I have a question concerning Picard's iterative method. I have created an algorithm which solves (first, second order and system of DEs) initial value problems (IVPs) using the Symbolic toolbox in MATLAB.
Upon running the algorithm the iterations produce a Taylor series which approximates the exact solution of the IVP. My question is what error would I use to show that function converges closer and closer to the exact solution?
I understand that the relative error can be used, but are there any other errors I can use for the same purpose?
Thank you for the help everyone!
Assume $y^n(t)$ is the $n$th Picard iteration. If it happens also that $y^n(t)$ is the same $n$th Taylor polynomial of the exact solution, say $y(t)$, then from Taylor's theorem the error is: $$y^n(t)-y(t)=\frac{y^{(n+1)}(t_0)}{(n+1)!}(t-t_0).$$