Math 6610 • Analysis of Numerical Methods I

{Iterative Methods}
Consider the fixed-point problem \(x=Tx+c\), where \(T\) is a real \(N\times N\) matrix and
\(x\) and \(c\) are real \(N\)-vectors.

\textbf{a)} Show that the fixed point iteration
\[
  x^{(k+1)}=Tx^{(k)}+c
\]
converges for any choice of initial vector \(x^{(0)}\) if and only if the spectral
radius of \(T\), \(\rho(T)\), satisfies \(\rho(T)<1\).

\textbf{b)} Consider the linear system \(Ax=b\) where \(A\) is a strictly diagonally
dominant matrix, that is, \(a_{ii}>\sum_{j\ne i}\lvert a_{ij}\rvert\) for all \(i\), and
strict inequality holds for at least one value of \(i\). Show that the Jacobi iterative
scheme converges for this problem.