Math 6610 • Analysis of Numerical Methods I

{Iterative Methods for Linear Systems} Consider the square linear
system \(Ax=b\) and the Gauss-Jacobi Method
\[
  x_i^{[k+1]}=
  \frac{1}{a_{ii}}\left(
    b_i-\sum_{j=1}^{i-1}a_{ij}x_j^{[k]}
      -\sum_{j=i+1}^{n}a_{ij}x_j^{[k]}
  \right)
\]
where \(i=1,\ldots,n\) and \(k=0,1,\ldots\). Show that this method converges for all
starting vectors \(x^{[0]}\) if \(A\) is diagonally dominant, i.e.,
\[
  \sum_{j=1}^{k}\lvert a_{ij}\rvert+
  \sum_{j=k+1}^{n}\lvert a_{ij}\rvert
  < \lvert a_{ii}\rvert
  \qquad\text{for all } i=1,2,\ldots,n.
\]