numpy.trapz#

numpy.trapz(y, x=None, dx=1.0, axis=- 1)[source]#

Integrate along the given axis using the composite trapezoidal rule.

If x is provided, the integration happens in sequence along its elements - they are not sorted.

Integrate y (x) along each 1d slice on the given axis, compute \(\int y(x) dx\). When x is specified, this integrates along the parametric curve, computing \(\int_t y(t) dt = \int_t y(t) \left.\frac{dx}{dt}\right|_{x=x(t)} dt\).

Parameters

yarray_like: Input array to integrate.
xarray_like, optional: The sample points corresponding to the y values. If x is None, the sample points are assumed to be evenly spaced dx apart. The default is None.
dxscalar, optional: The spacing between sample points when x is None. The default is 1.
axisint, optional: The axis along which to integrate.

Returns

trapzfloat or ndarray: Definite integral of y = n-dimensional array as approximated along a single axis by the trapezoidal rule. If y is a 1-dimensional array, then the result is a float. If n is greater than 1, then the result is an n-1 dimensional array.

See also

sum, cumsum

Notes

Image [2] illustrates trapezoidal rule – y-axis locations of points will be taken from y array, by default x-axis distances between points will be 1.0, alternatively they can be provided with x array or with dx scalar. Return value will be equal to combined area under the red lines.

References

1: Wikipedia page: https://en.wikipedia.org/wiki/Trapezoidal_rule
2: Illustration image: https://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png

Examples

>>> np.trapz([1,2,3])
4.0
>>> np.trapz([1,2,3], x=[4,6,8])
8.0
>>> np.trapz([1,2,3], dx=2)
8.0

Using a decreasing x corresponds to integrating in reverse:

>>> np.trapz([1,2,3], x=[8,6,4])
-8.0

More generally x is used to integrate along a parametric curve. This finds the area of a circle, noting we repeat the sample which closes the curve:

>>> theta = np.linspace(0, 2 * np.pi, num=1000, endpoint=True)
>>> np.trapz(np.cos(theta), x=np.sin(theta))
3.141571941375841

>>> a = np.arange(6).reshape(2, 3)
>>> a
array([[0, 1, 2],
       [3, 4, 5]])
>>> np.trapz(a, axis=0)
array([1.5, 2.5, 3.5])
>>> np.trapz(a, axis=1)
array([2.,  8.])

numpy.cross

numpy.exp