Quadpy

<p align="center"> <a href="https://github.com/sigma-py/quadpy"><img alt="quadpy" src="https://raw.githubusercontent.com/sigma-py/quadpy/main/logo/logo-with-text.svg" width="60%"></a> <p align="center">Your one-stop shop for numerical integration in Python.</p> </p>

More than 1500 numerical integration schemes for line segments, circles, disks, triangles, quadrilaterals, spheres, balls, tetrahedra, hexahedra, wedges, pyramids, n-spheres, n-balls, n-cubes, n-simplices, the 1D half-space with weight functions exp(-r), the 2D space with weight functions exp(-r), the 3D space with weight functions exp(-r), the nD space with weight functions exp(-r), the 1D space with weight functions exp(-r²), the 2D space with weight functions exp(-r²), the 3D space with weight functions exp(-r²), and the nD space with weight functions exp(-r²), for fast integration of real-, complex-, and vector-valued functions.

Installation

Install quadpy from PyPI with

pip install quadpy

See here on how to get a license.

Using quadpy

Quadpy provides integration schemes for many different 1D, 2D, even nD domains.

To start off easy: If you'd numerically integrate any function over any given 1D interval, do

import numpy as np
import quadpy


def f(x):
    return np.sin(x) - x


val, err = quadpy.quad(f, 0.0, 6.0)

This is like scipy with the addition that quadpy handles complex-, vector-, matrix-valued integrands, and "intervals" in spaces of arbitrary dimension.

To integrate over a triangle, do

import numpy as np
import quadpy


def f(x):
    return np.sin(x[0]) * np.sin(x[1])


triangle = np.array([[0.0, 0.0], [1.0, 0.0], [0.7, 0.5]])

# get a "good" scheme of degree 10
scheme = quadpy.t2.get_good_scheme(10)
val = scheme.integrate(f, triangle)

Most domains have get_good_scheme(degree). If you would like to use a particular scheme, you can pick one from the dictionary quadpy.t2.schemes.

All schemes have

<!--pytest.mark.skip-->

scheme.points
scheme.weights
scheme.degree
scheme.source
scheme.test_tolerance

scheme.show()
scheme.integrate(
    # ...
)

and many have

<!--pytest.mark.skip-->

scheme.points_symbolic
scheme.weights_symbolic

You can explore schemes on the command line with, e.g.,

quadpy info s2 rabinowitz_richter_3

<quadrature scheme for S2>
  name:                 Rabinowitz-Richter 2
  source:               Perfectly Symmetric Two-Dimensional Integration Formulas with Minimal Numbers of Points
                        Philip Rabinowitz, Nira Richter
                        Mathematics of Computation, vol. 23, no. 108, pp. 765-779, 1969
                        https://doi.org/10.1090/S0025-5718-1969-0258281-4
  degree:               9
  num points/weights:   21
  max/min weight ratio: 7.632e+01
  test tolerance:       9.417e-15
  point position:       outside
  all weights positive: True

Also try quadpy show!

quadpy is fully vectorized, so if you like to compute the integral of a function on many domains at once, you can provide them all in one integrate() call, e.g.,

<!--pytest.mark.skip-->

# shape (3, 5, 2), i.e., (corners, num_triangles, xy_coords)
triangles = np.stack(
    [
        [[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]],
        [[1.2, 0.6], [1.3, 0.7], [1.4, 0.8]],
        [[26.0, 31.0], [24.0, 27.0], [33.0, 28]],
        [[0.1, 0.3], [0.4, 0.4], [0.7, 0.1]],
        [[8.6, 6.0], [9.4, 5.6], [7.5, 7.4]],
    ],
    axis=-2,
)

The same goes for functions with vectorized output, e.g.,

def f(x):
    return [np.sin(x[0]), np.sin(x[1])]

More examples under test/examples_test.py.

Read more about the dimensionality of the input/output arrays in the wiki.

Advanced topics:

• Adaptive quadrature
• Creating your own Gauss scheme
• tanh-sinh quadrature

Schemes

Line segment (C₁)

<img src="https://raw.githubusercontent.com/sigma-py/quadpy/main/plots/line-segment-gauss-legendre-20.svg" width="50%">

• Chebyshev-Gauss (type 1 and 2, arbitrary degree)
• Clenshaw-Curtis (arbitrary degree)
• Fejér (type 1 and 2, arbitrary degree)
• Gauss-Jacobi (arbitrary degree)
• Gauss-Legendre (arbitrary degree)
• Gauss-Lobatto (arbitrary degree)
• Gauss-Kronrod (arbitrary degree)
• Gauss-Patterson (9 nested schemes up to degree 767)
• Gauss-Radau (arbitrary degree)
• Newton-Cotes (open and closed, arbitrary degree)

See here for how to generate Gauss formulas for your own weight functions.

Example:

import numpy as np
import quadpy

scheme = quadpy.c1.gauss_patterson(5)
scheme.show()
val = scheme.integrate(lambda x: np.exp(x), [0.0, 1.0])

1D half-space with weight function exp(-r) (E₁^r)

<img src="https://raw.githubusercontent.com/sigma-py/quadpy/main/plots/e1r-gauss-laguerre-3.svg" width="50%">

• Generalized Gauss-Laguerre

Example:

import quadpy

scheme = quadpy.e1r.gauss_laguerre(5, alpha=0)
scheme.show()
val = scheme.integrate(lambda x: x**2)

1D space with weight function exp(-r²) (E₁^r²)

<img src="https://raw.githubusercontent.com/sigma-py/quadpy/main/plots/e1r2-gauss-hermite-8.svg" width="50%">

• Gauss-Hermite (arbitrary degree)
• Genz-Keister (1996, 8 nested schemes up to degree 67)

Example:

import quadpy

scheme = quadpy.e1r2.gauss_hermite(5)
scheme.show()
val = scheme.integrate(lambda x: x**2)

Circle (U₂)

<img src="https://raw.githubusercontent.com/sigma-py/quadpy/main/plots/circle-krylov-30.svg" width="25%">

• Krylov (1959, arbitrary degree)

Example:

import numpy as np
import quadpy

scheme = quadpy.u2.get_good_scheme(7)
scheme.show()
val = scheme.integrate(lambda x: np.exp(x[0]), [0.0, 0.0], 1.0)

Triangle (T₂)

<img src="https://raw.githubusercontent.com/sigma-py/quadpy/main/plots/triangle-dunavant-15.svg" width="25%">

Apart from the classical centroid, vertex, and seven-point schemes we have

• Hammer-Marlowe-Stroud (1956, 5 schemes up to degree 5)
• Albrecht-Collatz (1958, degree 3)
• Stroud (1971, conical product scheme of degree 7)
• Franke (1971, 2 schemes of degree 7)
• Strang-Fix/Cowper (1973, 10 schemes up to degree 7),
• Lyness-Jespersen (1975, 21 schemes up to degree 11, two of which are used in TRIEX),
• Lether (1976, degree 2n-2, arbitrary n, not symmetric),
• Hillion (1977, 10 schemes up to degree 3),
• Laursen-Gellert (1978, 17 schemes up to degree 10),
• CUBTRI (Laurie, 1982, degree 8),
• Dunavant (1985, 20 schemes up to degree 20),
• Cools-Haegemans (1987, degrees 8 and 11),
• Gatermann (1988, degree 7)
• Berntsen-Espelid (1990, 4 schemes of degree 13, the first one being DCUTRI),
• Liu-Vinokur (1998, 13 schemes up to degree 5),
• Griener-Schmid, (1999, 2 schemes of degree 6),
• Walkington (2000, 5 schemes up to degree 5),
• Wandzura-Xiao (2003, 6 schemes up to degree 30),
• Taylor-Wingate-Bos (2005, 5 schemes up to degree 14),
• Zhang-Cui-Liu (2009, 3 schemes up to degree 20),
• Xiao-Gimbutas (2010, 50 schemes up to degree 50),
• Vioreanu-Rokhlin (2014, 20 sche