scipy.integrate.simpson#
- scipy.integrate.simpson(y, *, x=None, dx=1.0, axis=-1, even=<object object>)[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:
use the first N-2 intervals with a trapezoidal rule on the last interval and
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 1.14.0. 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