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
cumulative_simpson
cumulative integration using Simpson’s 1/3 rule
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=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
>>> integrate.simpson(y, x=x, even='first') 1644.5