The n n Hilbert matrix H(n) defined by Is an ill-conditioned matrix that arises in solving the normal equations for the coefficients of the

The n × n Hilbert matrix H(n) defined by

Is an ill-conditioned matrix that arises in solving the normal equations for the coefficients of the least-squares polynomial
a. Show that

And compute Kˆž(H(4)).
b. Show that

And compute Kˆž(H(5)).
c. Solve the linear system

Using five-digit rounding arithmetic, and compare the actual error to that estimated in (7.25).

s n. +i-1 16 120 240140 1200 [H(4)1-120 2402700 6480-4200 140 1680-4200 2800 10501400 630 300 4800 -18900 26880 -12600 [H(51-I = | 1050-18900 79380-117600 56700 -1400 26880-117600 179200-88200 630-12600 5670088200 44100 25 300 1001