integral.h File Reference

Integral approximation. More...

#include "../core/constants.h"
#include "../core/function.h"
#include "../polynomial/polynomial.h"
#include "../polynomial/orthogonal.h"
#include "./gauss.h"

Go to the source code of this file.

Namespaces
namespace	theoretica
	Main namespace of the library which contains all functions and objects.

Functions
template<typename Field = real>
polynomial< Field >	theoretica::integral (const polynomial< Field > &p)
	Compute the indefinite integral of a polynomial.
real	theoretica::integral (const polynomial< real > &p, real a, real b)
	Compute the definite integral of a polynomial over an interval.
template<typename RealFunction >
real	theoretica::integral_midpoint (RealFunction f, real a, real b, unsigned int steps=CALCULUS_INTEGRAL_STEPS)
	Approximate the definite integral of an arbitrary function using the midpoint method.
template<typename RealFunction >
real	theoretica::integral_trapezoid (RealFunction f, real a, real b, unsigned int steps=CALCULUS_INTEGRAL_STEPS)
	Approximate the definite integral of an arbitrary function using the trapezoid method.
template<typename RealFunction >
real	theoretica::integral_simpson (RealFunction f, real a, real b, unsigned int steps=CALCULUS_INTEGRAL_STEPS)
	Approximate the definite integral of an arbitrary function using Simpson's method.
template<typename RealFunction >
real	theoretica::integral_romberg (RealFunction f, real a, real b, unsigned int iter=8)
	Approximate the definite integral of an arbitrary function using Romberg's method accurate to the given order.
template<typename RealFunction >
real	theoretica::integral_romberg_tol (RealFunction f, real a, real b, real tolerance=CALCULUS_INTEGRAL_TOL)
	Approximate the definite integral of an arbitrary function using Romberg's method to the given tolerance.
template<typename RealFunction >
real	theoretica::integral_gauss (RealFunction f, const std::vector< real > &x, const std::vector< real > &w)
	Use Gaussian quadrature using the given points and weights.
template<typename RealFunction >
real	theoretica::integral_gauss (RealFunction f, real *x, real *w, unsigned int n)
	Use Gaussian quadrature using the given points and weights.
template<typename RealFunction >
real	theoretica::integral_gauss (RealFunction f, real *x, real *w, unsigned int n, real_function Winv)
	Use Gaussian quadrature using the given points and weights.
template<typename RealFunction >
real	theoretica::integral_legendre (RealFunction f, real a, real b, real *x, real *w, unsigned int n)
	Use Gauss-Legendre quadrature of arbitrary degree to approximate a definite integral providing the roots of the n degree Legendre polynomial.
template<typename RealFunction >
real	theoretica::integral_legendre (RealFunction f, real a, real b, const std::vector< real > &x, const std::vector< real > &w)
	Use Gauss-Legendre quadrature of arbitrary degree to approximate a definite integral providing the roots of the n degree Legendre polynomial.
template<typename RealFunction >
real	theoretica::integral_legendre (RealFunction f, real a, real b, const std::vector< real > &x)
	Use Gauss-Legendre quadrature of arbitrary degree to approximate a definite integral providing the roots of the n degree Legendre polynomial.
template<typename RealFunction >
real	theoretica::integral_legendre (RealFunction f, real a, real b, unsigned int n=16)
	Use Gauss-Legendre quadrature of degree 2, 4, 8 or 16, using pre-computed values, to approximate an integral over [a, b].
template<typename RealFunction >
real	theoretica::integral_laguerre (RealFunction f, const std::vector< real > &x)
	Use Gauss-Laguerre quadrature of arbitrary degree to approximate an integral over [0, +inf) providing the roots of the n degree Legendre polynomial.
template<typename RealFunction >
real	theoretica::integral_laguerre (RealFunction f, real a, real b, const std::vector< real > &x)
	Use Gauss-Laguerre quadrature of arbitrary degree to approximate an integral over [a, b] providing the roots of the n degree Legendre polynomial.
template<typename RealFunction >
real	theoretica::integral_laguerre (RealFunction f, unsigned int n=16)
	Use Gauss-Laguerre quadrature of degree 2, 4, 8 or 16, using pre-computed values, to approximate an integral over [0, +inf).
template<typename RealFunction >
real	theoretica::integral_hermite (RealFunction f, const std::vector< real > &x)
	Use Gauss-Hermite quadrature of arbitrary degree to approximate an integral over (-inf, +inf) providing the roots of the n degree Hermite polynomial.
template<typename RealFunction >
real	theoretica::integral_hermite (RealFunction f, unsigned int n=16)
	Use Gauss-Hermite quadrature of degree 2, 4, 8 or 16, using pre-computed values, to approximate an integral over (-inf, +inf).
real	theoretica::integral_inf_riemann (real_function f, real a, real step_sz=1, real tol=CALCULUS_INTEGRAL_TOL, unsigned int max_iter=100)
	Integrate a function from a point up to infinity by integrating it by steps, stopping execution when the variation of the integral is small enough or the number of steps reaches a maximum value.
template<typename RealFunction >
real	theoretica::integral (RealFunction f, real a, real b)
	Use the best available algorithm to approximate the definite integral of a real function.
template<typename RealFunction >
real	theoretica::integral (RealFunction f, real a, real b, real tol)
	Use the best available algorithm to approximate the definite integral of a real function to a given tolerance.

Detailed Description

Integral approximation.