Math 6610 • Analysis of Numerical Methods I

{Iterative Methods for Linear Systems}
Consider the fixed-point iteration
\[
  u^{(k+1)}=Tu^{(k)}+c
\]
for finding a solution of the problem
\[
  u=Tu+c
\]
where \(T\) is an \(m\times m\) real matrix and \(c\) is a real \(m\)-vector.

\textbf{a)} Show that the fixed point iteration will converge for an arbitrary
initial guess \(u^{(0)}\) if and only if the spectral radius of \(T\), \(\rho(T)\), is
less than 1.

\textbf{b)} Consider the boundary value problem
\[
  -u''(x)=f(x),\qquad \text{for }0\le x\le1
\]
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 the Jacobi iterative method for this problem.
In particular, 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 of \(\delta\) depend on \(h\)? In practice, would you use this method to
solve the given problem? If so, explain why this is a good idea? If not, how would
you solve it in practice?