Math 6610 • Analysis of Numerical Methods I

{Iterative Methods:}

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

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

\partitem{b} Consider the boundary value problem
\[
  -u''(x)=f(x),\qquad \text{for }0\leq x\leq1
\]
with \(u(0)=u(1)=0\), and the following discretization of it:
\[
  -U_{j-1}+2\,U_j-U_{j+1}=F_j
\]
for \(j=1,2,\ldots,N-1\) where \(Nh=1\) and \(F_j=h^2 f(jh)\).

Show that the Jacobi iterative method will converge for this problem for any
choice of initial guess. 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 \(h\)? 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?

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