2. Heath Computer Problem 3.12 parts (a), (b), (d), (e) (for the Hilbert matrix, run CGS, CGS2, MGS, MGS2, Householder, Normal equations-based least squares and compare the relative quality of results obtained.) 3.12. (a) Implement both the classical and mod- ified Gram-Schmidt procedures and use each to generate an orthogonal matrix Q whose columns form an orthonormal basis for the column space of the Hilbert matrix H, with entries hij = 1/(i+j- 1), for n = 2, ..., 12 (see Computer Problem 2.6). As a measure of the quality of the results (specifi- cally, the potential loss of orthogonality), plot the quantity log10 (||I - Q7Q|I), which can be in- terpreted as "digits of accuracy, for each method as a function of n. In addition, try applying the classical procedure twice (i.e., apply your classi- cal 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? (6) Repeat the previous experiment, but this time use the Householder method, that is, use the ex- plicitly computed orthogonal matrix Q resulting from Householder QR factorization of the Hilbert matrix. Note that if the routine you use for House- holder QR factorization does not form explic- itly, then you can obtain Q by multiplying the sequence of Householder transformations times a matrix that is initialized to the identity matrix I (see previous exercise). Again, plot the departure from orthogonality for this method and compare it with that of the previous methods. (c) Repeat the previous experiment, but this time use the SVD to obtain the orthonormal basis (see Section 3.6.1). (d) Yet another way to compute an orthonormal basis is to use the normal equations. If we form the cross-product matrix and compute its Cholesky 2. Heath Computer Problem 3.12 parts (a), (b), (d), (e) (for the Hilbert matrix, run CGS, CGS2, MGS, MGS2, Householder, Normal equations-based least squares and compare the relative quality of results obtained.) 3.12. (a) Implement both the classical and mod- ified Gram-Schmidt procedures and use each to generate an orthogonal matrix Q whose columns form an orthonormal basis for the column space of the Hilbert matrix H, with entries hij = 1/(i+j- 1), for n = 2, ..., 12 (see Computer Problem 2.6). As a measure of the quality of the results (specifi- cally, the potential loss of orthogonality), plot the quantity log10 (||I - Q7Q|I), which can be in- terpreted as "digits of accuracy, for each method as a function of n. In addition, try applying the classical procedure twice (i.e., apply your classi- cal 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? (6) Repeat the previous experiment, but this time use the Householder method, that is, use the ex- plicitly computed orthogonal matrix Q resulting from Householder QR factorization of the Hilbert matrix. Note that if the routine you use for House- holder QR factorization does not form explic- itly, then you can obtain Q by multiplying the sequence of Householder transformations times a matrix that is initialized to the identity matrix I (see previous exercise). Again, plot the departure from orthogonality for this method and compare it with that of the previous methods. (c) Repeat the previous experiment, but this time use the SVD to obtain the orthonormal basis (see Section 3.6.1). (d) Yet another way to compute an orthonormal basis is to use the normal equations. If we form the cross-product matrix and compute its Cholesky