Adaptive Simpson and Romberg integration methods

[robve]

My guess is that VA uses Guass Kronrod with pre-computed tables of weights and abscissas that can give at up to 100-decimal digit precision. The highlighted part I saw somewhere in his comments, so...

Gauss Kronrod for I1 gives -0.946030174332872 with 735 evaluations and \( 10^{-15} \) precision:

auto f2 = [](double x) { return cos(x)*log(x); }
double Q = gauss_kronrod<double, 15>::integrate(f2, 0, 1, 5, 1E-15, &error)

See NR2ed Ch.4.5 and Boost Math which I've used for this example.

PS: for clarity, 100 digits refers to the internal Gauss-Kronrod points, not the integral precision.

- Rob

Edit: fixed ugly "Gauss" typo.