One-dimensional numerical integration using the double exponential formula
Author machakann
6 Stars
Updated Last
1 Year Ago
Started In
December 2019


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This package provides functions for one-dimensional numerical integration using the double exponential formula [1,2] also known as the tanh-sinh quadrature and its variants.


Press ] on a Julia REPL to enter the Pkg mode and run the following command.



Handy interface with Float64 precision

    I, E = quadde(f::Function, a::Real, b::Real, c::Real...;
                  rtol::Real=atol>0 ? zero(Float64) : sqrt(eps(Float64)))

The quadde function provides a handy way to integrate a function f(x) over an arbitrary interval.

using DoubleExponentialFormulas
using LinearAlgebra: norm

f(x) = 1/(1 + x^2)
I, E = quadde(f, -1, 1)

I  π/2    # true
E  sqrt(eps(Float64))*norm(I)  # true

The above example computes ∫ 1/(1+x^2) dx in [-1, 1]. The I is the obtained integral value and the E is an estimated numerical error. The E is not exactly equal to the difference from the true value. However, one can expect that the integral value I is converged if E <= max(atol, rtol*norm(I)) is true. Otherwise, the obtained I would be unreliable; the number of repetitions exceeds the maxlevel before converged.

Half-infinite intervals and the infinite interval are also valid, as far as the integral is convergent.

# Computes ∫ 1/(1+x^2) dx in [0, ∞)
I, E = quadde(x -> 1/(1 + x^2), 0, Inf)
I  π/2    # true

# Computes ∫ 1/(1+x^2) dx in (-∞, ∞)
I, E = quadde(x -> 1/(1 + x^2), -Inf, Inf)
I  π      # true

Optionally, one can divide the integral interval [a, b, c, ...], which returns ∫f(x)dx in [a, b] + ∫f(x)dx in [b, c] + ⋯. It is worth noting that discontinuity or singularity is allowed at the endpoints.

# Computes ∫ 1/sqrt(|x|) dx in (-∞, ∞)
# The integrand has a singular point at x = 0
I, E = quadde(x -> 1/sqrt(abs(x)), -1, 0, 1)
I  4    # true

Optimized numerical integrators

User can get an optimized integrators, for example, for better accuracy; QuadDE will provides the functionality.

qde = QuadDE(BigFloat; h0=one(BigFloat)/8, maxlevel=10)
qde(x -> 2/(1 + x^2), -1,  1)

User can specify the required precision as a type (T<:AbstractFloat), the starting step size h0 and the maximum number of repetition maxlevel. The h0 and maxlevel shown above are the default values, so it can be omitted. QuadDE instance is an callable object which has the same interface of quadde, actually quadde is an alias to QuadDE(Float64)(...) with a precalculated instance.

QuadDE tries to calculate integral values maxlevel times at a maximum; the step size of a trapezoid is started from h0 and is halved in each following repetition for finer accuracy. The repetition is terminated when the difference from the previous estimation gets smaller than a certain threshold. The threshold is determined by the runtime parameters, atol or rtol.

Using smaller h0 may help if the integrand f(x) includes fine structure, such as spikes, in the integral interval. However, it seems that the subdivision of the interval would be more effective in many cases. Try subdivision first, and then think of an optimized integrator.

See documentation for more details.

Numerical integrator for decaying oscillatory integrands

    quaddeo(f::Function, ω::Real, θ::Real, a::Real, b::Real;
            h0::Real=one(ω)/5, maxlevel::Integer=12,
            rtol::Real=atol>0 ? zero(atol) : sqrt(eps(typeof(atol))))

The quaddeo function is specialized for the decaying oscillatory integrands, [3-5]

f(x) = g(x)sin(ωx + θ),

where g(x) is a decaying algebraic function. ω and θ are the frequency and the phase of the oscillatory part of the integrand. If the oscillatory part is sin(ωx), then θ = 0.0; if it is cos(ωx) instead, then θ = π/(2ω).

using DoubleExponentialFormulas
using LinearAlgebra: norm

f(x) = sin(x)/x;
I, E = quaddeo(f, 1.0, 0.0, 0.0, Inf);
I  π/2  # true


  1. Takahasi, H.; Mori, M. Double Exponential Formulas for Numerical Integration. Publ. Res. Inst. Math. Sci. 1973, 9 (3), 721–741. 10.2977/prims/1195192451.

  2. Mori, M.; Sugihara, M. The Double-Exponential Transformation in Numerical Analysis. J. Comput. Appl. Math. 2001, 127 (1–2), 287–296. 10.1016/S0377-0427(00)00501-X.

  3. Ooura, T.; Mori, M. The double exponential formula for oscillatory functions over the half infinite interval. J. Comput. Appl. Math. 1991, 38, 353–360. 10.1016/0377-0427(91)90181-I