DECUHR.jl

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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 ​

julia
import Pkg; Pkg.add("DECUHR")

Quick start ​

julia
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   at the lower-left corner of the domain.

  • Automatic estimation of the singularity exponent 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.

DECUHR.jl is an independent port: it is not affiliated with, endorsed by, or supported by the authors of the original DECUHR routines. Please report problems with this package here rather than to them. The package logo is an original work made for this port.