Quadrature rules

The online encylopedia of quadrature rules is a reference website that lists a number of quadrature rules.

Quadrature rules are sets of points and weights that are used to approximate integrals. If ${p_0,\dots,p_{n-1}}\subset\mathbb{R}^d$ and ${w_0,\dots,w_{n-1}}\subset\mathbb{R}$ are the points and weights (repectively) of the quadrature rule for a single integral, then:

$$\int f(x),\mathrm{d}x \approx \sum_{i=0}^{n-1}f(p_i)w_i$$

Website

Before building the online encylopedia of quadrature rules, you must first install qrtools from the python directory:

cd python
python3 -m pip install .

The online encylopedia of quadrature rules website can then be built by running:

cd website
python3 build.py

Libraries

All of the quadrature rules included in the online encylopedia of quadrature rules are included in the quadraturerules library, which is available in the following languages:

Language	Badges
C
C++
Fortran
FORTRAN 77
Julia
Python
Rust

Before building any of the libraries, you must first install qrtools from the python directory:

cd python
python3 -m pip install .

You can then build the libraries using the build.py script in the library directory. For example, to build the python library, you can run:

cd library
python build.py python

and to build the rust library, you can run:

cd library
python build.py rust

C++ library

The source code of the quadraturerules C++ library can be downloaded from the latest release on GitHub. It can be installed by running:

wget https://github.com/quadraturerules/quadraturerules/releases/download/v{{VERSION}}/quadraturerules-cpp-v{{VERSION}}.tar.gz
mkdir src
tar -xvf quadraturerules-cpp-v{{VERSION}}.tar.gz -C src
mkdir build
cd build
cmake ../src
make
make install

Once the library is installed, you can run the tests by running:

python src/test/run_tests.py

Or you can run individual tests:

cd src/test/{TEST_NAME}
cmake .
make .
./{TEST_NAME}

Usage

The library's function single_integral_quadrature can be used to get the points and weights of quadrature rules for a single integral. For example the following snippet will create an order 3 Xiao--Gimbutas rule on a triangle:

#include <quadraturerules.h>

using quadraturerules;

auto [points, weights] = single_integral_quadrature(
    QuadratureRule::XiaoGimbutas,
    Domain::Triangle,
    3
);

Note that the points returned by the library are represented using barycentric coordinates.

Generating the library

The C++ quadraturerules library can be generated from the templates in the online encyclopedia of quadrature rules GitHub repo. First clone the repo and move into the library directory:

git clone https://github.com/quadraturerules/quadraturerules.git
cd quadraturerules/library

The C++ library can then be generated by running:

python build.py cpp

This will create a directory called cpp.build containing the C++ source code.