Iterative Methods for Linear Systems Consider the boundary value problem \[ -u''(x)=f(x),\qquad \text {for }0\le x\le 1 \] with \(u(0)=u(1)=0\), and the following discretization of it: \[ -U_{j-1}+2U_j-U_{j+1}=F_j \] for \(j=1,2,\ldots ,N-1\) where \(Nh=1\) and \(F_j=h^2f(jh)\).
Analyze the convergence properties of both the Jacobi and Gauss Seidel iterative methods for this problem. In particular, for each, express the speed of convergence as a function of the discretization stepsize \(h\). How does the number of iterations required to reduce the initial error by a factor \(\delta \) depend on \(h\) for each method? In practice, would you use one of these methods to solve the given problem? If so, explain why this is a good idea? If not, how would you solve it in practice?