According to Example 3.33. the n × n Hilbert matrix Hn is positive definite, and hence we can apply the conjugate gradient method to solve the linear system Hnu = f. For the values n = 5. 10. 30, let u* ∈ Kn be the vector w ith all entries equal to 1.
(a) Compute f = Hnu*.
(b) Use Gaussian Elimination to solve Hnu = f. How close is your solution to u*?
(c) Does pivoting improve the solution in part (b)?
(d) Does the conjugate gradient algorithm do any better?