Numerical Integration

Functions for numerical integration using Trapezoidal, Gauss-Legendre, and Gauss-Kronrod methods.

Trapezoidal Quadrature: Calculates the integral of a function \(f\) using the trapezoidal rule. If the integrand is periodic and integrated over a full period, the trapezoidal rule converges faster than any power of the step size \(h\) (exponential convergence). For non-periodic twice continuously differentiable functions, the error is \(O(h^2)\). Checks for convergence by halving the interval until the tolerance is met or max_refinements is reached. Useful for periodic functions, bump functions, and bell-shaped integrals over infinite intervals.

Gauss-Legendre Quadrature: Performs "one-shot" non-adaptive integration on \((a, b)\) using a fixed number of points. Very efficient for smooth "bell-like" functions and functions with rapidly convergent power series. Does not provide an error estimate.

Gauss-Kronrod Quadrature: An adaptive extension of Gaussian quadrature. Adds \(n+1\) nodes (Kronrod points) to an \(n\)-point Gaussian quadrature to provide an a posteriori error estimate (\(O(h^{3n+1})\) vs \(O(h^{2n-1})\)). Preserves exponential convergence for smooth functions. Best suited for smooth functions with no end-point singularities.

Usage

trapezoidal(f, a, b, tol = sqrt(.Machine$double.eps), max_refinements = 12)

gauss_legendre(f, a, b, points = 7)

gauss_kronrod(
  f,
  a,
  b,
  points = 15,
  max_depth = 15,
  tol = sqrt(.Machine$double.eps)
)

Arguments

f

A function to integrate. It should accept a single numeric value and return a single numeric value.

a

The lower limit of integration.

b

The upper limit of integration.

tol

The tolerance for the approximation. For trapezoidal, default is sqrt(.Machine$double.eps). For Gauss-Kronrod, default is sqrt(.Machine$double.eps).

max_refinements

The maximum number of refinements to apply. Default is 12.

points

The number of evaluation points to use in the Gauss-Legendre or Gauss-Kronrod quadrature.

max_depth

Sets the maximum number of interval splittings for Gauss-Kronrod permitted before stopping. Set this to zero for non-adaptive quadrature.

Value

A single numeric value with the computed integral.

See also

numerical_differentiation, double_exponential_quadrature

Examples

# Trapezoidal rule integration of sin(x) from 0 to pi (Periodic over 0 to 2*pi)
trapezoidal(sin, 0, pi)
#> [1] 2

# Gauss-Legendre integration of exp(x) from 0 to 1
gauss_legendre(exp, 0, 1, points = 7)
#> [1] 1.718282

# Adaptive Gauss-Kronrod integration of log(x) from 1 to 2
gauss_kronrod(log, 1, 2, points = 15, max_depth = 10)
#> [1] 0.3862944