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

scipy.integrate.cumulative_trapezoid

scipy.integrate.romb