The damped Jacobi (or under-relaxed Jacobi) method is briefly described in Sections 7.2 and 7.6. Consider the system Ax = b, and let D be the diagonal part of A.
(a) Write down the corresponding splitting in terms of D, A, and ω.
(b) Suppose A is again the same block tridiagonal matrix arising from discretization of the two-dimensional Poisson equation.
i. Find the eigenvalues of the iteration matrix of the damped Jacobi scheme.
ii. Find the values of ω > 0 for which the scheme converges.
iii. Determine whether there are values ω ≠ 1 for which the performance is better than the performance of standard Jacobi, i.e., with ω = 1.