Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 3 / 16B / b / Solution

Write $A=D+L+U$, with $D$ diagonal, and use only old iterates on the right side. The Jacobi method isStrict row diagonal dominance ensures every $a_{ii}\ne 0$. The exact solution satisfies the same coordinate relation. For $e^k=x^k-x^{\ast }$, subtraction givesTaking the maximum and iterating provesThis direct contraction proof requires no eigenvector assumption and applies to every strictly diagonally dominant matrix with the stated uniform $\gamma <1$.