Consider the 2×2 matrix
and suppose we are required to solve Ax = b.
(a) Write down explicitly the iteration matrices corresponding to the Jacobi, Gauss-Seidel, and SOR schemes.
(b) Find the spectral radius of the Jacobi and Gauss-Seidel iteration matrices and the asymptotic rates of convergence for these two schemes.
(c) Plot a graph of the spectral radius of the SOR iteration matrix vs. the relaxation parameter ω for 0 ≤ ω ≤ 2.
(d) Find the optimal SOR parameter, ω∗. What is the spectral radius of the corresponding iteration matrix? Approximately how much faster would SOR with ω∗ converge compared to Jacobi?