered Jacobian Newton Iterative Method for solving system nonlinear equations formed from discretization of nonlinear elliptic problems.Presented numerical work shows
that the Altered Jacobian Newton Iterative Method is robust with respect to the initialization.