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DECUHR.jl

Pure-Julia port of the DECUHR algorithm (Espelid & Genz, 1994) for automatic adaptive integration of functions with homogeneous vertex singularities over hyper-rectangular regions.

The package exposes a single algorithm type, DecuhrAlgorithm, that plugs directly into the Integrals.jl / SciML ecosystem through the standard SciMLBase.AbstractIntegralAlgorithm interface.

Installation

using Pkg
Pkg.develop(path = "path/to/DECUHR.jl")   # local development
# or, once registered:
# Pkg.add("DECUHR")

Quick start

using Integrals, DECUHR

# ∫₀¹∫₀¹ (x·y)^{-1/2} dx dy = 4
f    = (u, p) -> (u[1] * u[2])^(-0.5)
prob = IntegralProblem(f, (zeros(2), ones(2)))
sol  = solve(prob, DecuhrAlgorithm(singul=2, alpha=-0.5); abstol=1e-8)

println(sol.u)       # ≈ 4.0
println(sol.retcode) # Success

Features

  • Vertex singularities of the form $f(\mathbf{x}) \sim \prod_{i=1}^{s} x_i^{\alpha}$ at the lower-left corner of the domain.
  • Automatic estimation of the singularity exponent $\alpha$ and detection of logarithmic factors (when alpha ≤ -singul, the default).
  • Vector-valued integrands (NUMFUN > 1): all components are integrated simultaneously.
  • Four built-in integration rules: degree-13 (2D, 65 pts), degree-11 (3D, 127 pts), degree-9 (nD), degree-7 (nD); auto-selected by default.
  • Generic value type TV: works with dual numbers (ForwardDiff) for automatic differentiation with respect to parameters p.

References

  • T. O. Espelid and A. Genz, DECUHR: An Algorithm for Automatic Integration of Singular Functions over a Hyperrectangular Region, Numerical Algorithms 8 (1994), pp. 201–220.
  • T. O. Espelid, On Integrating Vertex Singularities using Extrapolation, BIT 34 (1994), pp. 62–79.

License and upstream notice

The Julia port is MIT-licensed. Because it is a translation and modification of the original Fortran 77 DECUHR routines, the upstream copyright notice of Espelid & Genz must be preserved in every copy and every derivative work. See the License page for the full text of both the MIT License and the upstream notice.