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 experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. 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