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.
Install quadpy from PyPI with
pip install quadpy
See here on how to get a license.
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:
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])
Example:
import quadpy
scheme = quadpy.e1r.gauss_laguerre(5, alpha=0)
scheme.show()
val = scheme.integrate(lambda x: x**2)
Example:
import quadpy
scheme = quadpy.e1r2.gauss_hermite(5)
scheme.show()
val = scheme.integrate(lambda x: x**2)
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)
Apart from the classical centroid, vertex, and seven-point schemes we have
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]])
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)
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.
Example:
import quadpy
scheme = quadpy.e2r.get_good_scheme(5)
scheme.show()
val = scheme.integrate(lambda x: x[0] ** 2)
Example:
import quadpy
scheme = quadpy.e2r2.get_good_scheme(3)
scheme.show()
val = scheme.integrate(lambda x: x[0] ** 2)
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)
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)
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]],
)
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]),
)
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],
],
)
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]],
],
)
Example:
import quadpy
scheme = quadpy.e3r.get_good_scheme(5)
# scheme.show()
val = scheme.integrate(lambda x: x[0] ** 2)
Example:
import quadpy
scheme = quadpy.e3r2.get_good_scheme(6)
# scheme.show()
val = scheme.integrate(lambda x: x[0] ** 2)
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],
]
),
)
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)
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)
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]),
)
Example:
import quadpy
dim = 4
scheme = quadpy.enr.stroud_enr_5_4(dim)
val = scheme.integrate(lambda x: x[0] ** 2)
Example:
import quadpy
dim = 4
scheme = quadpy.enr2.stroud_enr2_5_2(dim)
val = scheme.integrate(lambda x: x[0] ** 2)
FAQs
Numerical integration, quadrature for various domains
We found that quadpy demonstrated a healthy version release cadence and project activity because the last version was released less than a year ago. It has 1 open source maintainer collaborating on the project.
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