Crate integrate 

A lightweight Rust library for numerical integration of real-valued functions.

integrate approximates definite integrals of the form

$$\int_a^b f(x)\, dx$$

using Newton-Cotes formulas, Gaussian quadrature, Romberg’s method, and adaptive techniques.

Methods

Newton-Cotes (newton_cotes)

Function	Description
newton_cotes::rectangle_rule	Midpoint rectangle rule
newton_cotes::trapezoidal_rule	Trapezoidal rule
newton_cotes::simpson_rule	Simpson’s 1/3 rule
newton_cotes::newton_rule	Newton’s 3/8 rule

Gaussian Quadrature (gauss_quadrature)

Function	Description
gauss_quadrature::legendre_rule	Gauss-Legendre rule
gauss_quadrature::gauss_laguerre_rule	Gauss-Laguerre rule
gauss_quadrature::gauss_hermite_rule	Gauss-Hermite rule
gauss_quadrature::gauss_first_kind_chebyshev_rule	Gauss-Chebyshev rule (first kind)
gauss_quadrature::gauss_second_kind_chebyshev_rule	Gauss-Chebyshev rule (second kind)

Adaptive Quadrature (adaptive_quadrature)

Function	Description
adaptive_quadrature::adaptive_simpson_method	Adaptive Simpson’s method

Romberg (romberg)

Function	Description
romberg::romberg_method	Romberg integration

Quick Start

use integrate::prelude::*;
let result = trapezoidal_rule(|x: f64| x.exp(), 0.0, 1.0, 1000_usize);
assert!((result - (std::f64::consts::E - 1.0)).abs() < 1e-6);

Caveats

All of the numerical integration techniques listed above assume that the integrand is continuous on the interval of integration. The error estimates generally require that the integrands are differentiable of whatever order is required so that the formula for the error estimate makes sense. Below is a checklist to verify before using any of these algorithms.

Integrand checklist

• Are there any singularities of the integrand in the interval of integration?

  • If there are, then can they be removed?

  • A function which has jump discontinuities can be integrated by splitting the interval of integration into subintervals on which the function is continuous, then numerically integrating over each subinterval and summing the results.

  • Using integration by parts, certain types of singular integrals can be expressed as the sum of a closed-form term and a non-singular integral.

  • In other cases, an approximation of the integral in a small neighborhood of the singularity can be obtained analytically, and the interval of integration can be split so that numerical integration is used on the non-singular parts.

• Does the function oscillate over the region of integration? If so, make sure that the step size is smaller than the wavelength of the function. The interval can also be split into subintervals of half a wavelength each.

Modules

adaptive_quadrature

Adaptive Simpson quadrature for numerical integration.

gauss_quadrature

Gaussian quadrature rules for numerical integration.

newton_cotes

Newton-Cotes quadrature rules for numerical integration.

prelude

Convenience re-exports of all public integration functions.

romberg

Romberg’s method for numerical integration via Richardson extrapolation.