Implement both the classical and modied Gram-Schmidt procedures in python and use each to generate an orthogonal matrix Q whose columns form an orthogonal basis for the column space of the Hilbert matrix H, with entries h ij = 1/(i + j 1), for n = 2, , 12. As a measure of the quality of the results (specically, the potential loss of orthogonality), plot the quantity log 10 (||I Q T Q|| ), which can be interpreted as digits of accuracy for each method as function of n. In addition, try applying the classical procedure twice (i.e. apply your classical Gram-Schmidt routine to its own output Q to obtain a new Q) and again plot the resulting departure from orthogonality. How do the three methods compare in speed, storage, and accuracy?

Please hand in a copy of your (commented) python code and the resulting plots.