Consider the linear system Au = e1 in which A is the 8 × 8 tridiagonal matrix with all 2's on the main diagonal and all - l's on the sub- and superdiagonal.
(a) Use Exercise 8.2.47 to find the spectral radius of the Jacobi iteration method to solve A u = b. Does the Jacobi scheme converge?
(b) What is the optimal value of the SOR parameter based on (10.86)? How many Jacobi iterations are needed to match the effect of a single SOR step?
(c) Test out your conclusions by using both Jacobi and SOR to approximate the solution to 3 decimal places.