Let A be the n × n tridiagonal matrix with all its diagonal entries equal to c and all 1 's on the sub- and super-diagonals.
(a) For which values of c is A diagonally dominant?
(b) For which values of c does the Jacobi iteration for A u = b converge to the solution? What is the rate of convergence? Use Exercise 8.2.48.
(c) Set c = 2 and use the Jacobi method to solve the linear systems Ku = e1, for n = 5, 10, and 20. Starting with an initial guess of 0, how many Jacobi iterations does it take to obtain 3 decimal place accuracy? Does the convergence rate agree with what you computed in part (c)?