scipy.integrate.

simpson#

scipy.integrate.simpson(y, x=None, *, dx=1.0, axis=-1)[source]#

Integrate y(x) using samples along the given axis and the composite Simpson’s rule. If x is None, spacing of dx is assumed.

Parameters:

yarray_like: Array to be integrated.
xarray_like, optional: If given, the points at which y is sampled.
dxfloat, optional: Spacing of integration points along axis of x. Only used when x is None. Default is 1.
axisint, optional: Axis along which to integrate. Default is the last axis.

Returns:

float: The estimated integral computed with the composite Simpson’s rule.

See also

quad: adaptive quadrature using QUADPACK
fixed_quad: fixed-order Gaussian quadrature
dblquad: double integrals
tplquad: triple integrals
romb: integrators for sampled data
cumulative_trapezoid: cumulative integration for sampled data
cumulative_simpson: cumulative integration using Simpson’s 1/3 rule

Notes

For an odd number of samples that are equally spaced the result is exact if the function is a polynomial of order 3 or less. If the samples are not equally spaced, then the result is exact only if the function is a polynomial of order 2 or less.

Array API Standard Support

simpson has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	✅	✅
Dask	✅	n/a

See Support for the array API standard for more information.

References

[1]

Cartwright, Kenneth V. Simpson’s Rule Cumulative Integration with MS Excel and Irregularly-spaced Data. Journal of Mathematical Sciences and Mathematics Education. 12 (2): 1-9

Examples

>>> from scipy import integrate
>>> import numpy as np
>>> x = np.arange(0, 10)
>>> y = np.arange(0, 10)

>>> integrate.simpson(y, x=x)
40.5

>>> y = np.power(x, 3)
>>> integrate.simpson(y, x=x)
1640.5
>>> integrate.quad(lambda x: x**3, 0, 9)[0]
1640.25