scipy.integrate.simpson#

scipy.integrate.simpson(y, x=None, dx=1.0, axis=-1, even=None)[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.

If there are an even number of samples, N, then there are an odd number of intervals (N-1), but Simpson’s rule requires an even number of intervals. The parameter ‘even’ controls how this is handled.

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.

even{None, ‘simpson’, ‘avg’, ‘first’, ‘last’}, optional
‘avg’Average two results:

  1. use the first N-2 intervals with a trapezoidal rule on the last interval and

  2. use the last N-2 intervals with a trapezoidal rule on the first interval.

‘first’Use Simpson’s rule for the first N-2 intervals with

a trapezoidal rule on the last interval.

‘last’Use Simpson’s rule for the last N-2 intervals with a

trapezoidal rule on the first interval.

None : equivalent to ‘simpson’ (default)

‘simpson’Use Simpson’s rule for the first N-2 intervals with the

addition of a 3-point parabolic segment for the last interval using equations outlined by Cartwright [1]. If the axis to be integrated over only has two points then the integration falls back to a trapezoidal integration.

New in version 1.11.0.

Changed in version 1.11.0: The newly added ‘simpson’ option is now the default as it is more accurate in most situations.

Deprecated since version 1.11.0: Parameter even is deprecated and will be removed in SciPy 1.13.0. After this time the behaviour for an even number of points will follow that of even=’simpson’.

Returns:
float

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

See also

quad

adaptive quadrature using QUADPACK

romberg

adaptive Romberg quadrature

quadrature

adaptive Gaussian quadrature

fixed_quad

fixed-order Gaussian quadrature

dblquad

double integrals

tplquad

triple integrals

romb

integrators for sampled data

cumulative_trapezoid

cumulative integration for sampled data

ode

ODE integrators

odeint

ODE integrators

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.

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)
40.5

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

>>> integrate.simpson(y, x, even='first')
1644.5