quadpy 0.17.32

Numerical integration, quadrature for various domains

Your one-stop shop for numerical integration in Python.

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

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

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

and many have

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.,

# 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)

• 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)

• 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)

• 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)

• 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)

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 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),
• all schemes for the n-simplex.

Example:

import numpy as np
import quadpy

scheme = quadpy.t2.get_good_scheme(12)
scheme.show()
val = scheme.integrate(lambda x: np.exp(x[0]), [[0.0, 0.0], [1.0, 0.0], [0.5, 0.7]])

Disk (S₂)

• Radon (1948, degree 5)
• Peirce (1956, 3 schemes up to degree 11)
• Peirce (1957, arbitrary degree)
• 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)
• Piessens-Haegemans (1975, 1 scheme of degree 9)
• Haegemans-Piessens (1977, degree 9)
• Cools-Haegemans (1985, 4 schemes up to degree 13)
• Wissmann-Becker (1986, 3 schemes up to degree 8)
• Kim-Song (1997, 15 schemes up to degree 17)
• Cools-Kim (2000, 3 schemes up to degree 21)
• Luo-Meng (2007, 6 schemes up to degree 17)
• Takaki-Forbes-Rolland (2022, 19 schemes up to degree 77)
• all schemes from the n-ball

Example:

import numpy as np
import quadpy

scheme = quadpy.s2.get_good_scheme(6)
scheme.show()
val = scheme.integrate(lambda x: np.exp(x[0]), [0.0, 0.0], 1.0)

Quadrilateral (C₂)

• Maxwell (1890, degree 7)
• Burnside (1908, degree 5)
• Irwin (1923, 3 schemes up to degree 5)
• Tyler (1953, 3 schemes up to degree 7)
• Hammer-Stroud (1958, 3 schemes up to degree 7)
• Albrecht-Collatz (1958, 4 schemes up to degree 5)
• Miller (1960, degree 1, degree 11 for harmonic integrands)
• Meister (1966, degree 7)
• Phillips (1967, degree 7)
• Rabinowitz-Richter (1969, 6 schemes up to degree 15)
• Franke (1971, 10 schemes up to degree 9)
• Piessens-Haegemans (1975, 2 schemes of degree 9)
• Haegemans-Piessens (1977, degree 7)
• Schmid (1978, 3 schemes up to degree 6)
• Cools-Haegemans (1985, 6 schemes up to degree 17)
• Dunavant (1985, 11 schemes up to degree 19)
• Morrow-Patterson (1985, 2 schemes up to degree 20, single precision)
• Cohen-Gismalla, (1986, 2 schemes up to degree 3)
• Wissmann-Becker (1986, 6 schemes up to degree 8)
• Cools-Haegemans (1988, 2 schemes up to degree 13)
• Waldron (1994, infinitely many schemes of degree 3)
• Sommariva (2012, 55 schemes up to degree 55)
• Witherden-Vincent (2015, 11 schemes up to degree 21)
• products of line segment schemes
• all schemes from the n-cube

Example:

import numpy as np
import quadpy

scheme = quadpy.c2.get_good_scheme(7)
val = scheme.integrate(
    lambda x: np.exp(x[0]),
    [[[0.0, 0.0], [1.0, 0.0]], [[0.0, 1.0], [1.0, 1.0]]],
)

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 c2 has its sides aligned with the coordinate axes, you can use the convenience function

quadpy.c2.rectangle_points([x0, x1], [y0, y1])

to generate the array.

2D space with weight function exp(-r) (E₂^r)

• Stroud-Secrest (1963, 2 schemes up to degree 7)
• Rabinowitz-Richter (1969, 4 schemes up to degree 15)
• Stroud (1971, degree 4)
• Haegemans-Piessens (1977, 2 schemes up to degree 9)
• Cools-Haegemans (1985, 3 schemes up to degree 13)
• all schemes from the nD space with weight function exp(-r)

Example:

import quadpy

scheme = quadpy.e2r.get_good_scheme(5)
scheme.show()
val = scheme.integrate(lambda x: x[0] ** 2)

2D space with weight function exp(-r²) (E₂^r2)

• Stroud-Secrest (1963, 2 schemes up to degree 7)
• Rabinowitz-Richter (1969, 5 schemes up to degree 15)
• Stroud (1971, 3 schemes up to degree 7)
• Haegemans-Piessens (1977, 2 schemes of degree 9)
• Cools-Haegemans (1985, 3 schemes up to degree 13)
• Van Zandt (2019, degree 6)
• all schemes from the nD space with weight function exp(-r²)

Example:

import quadpy

scheme = quadpy.e2r2.get_good_scheme(3)
scheme.show()
val = scheme.integrate(lambda x: x[0] ** 2)

Sphere (U₃)

• Albrecht-Collatz (1958, 5 schemes up to degree 7)
• McLaren (1963, 10 schemes up to degree 14)
• Lebedev (1976, 34 schemes up to degree 131)
• Bažant-Oh (1986, 3 schemes up to degree 13)
• Heo-Xu (2001, 27 schemes up to degree 39)
• Fliege-Maier (2007, 4 schemes up to degree 4, single-precision)
• all schemes from the n-sphere

Example:

import numpy as np
import quadpy

scheme = quadpy.u3.get_good_scheme(19)
# scheme.show()
val = scheme.integrate(lambda x: np.exp(x[0]), [0.0, 0.0, 0.0], 1.0)

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

import numpy as np
import quadpy


def f(theta_phi):
    theta, phi = theta_phi
    return np.sin(phi) ** 2 * np.sin(theta)


scheme = quadpy.u3.get_good_scheme(19)
val = scheme.integrate_spherical(f)

Ball (S₃)

• Ditkin (1948, 3 schemes up to degree 7)
• Hammer-Stroud (1958, 6 schemes up to degree 7)
• Mysovskih (1964, degree 7)
• Stroud (1971, 2 schemes up to degree 14)
• Van Zandt (2020, degree 4)
• all schemes from the n-ball

Example:

import numpy as np
import quadpy

scheme = quadpy.s3.get_good_scheme(4)
# scheme.show()
val = scheme.integrate(lambda x: np.exp(x[0]), [0.0, 0.0, 0.0], 1.0)

Tetrahedron (T₃)

• Hammer-Marlowe-Stroud (1956, 3 schemes up to degree 3, also appearing in Hammer-Stroud)
• Stroud (1971, degree 7)
• Yu (1984, 5 schemes up to degree 6)
• Keast (1986, 10 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)
• 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)
• Jaśkowiec-Sukumar (2020, 21 schemes up to degree 20)
• all schemes for the n-simplex.

Example:

import numpy as np
import quadpy

scheme = quadpy.t3.get_good_scheme(5)
# scheme.show()
val = scheme.integrate(
    lambda x: np.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]],
)

Hexahedron (C₃)

• 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)
• Witherden-Vincent (2015, 7 schemes up to degree degree 11)
• all schemes from the n-cube
• Product schemes derived from line segment schemes

Example:

import numpy as np
import quadpy

scheme = quadpy.c3.product(quadpy.c1.newton_cotes_closed(3))
# scheme.show()
val = scheme.integrate(
    lambda x: np.exp(x[0]),
    quadpy.c3.cube_points([0.0, 1.0], [-0.3, 0.4], [1.0, 2.1]),
)

Pyramid (P₃)

• Felippa (2004, 9 schemes up to degree 5)

Example:

import numpy as np
import quadpy

scheme = quadpy.p3.felippa_5()

val = scheme.integrate(
    lambda x: np.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],
    ],
)

Wedge (W₃)

• Felippa (2004, 6 schemes up to degree 6)
• Kubatko-Yeager-Maggi (2013, 21 schemes up to degree 9)

Example:

import numpy as np
import quadpy

scheme = quadpy.w3.felippa_3()
val = scheme.integrate(
    lambda x: np.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]],
    ],
)

3D space with weight function exp(-r) (E₃^r)

• Stroud-Secrest (1963, 5 schemes up to degree 7)
• all schemes from the nD space with weight function exp(-r)

Example:

import quadpy

scheme = quadpy.e3r.get_good_scheme(5)
# scheme.show()
val = scheme.integrate(lambda x: x[0] ** 2)

3D space with weight function exp(-r²) (E₃^r2)

• Stroud-Secrest (1963, 7 schemes up to degree 7)
• Stroud (1971, scheme of degree 14)
• Van Zandt (2020, degree 4)
• all schemes from the nD space with weight function exp(-r²)

Example:

import quadpy

scheme = quadpy.e3r2.get_good_scheme(6)
# scheme.show()
val = scheme.integrate(lambda x: x[0] ** 2)

n-Simplex (T_n)

• Lauffer (1955, 5 schemes up to degree 5)
• Hammer-Stroud (1956, 3 schemes up to degree 3)
• Stroud (1964, degree 3)
• Stroud (1966, 7 schemes of degree 3)
• Stroud (1969, degree 5)
• Silvester (1970, arbitrary degree),
• Grundmann-Möller (1978, arbitrary degree)
• Walkington (2000, 5 schemes up to degree 7)

Example:

import numpy as np
import quadpy

dim = 4
scheme = quadpy.tn.grundmann_moeller(dim, 3)
val = scheme.integrate(
    lambda x: np.exp(x[0]),
    np.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],
        ]
    ),
)

n-Sphere (U_n)

• Stroud (1967, degree 7)
• Stroud (1969, 3 <= n <= 16, degree 11)
• Stroud (1971, 6 schemes up to degree 5)
• Dobrodeev (1978, n >= 2, degree 5)
• Mysovskikh (1980, 2 schemes up to degree 5)

Example:

import numpy as np
import quadpy

dim = 4
scheme = quadpy.un.dobrodeev_1978(dim)
val = scheme.integrate(lambda x: np.exp(x[0]), np.zeros(dim), 1.0)

n-Ball (S_n)

• 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)
• McNamee-Stenger (1967, 6 schemes up to degree 9)
• Dobrodeev (1970, n >= 3, degree 7)
• Dobrodeev (1978, 2 <= n <= 20, degree 5)
• Stoyanova (1997, n >= 5, degree 7)

Example:

import numpy as np
import quadpy

dim = 4
scheme = quadpy.sn.dobrodeev_1970(dim)
val = scheme.integrate(lambda x: np.exp(x[0]), np.zeros(dim), 1.0)

n-Cube (C_n)

• 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)
• McNamee-Stenger (1967, 6 schemes up to degree 9)
• Stroud (1968, degree 5)
• Dobrodeev (1970, n >= 5, degree 7)
• Dobrodeev (1978, n >= 2, degree 5)
• Cools-Haegemans (1994, 2 schemes up to degree 5)

Example:

import numpy as np
import quadpy

dim = 4
scheme = quadpy.cn.stroud_cn_3_3(dim)
val = scheme.integrate(
    lambda x: np.exp(x[0]),
    quadpy.cn.ncube_points([0.0, 1.0], [0.1, 0.9], [-1.0, 1.0], [-1.0, -0.5]),
)

nD space with weight function exp(-r) (E_n^r)

• Stroud-Secrest (1963, 4 schemes up to degree 5)
• McNamee-Stenger (1967, 6 schemes up to degree 9)
• Stroud (1971, 2 schemes up to degree 5)

Example:

import quadpy

dim = 4
scheme = quadpy.enr.stroud_enr_5_4(dim)
val = scheme.integrate(lambda x: x[0] ** 2)

nD space with weight function exp(-r²) (E_n^r2)

• Stroud-Secrest (1963, 4 schemes up to degree 5)
• McNamee-Stenger (1967, 6 schemes up to degree 9)
• 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)
• Stroud (1971, 5 schemes up to degree 5)
• Phillips (1980, degree 5)
• Cools-Haegemans (1994, 3 schemes up to degree 7)
• Lu-Darmofal (2004, degree 5)
• Xiu (2008, degree 2)

Example:

import quadpy

dim = 4
scheme = quadpy.enr2.stroud_enr2_5_2(dim)
val = scheme.integrate(lambda x: x[0] ** 2)