Analysis Examples

Analysis routines provide scalar quadrature and root finding. They are intended for small numerical subproblems inside larger solvers and experiments.

Composite Quadrature

#include <numerics.hpp>
 
auto f = [](double x) {
    return std::exp(-x * x);
};
 
double q1 = num::trapz(f, 0.0, 1.0, 4096);
double q2 = num::simpson(f, 0.0, 1.0, 4096);
double q3 = num::romberg(f, 0.0, 1.0, 1e-12);

Backend::omp parallelizes the panel sums in trapz and simpson.

double q = num::simpson(f, 0.0, 1.0, 1 << 20, num::Backend::omp);

Gauss-Legendre Rule

auto p = [](double x) {
    return x * x * x * x;
};
 
double exact_for_degree_4 = num::gauss_legendre(p, -1.0, 1.0, 3);

With p points, the rule is exact for polynomials through degree (2p-1).

Bracketed Roots

auto g = [](double x) {
    return std::cos(x) - x;
};
 
num::RootResult r = num::brent(g, 0.0, 1.0, 1e-12);

Use Brent's method when a sign-changing bracket is available.

Newton Iteration

auto f0 = [](double x) { return x * x - 2.0; };
auto df = [](double x) { return 2.0 * x; };
 
num::RootResult r = num::newton(f0, df, 1.0, 1e-12);

Newton iteration is appropriate when the derivative is available and the starting value is in the local basin of attraction.