Condiționarea matricei Vandermonde

Matricea Vandermonde , unde t este un vector fixat, de lungime n, are elementele , .
Considerăm două cazuri
Cazul 1. Elemente echidistate în [-1,1], . În acest caz avem estimarea teoretică
warning off
fprintf(' n cond_inf cond.estimate theoretical\n')
n cond_inf cond.estimate theoretical
for n=[10,20,40,80]
t=linspace(-1,1,n);
V=vander(t);
et=1/pi*exp(-pi/4)*exp(n*(pi/4+1/2*log(2)));
x=[n, norm(V,inf)*norm(inv(V),inf), condest(V), et];
fprintf('%3d %e %e %e\n',x)
end
10 2.056171e+04 1.362524e+04 1.196319e+04 20 1.751063e+09 1.053490e+09 9.861382e+08 40 1.208386e+19 6.926936e+18 6.700689e+18 80 2.665120e+38 1.894750e+38 3.093734e+38
warning on
Cazul 2. :
warning off
fprintf(' n cond_inf cond.estimate theoretical\n')
n cond_inf cond.estimate theoretical
for n=10:15
t=1./(1:n);
V=vander(t);
x=[n, norm(V,inf)*norm(inv(V),inf), condest(V), n^(n+1)];
fprintf('%3d %e %e %e\n',x)
end
10 5.792417e+11 5.905580e+11 1.000000e+11 11 2.382382e+13 2.278265e+13 3.138428e+12 12 1.060780e+15 9.692982e+14 1.069932e+14 13 5.087470e+16 4.732000e+16 3.937376e+15 14 2.615990e+18 2.419007e+18 1.555681e+17 15 1.436206e+20 1.294190e+20 6.568408e+18
warning on