(i) Write a routine that, given a matrix Am×m, and a number n < m, computes the matrix Qn and the matrix Sn that appear

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(i) Write a routine that, given a matrix Am×m, and a number n
< m, computes the matrix Qn and the matrix Sn that appear in the
Arnoldi Method (which is the algorithm that underlines the GMRES).
Note that Sn is (n + 1) × n. Next, let I1, I2, I3 be three
intervals of length ε centered around 3, 6 and 10 respectively.
Choose randomly 80 eigenvalues from the union of the three
intervals, and create a matrix A whose spectrum are these 80
numbers (first, create a diagonal matrix D with the above
eigenvalues, and then define A = P DP −1, for some random,
well-conditioned, P .) Let b ∈ IR80 be a randomly chosen vector.
Use your routine in (i) in order to find the GMRES solution of the
system Ax = b: (ii) Use first matlab and find x directly (iii)
Apply GMRES for n = 2, 3, 8, 9, 10. For each case, check accuracy.
(iv) Turn in your code, the output, and your observations. Try to
include in your observations the speed of convergence, and to
compare the results, if possible, to the theory studied in class.
Note: there is a parameter ε here. You can play with ε and see how
its choice affects the speed of convergence. If for your given ε
there is not much to observe, choose a smaller one: Remember that
when ε = 0, you have only three e-values, and you are going to
converge after 3 iterations!