Conditionarea matricei lui Hilbert

Calculeaza conditionarea matricei Hilbert $cond_{2}(H_{n})$ cu ajutorul valorilor proprii. Se va utiliza formula

$$\mathrm{cond}_{2}(H_n) = \lambda_{\max}(H_n)\lambda_{\max}(H_{n}^{-1})$$

O estimare teoretica, pentru $n$ mare, datorata lui Szego, este

$$cond_{2}(H_n) = \frac{(1+\sqrt{2})^{4n+4}}{2^{15/4}\sqrt{\pi n}}$$

format short e
for n=[10:10:100]
    c = max(eig(hilb(n)))*max(eig(invhilb(n)));
    th = (sqrt(2)+1)^(4*n+4)/(2^(15/4)*sqrt(pi*n));
    fprintf('%3d, %11.4e, %11.4e\n', n,c,th)
end
 10,  1.6026e+13,  9.2192e+14
 20,  2.4522e+28,  1.3342e+30
 30,  4.2278e+43,  2.2294e+45
 40,  7.6529e+58,  3.9513e+60
 50,  1.4229e+74,  7.2330e+75
 60,  2.6913e+89,  1.3513e+91
 70, 5.1508e+104, 2.5604e+106
 80, 9.9444e+119, 4.9016e+121
 90, 1.9328e+135, 9.4578e+136
100, 3.7765e+150, 1.8363e+152