SciPy

scipy.integrate.simpson

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

evenstr {‘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.

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.

Examples

>>> from scipy import integrate
>>> 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)
1642.5
>>> integrate.quad(lambda x: x**3, 0, 9)[0]
1640.25
>>> integrate.simpson(y, x, even='first')
1644.5

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