Given a first order ODE $y'(x)=f(x,y)$ with the initial condition $y(x_0)=x_0$ such that it satisfies Picard thoerem of existence and uniquness, one can compute the solution by Picard iterations :$$ y_0 (x) \equiv y_0 \, ,$$ and$$ y_{n+1}(x) = y_0 + \int\limits_{x_0}^x f(t,y_n (t)) \, dt \, .$$Computationally speaking, how fast is this method? Do we have a theorem about its convergence rate? Do we have particulary bad examples for its convergence?Generally speaking, I know this method is not the usual go-to for numerical solutions of ODE's, but why?
Convergence rate of Picard iterations
numerical analysis;ode;iterative method;convergence;fixed point