Double Exponential Quadrature

Numerical integration using double exponential quadrature methods (Tanh-Sinh, Sinh-Sinh, Exp-Sinh). These methods use variable transformations to achieve high-order convergence, often optimal for functions in the Hardy space (holomorphic in the unit disk).

Tanh-Sinh Quadrature: Best for integration over a finite interval \((a, b)\). Can handle singularities at the endpoints of the integration domain. Converges rapidly for holomorphic integrands.

Sinh-Sinh Quadrature: Designed for integration over the entire real line \((-\infty, \infty)\). Handles integrands with large features or decay properties.

Exp-Sinh Quadrature: Designed for integration over a semi-infinite interval, typically \((0, \infty)\), or ranges like \((a, \infty)\) or \((-\infty, b)\). Supports endpoint singularities.

Usage

tanh_sinh(f, a, b, tol = sqrt(.Machine$double.eps), max_refinements = 15)

sinh_sinh(f, tol = sqrt(.Machine$double.eps), max_refinements = 9)

exp_sinh(f, a, b, tol = sqrt(.Machine$double.eps), max_refinements = 9)

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. Default is sqrt(.Machine$double.eps).

max_refinements

The maximum number of refinements to apply. Default is 15 for tanh-sinh and 9 for sinh-sinh and exp-sinh.

Value

A single numeric value with the computed integral.

See also

numerical_integration

Examples

# Tanh-sinh quadrature of log(x) from 0 to 1 (Endpoint singularity)
tanh_sinh(function(x) { log(x) * log1p(-x) }, a = 0, b = 1)
#> [1] 0.3550659

# Sinh-sinh quadrature of exp(-x^2) over (-Inf, Inf)
sinh_sinh(function(x) { exp(-x * x) })
#> [1] 1.772454

# Exp-sinh quadrature of exp(-3*x) from 0 to Inf
exp_sinh(function(x) { exp(-3 * x) }, a = 0, b = Inf)
#> [1] 0.3333333