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2. (Hilbert Matrix, 10 points) As discussed in class and Chapter 12 of AFCNM, the Hilbert matrix arises from computing the best least squares polynomial approximant to a function f using the monomial basis. The order n Hilbert matrix is given as Hn=121n12131n+113141n+21n1n+112n11. Compute the max-norm condition numbers of Hn for n=5,6,,12 and compare these to the estimate cond (Hn)=O((1+2)4n). Report your results in a nice table. The Hilbert matrix can be constructed in MATLAB using the hilb function. A closed-form expression for the entries of the inverse of the Hilbert matrix can be worked out using a more general result on Cauchy Matrices. The entries in the inverse are given by (Hn1)i,j=(1)i+j(i+j1)(n+i1nj)(n+j1ni)(i+j2i1)2. The condition number estimate quoted above can be obtained from this result. 2. (Hilbert Matrix, 10 points) As discussed in class and Chapter 12 of AFCNM, the Hilbert matrix arises from computing the best least squares polynomial approximant to a function f using the monomial basis. The order n Hilbert matrix is given as Hn=121n12131n+113141n+21n1n+112n11. Compute the max-norm condition numbers of Hn for n=5,6,,12 and compare these to the estimate cond (Hn)=O((1+2)4n). Report your results in a nice table. The Hilbert matrix can be constructed in MATLAB using the hilb function. A closed-form expression for the entries of the inverse of the Hilbert matrix can be worked out using a more general result on Cauchy Matrices. The entries in the inverse are given by (Hn1)i,j=(1)i+j(i+j1)(n+i1nj)(n+j1ni)(i+j2i1)2. The condition number estimate quoted above can be obtained from this result

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