Convergence

This page demonstrates the convergence properties of Tanh-Sinh quadrature for challenging test functions from the numerical integration literature.

Reference

The Tanh-Sinh (Double Exponential) quadrature method was introduced by:

Double Exponential Formulas for Numerical Integration Hidetosi Takahasi, Masatake Mori Publ. Res. Inst. Math. Sci. 9 (1973), no. 3, pp. 721–741 DOI: 10.2977/PRIMS/1195192451

The last two test functions below are taken directly from this foundational paper.

Convergence Plot

The following plot shows how the absolute integration error decreases as the number of quadrature points increases. The test functions include:

$\log(1+x)$ — Logarithmic function with mild endpoint behavior
$1/(1+25x^2)$ — The Runge function, known to be challenging for polynomial-based methods
$\frac{1}{(x-2)(1-x)^{1/4}(1+x)^{3/4}}$ — A challenging singular integrand (from Takahasi-Mori)
$\frac{\cos(\pi x)}{\sqrt{1-x}}$ — Oscillatory function with endpoint singularity (from Takahasi-Mori)

Convergence of Tanh-Sinh Quadrature

Key Observations

Double Exponential Convergence: For all test functions, the error decreases exponentially (linear on the log-scale plot) as the number of points increases.
Singularity Handling: Even for functions with algebraic singularities at endpoints (like $(1-x)^{1/4}$ or $\sqrt{1-x}$), Tanh-Sinh quadrature maintains excellent convergence.
High Precision: With 256-bit BigFloat precision, the quadrature achieves errors below $10^{-60}$ with only a few hundred points.

Test Functions

The exact values were computed using arbitrary-precision arithmetic. Here are the test integrals:

Function	Domain	Exact Value	Source
$\log(1+x)$	$[-1, 1]$	$2\log(2) - 2 \approx -0.6137$	Standard
$1/(1+25x^2)$	$[-1, 1]$	$(2/5)\arctan(5) \approx 0.5493$	Runge
$\frac{1}{(x-2)(1-x)^{1/4}(1+x)^{3/4}}$	$[-1, 1]$	$\approx -1.9491$	Takahasi-Mori (1973)
$\frac{\cos(\pi x)}{\sqrt{1-x}}$	$[-1, 1]$	$\approx -0.6905$	Takahasi-Mori (1973)

Reproducing the Plot

The convergence plot can be regenerated using the script docs/convergence_plots.jl:

include("docs/convergence_plots.jl")

This requires the packages: FastTanhSinhQuadrature, DoubleFloats, GaussQuadrature, CairoMakie, LaTeXStrings.