The n n Hilbert matrix H(n) defined by H(n) i j = 1 i + j 1 , 1 i, j n, is an
Chapter 7, Problem 7.4.11(choose chapter or problem)
The n n Hilbert matrix H(n) defined by H(n) i j = 1 i + j 1 , 1 i, j n, is an ill-conditioned matrix that arises in solving the normal equations for the coefficients of the least-squares polynomial (see Example 1 of Section 8.2). a. Show that [H(4) ] 1 = 16 120 240 140 120 1200 2700 1680 240 2700 6480 4200 140 1680 4200 2800 , and compute K(H(4) ). b. Show that [H(5) ] 1 = 25 300 1050 1400 630 300 4800 18900 26880 12600 1050 18900 79380 117600 56700 1400 26880 117600 179200 88200 630 12600 56700 88200 44100 , and compute K(H(5) ). c. Solve the linear system H(4) x1 x2 x3 x4 = 1 0 0 1 using five-digit rounding arithmetic, and compare the actual error to that estimated in Eq. (7.24).
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