quadax.trapezoid

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:

y (array_like) – Input array to integrate.
x (array_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.
dx (scalar, optional) – The spacing between sample points when x is None. The default is 1.
axis (int, optional) – The axis along which to integrate.

Returns:

trapezoid (float 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.

Examples

Use the trapezoidal rule on evenly spaced points:

>>> import jax.numpy as jnp
>>> from quadax import trapezoid
>>> trapezoid([1, 2, 3])
4.0

The spacing between sample points can be selected by either the x or dx arguments:

>>> trapezoid([1, 2, 3], x=[4, 6, 8])
8.0
>>> trapezoid([1, 2, 3], dx=2)
8.0

Using a decreasing x corresponds to integrating in reverse:

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

More generally x is used to integrate along a parametric curve. We can estimate the integral \(\int_0^1 x^2 = 1/3\) using:

>>> x = jnp.linspace(0, 1, num=50)
>>> y = x**2
>>> trapezoid(y, x)
0.33340274885464394

Or estimate the area of a circle, noting we repeat the sample which closes the curve:

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

trapezoid can be applied along a specified axis to do multiple computations in one call:

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