Your one-stop shop for numerical integration in Python.

Hundreds of numerical integration schemes for line segments, circles, disks, triangles, quadrilaterals, spheres, balls, tetrahedra, hexahedra, wedges, pyramids, n-spheres, n-balls, n-cubes, n-simplices, and the 2D/3D/nD spaces with weight functions exp(-r) and exp(-r²).

To numerically integrate any function over any given triangle, do

import numpy
import quadpy

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

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

val = quadpy.triangle.integrate(f, triangle, quadpy.triangle.Strang(9))

This uses Strang's rule of degree 6.

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.,

triangles = numpy.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 [numpy.sin(x[0]), numpy.sin(x[1])]

More examples under test/examples_test.py.

quadpy can do adaptive quadrature for certain domains. Again, everything is fully vectorized, so you can provide multiple intervals and vector-valued functions.

val, error_estimate = quadpy.line_segment.adaptive_integrate(
        lambda x: x * sin(5 * x),
        [0.0, pi],
        1.0e-10
        )

val, error_estimate = quadpy.triangle.adaptive_integrate(
        lambda x: x[0] * sin(5 * x[1]),
        [[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]],
        1.0e-10
        )

ProTip: You can provide many triangles that together form a domain to get an approximation of the integral over the domain.

• Chebyshev-Gauss (both variants, arbitrary degree)
• Clenshaw-Curtis (after Waldvogel, arbitrary degree)
• Fejér-type-1 (after Waldvogel, arbitrary degree)
• Fejér-type-2 (after Waldvogel, arbitrary degree)
• Gauss-Hermite (via NumPy, arbitrary degree)
• Gauss-Laguerre (via NumPy, arbitrary degree)
• Gauss-Legendre (via NumPy, arbitrary degree)
• Gauss-Lobatto (arbitrary degree)
• Gauss-Kronrod (after Laurie, arbitrary degree)
• Gauss-Patterson (7 schemes up to degree 191)
• Gauss-Radau (arbitrary degree)
• closed Newton-Cotes (arbitrary degree)
• open Newton-Cotes (arbitrary degree)

You can use orthopy to generate Gauss formulas for your own weight functions.

Example:

val = quadpy.line_segment.integrate(
    lambda x: numpy.exp(x),
    [0.0, 1.0],
    quadpy.line_segment.GaussPatterson(5)
    )

• Krylov (1959, arbitrary degree)

Example:

val = quadpy.circle.integrate(
    lambda x: numpy.exp(x[0]),
    [0.0, 0.0], 1.0,
    quadpy.circle.Krylov(7)
    )

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

• Hammer-Marlowe-Stroud (1956, 5 schemes up to degree 5),
• Hammer-Stroud (1958, 2 schemes up to degree 3)
• open and closed Newton-Cotes schemes (1970, after Silvester, arbitrary degree),
• via Stroud (1971):
  • Albrecht-Collatz (1958, degree 3)
  • conical product scheme (degree 7)
• Strang/Cowper (1973, 10 schemes up to degree 7),
• Lyness-Jespersen (1975, 21 schemes up to degree 11),
• Lether (1976, degree 2n-2, arbitrary n, not symmetric)
• Hillion (1977, 8 schemes up to degree 3),
• Grundmann-Möller (1978, arbitrary degree),
• Laursen-Gellert (1978, 17 schemes up to degree 10),
• CUBTRI (Laurie, 1982, degree 8),
• TRIEX (de Doncker-Robinson, 1984, degrees 9 and 11),
• 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),
• 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 schemes up to degree 62),
• Williams-Shunn-Jameson (2014, 8 schemes up to degree 12),
• Witherden-Vincent (2015, 19 schemes up to degree 20),
• Papanicolopulos (2016, 27 schemes up to degree 25).

Example:

val = quadpy.triangle.integrate(
    lambda x: numpy.exp(x[0]),
    [[0.0, 0.0], [1.0, 0.0], [0.5, 0.7]],
    quadpy.triangle.XiaoGimbutas(5)
    )

• Peirce (1957, arbitrary degree)
• via Stroud:
  • Radon (1948, degree 5)
  • Peirce (1956, 3 schemes up to degree 11)
  • Albrecht-Collatz (1958, degree 3)
  • Hammer-Stroud (1958, 8 schemes up to degree 15)
  • Albrecht (1960, 8 schemes up to degree 17)
  • Mysovskih (1964, 3 schemes up to degree 15)
  • Rabinowitz-Richter (1969, 6 schemes up to degree 15)
• Lether (1971, arbitrary degree)
• Cools-Haegemans (1985, 3 schemes up to degree 9)
• Wissmann-Becker (1986, 3 schemes up to degree 8)
• Cools-Kim (2000, 3 schemes up to degree 21)

Example:

val = quadpy.disk.integrate(
    lambda x: numpy.exp(x[0]),
    [0.0, 0.0], 1.0,
    quadpy.disk.Lether(6)
    )

• Hammer-Stroud (1958, 3 schemes up to degree 7)
• via Stroud (1971, 15 schemes up to degree 15):
  • Maxwell (1890, degree 7)
  • Burnside (1908, degree 5)
  • Irwin (1923, 3 schemes up to degree 5)
  • Tyler (1953, 3 schemes up to degree 7)
  • Albrecht-Collatz (1958, 4 schemes up to degree 5)
  • Miller (1960, degree 1)
  • Meister (1966, degree 7)
  • Phillips (1967, degree 7)
  • Rabinowitz-Richter (1969, 6 schemes up to degree 15)
• Cools-Haegemans (1985, 3 schemes up to degree 13)
• Dunavant (1985, 11 schemes up to degree 19)
• Morrow-Patterson (1985, 2 schemes up to degree 20, single precision)
• Wissmann-Becker (1986, 6 schemes up to degree 8)
• Cools-Haegemans (1988, 2 schemes up to degree 13)
• products of line segment schemes
• all formulas from the n-cube

Example:

val = quadpy.quadrilateral.integrate(
    lambda x: numpy.exp(x[0]),
    [[[0.0, 0.0], [1.0, 0.0]], [[0.0, 1.0], [1.0, 1.0]]],
    quadpy.quadrilateral.Stroud('C2 7-2')
    )

The points are specified in an array of shape (2, 2, ...) such that arr[0][0] is the lower left corner, arr[1][1] the upper right. If your quadrilateral has its sides aligned with the coordinate axes, you can use the convenience function

quadpy.quadrilateral.rectangle_points([x0, x1], [y0, y1])

to generate the array.

• via Stroud (1971):
  • Stroud-Secrest (1963, 2 schemes up to degree 7)
  • Rabinowitz-Richter (1969, 4 schemes up to degree 15)
  • a scheme of degree 4

Example:

val = quadpy.e2r.integrate(
    lambda x: numpy.exp(x[0]),
    quadpy.e2r.RabinowitzRichter(5)
    )

• via Stroud (1971):
  • Stroud-Secrest (1963, 2 schemes up to degree 7)
  • Rabinowitz-Richter (1969, 5 schemes up to degree 15)
  • 3 schemes up to degree 7

Example:

val = quadpy.e2r2.integrate(
    lambda x: numpy.exp(x[0]),
    quadpy.e2r2.RabinowitzRichter(3)
    )

• via Stroud (1971):
  • Albrecht-Collatz (1958, 5 schemes up to degree 7)
  • McLaren (1963, 10 schemes up to degree 14)
• Lebedev (1976, 32 schemes up to degree 131)
• Heo-Xu (2001, 27 schemes up to degree 39, single-precision)

Example:

val = quadpy.sphere.integrate(
    lambda x: numpy.exp(x[0]),
    [0.0, 0.0, 0.0], 1.0,
    quadpy.sphere.Lebedev(19)
    )

Integration on the sphere can also be done for function defined in spherical coordinates:

val = quadpy.sphere.integrate_spherical(
    lambda phi_theta: numpy.sin(phi_theta[0])**2 * numpy.sin(phi_theta[1]),
    radius=1.0,
    rule=quadpy.sphere.Lebedev(19)
    )

Note that phi_theta[0] is the azimuthal, phi_theta[1] the polar angle here.

• Hammer-Stroud (1958, 6 schemes up to degree 7)
• via: Stroud (1971):
  • Ditkin (1948, 3 schemes up to degree 7)
  • Mysovskih (1964, degree 7)
  • 2 schemes up to degree 14

Example:

val = quadpy.ball.integrate(
    lambda x: numpy.exp(x[0]),
    [0.0, 0.0, 0.0], 1.0,
    quadpy.ball.HammerStroud('14-3a')
    )

• Hammer-Marlowe-Stroud (1956, 3 schemes up to degree 3)
• Hammer-Stroud (1958, 2 schemes up to degree 3)
• open and closed Newton-Cotes (1970, after Silvester) (arbitrary degree)
• Stroud (1971, degree 7)
• Grundmann-Möller (1978, arbitrary degree),
• Yu (1984, 5 schemes up to degree 6)
• Keast (1986, 11 schemes up to degree 8)
• Beckers-Haegemans (1990, degrees 8 and 9)
• Gatermann (1992, degree 5)
• Liu-Vinokur (1998, 14 schemes up to degree 5)
• Walkington (2000, 6 schemes up to degree 7)
• Zienkiewicz (2005, 2 schemes up to degree 3)
• Zhang-Cui-Liu (2009, 2 schemes up to degree 14)
• Xiao-Gimbutas (2010, 15 schemes up to degree 15)
• Shunn-Ham (2012, 6 schemes up to degree 7)
• Vioreanu-Rokhlin (2014, 10 schemes up to degree 13)
• Williams-Shunn-Jameson (2014, 1 scheme with degree 9)
• Witherden-Vincent (2015, 9 schemes up to degree 10)

Example:

val = quadpy.tetrahedron.integrate(
    lambda x: numpy.exp(x[0]),
    [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.5, 0.7, 0.0], [0.3, 0.9, 1.0]],
    quadpy.tetrahedron.Keast(10)
    )

• Product schemes derived from line segment schemes
• via: Stroud (1971):
  • Sadowsky (1940, degree 5)
  • Tyler (1953, 2 schemes up to degree 5)
  • Hammer-Wymore (1957, degree 7)
  • Albrecht-Collatz (1958, degree 3)
  • Hammer-Stroud (1958, 6 schemes up to degree 7)
  • Mustard-Lyness-Blatt (1963, 6 schemes up to degree 5)
  • Stroud (1967, degree 5)
  • Sarma-Stroud (1969, degree 7)
• all formulas from the n-cube

Example:

val = quadpy.hexahedron.integrate(
    lambda x: numpy.exp(x[0]),
    quadpy.hexahedron.cube_points([0.0, 1.0], [-0.3, 0.4], [1.0, 2.1]),
    quadpy.hexahedron.Product(quadpy.line_segment.NewtonCotesClosed(3))
    )

• Felippa (9 schemes up to degree 5)

Example:

val = quadpy.pyramid.integrate(
    lambda x: numpy.exp(x[0]),
    [
      [0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.5, 0.7, 0.0], [0.3, 0.9, 0.0],
      [0.0, 0.1, 1.0],
    ],
    quadpy.pyramid.Felippa(5)
    )

• Felippa (6 schemes up to degree 6)

Example:

val = quadpy.wedge.integrate(
    lambda x: numpy.exp(x[0]),
    [
      [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.5, 0.7, 0.0]],
      [[0.0, 0.0, 1.0], [1.0, 0.0, 1.0], [0.5, 0.7, 1.0]],
    ],
    quadpy.wedge.Felippa(3)
    )

• via Stroud (1971):
  • Stroud-Secrest (1963, 5 schemes up to degree 7)

Example:

val = quadpy.e2r.integrate(
    lambda x: numpy.exp(x[0]),
    quadpy.e2r.StroudSecrest('IX')
    )

• via Stroud (1971):
  • Stroud-Secrest (1963, 7 schemes up to degree 7)
  • scheme of degree 14

Example:

val = quadpy.e2r2.integrate(
    lambda x: numpy.exp(x[0]),
    quadpy.e2r2.RabinowitzRichter(3)
    )

• via Stroud:
  • Lauffer (1955, 5 schemes up to degree 5)
  • Hammer-Stroud (1956, 3 schemes up to degree 3)
  • Stroud (1966, 7 schemes of degree 3)
  • Stroud (1969, degree 5)
• Grundmann-Möller (1978, arbitrary degree)
• Walkington (2000, 5 schemes up to degree 7)

Example:

dim = 4
val = quadpy.simplex.integrate(
    lambda x: numpy.exp(x[0]),
    numpy.array([
        [0.0, 0.0, 0.0, 0.0],
        [1.0, 2.0, 0.0, 0.0],
        [0.0, 1.0, 0.0, 0.0],
        [0.0, 3.0, 1.0, 0.0],
        [0.0, 0.0, 4.0, 1.0],
        ]),
    quadpy.simplex.GrundmannMoeller(dim, 3)
    )

• via Stroud (1971):
  • Stroud (1967, degree 7)
  • Stroud (1969, 3 <= n <= 16, degree 11)
  • 6 schemes up to degree 5
• Dobrodeev (1978, n >= 2, degree 5)

Example:

dim = 4
quadpy.nsphere.integrate(
    lambda x: numpy.exp(x[0]),
    numpy.zeros(dim), 1.0,
    quadpy.nsphere.Dobrodeev1978(dim)
    )

• Dobrodeev (1970, n >= 3, degree 7)
• via Stroud (1971):
  • Stroud (1957, degree 2)
  • Hammer-Stroud (1958, 2 schemes up to degree 5)
  • Stroud (1966, 4 schemes of degree 5)
  • Stroud (1967, 4 <= n <= 7, 2 schemes of degree 5)
  • Stroud (1967, n >= 3, 3 schemes of degree 7)
  • Stenger (1967, 6 schemes up to degree 11)
• Dobrodeev (1978, 2 <= n <= 20, degree 5)

Example:

dim = 4
quadpy.nball.integrate(
    lambda x: numpy.exp(x[0]),
    numpy.zeros(dim), 1.0,
    quadpy.nball.Dobrodeev1970(dim)
    )

• Dobrodeev (1970, n >= 5, degree 7)
• via Stroud (1971):
  • Ewing (1941, degree 3)
  • Tyler (1953, degree 3)
  • Stroud (1957, 2 schemes up to degree 3)
  • Hammer-Stroud (1958, degree 5)
  • Mustard-Lyness-Blatt (1963, degree 5)
  • Thacher (1964, degree 2)
  • Stroud (1966, 4 schemes of degree 5)
  • Phillips (1967, degree 7, single precision)
  • Stroud (1968, degree 5)
• Dobrodeev (1978, n >= 2, degree 5)

Example:

dim = 4
quadpy.ncube.integrate(
    lambda x: numpy.exp(x[0]),
    quadpy.ncube.ncube_points(
        [0.0, 1.0], [0.1, 0.9], [-1.0, 1.0], [-1.0, -0.5]
        ),
    quadpy.ncube.Stroud(dim, 'Cn 3-3')
    )

• via Stroud (1971):
  • Stroud-Secrest (1963, 4 schemes up to degree 5)
  • 2 schemes up to degree 5

Example:

dim = 4
val = quadpy.enr.integrate(
    lambda x: numpy.exp(x[0]),
    quadpy.enr.Stroud(dim, '5-4')
    )

• via Stroud (1971):
  • Stroud-Secrest (1963, 4 schemes up to degree 5)
  • Stroud (1967, 2 schemes of degree 5)
  • Stroud (1967, 3 schemes of degree 7)
  • Stenger (1971, 6 schemes up to degree 11, varying dimensionality restrictions)
  • 5 schemes up to degree 5

Example:

dim = 4
val = quadpy.enr2.integrate(
    lambda x: numpy.exp(x[0]),
    quadpy.enr2.Stroud(dim, '5-2')
    )

quadpy is available from the Python Package Index, so with

pip install -U quadpy

you can install/upgrade.

To run the tests, just check out this repository and type

MPLBACKEND=Agg pytest

To create a new release

1. bump the __version__ number,

2. publish to PyPi and GitHub:

   $ make publish
   

quadpy is published under the MIT license.