dpnp.trapezoid

dpnp.trapezoid(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 \(\int_t y(t) dt = \int_t y(t) \left.\frac{dx}{dt}\right|_{x=x(t)} dt\).

For full documentation refer to numpy.trapezoid.

Parameters:

y ({dpnp.ndarray, usm_ndarray}) -- Input array to integrate.
x ({dpnp.ndarray, usm_ndarray, None}, optional) -- The sample points corresponding to the y values. If x is None, the sample points are assumed to be evenly spaced dx apart. Default: None.
dx (scalar, optional) -- The spacing between sample points when x is None. Default: 1.
axis (int, optional) -- The axis along which to integrate. Default: -1.

Returns:

out -- Definite integral of y = n-dimensional array as approximated along a single axis by the trapezoidal rule. The result is an n-1 dimensional array.

Return type:

dpnp.ndarray

See also

dpnp.sum: Sum of array elements over a given axis.
dpnp.cumsum: Cumulative sum of the elements along a given axis.

Examples

>>> import dpnp as np

Use the trapezoidal rule on evenly spaced points:

>>> y = np.array([1, 2, 3])
>>> np.trapezoid(y)
array(4.)

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

>>> y = np.array([1, 2, 3])
>>> x = np.array([4, 6, 8])
>>> np.trapezoid(y, x=x)
array(8.)
>>> np.trapezoid(y, dx=2)
array(8.)

Using a decreasing x corresponds to integrating in reverse:

>>> y = np.array([1, 2, 3])
>>> x = np.array([8, 6, 4])
>>> np.trapezoid(y, x=x)
array(-8.)

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 = np.linspace(0, 1, num=50)
>>> y = x**2
>>> np.trapezoid(y, x)
array(0.33340275)

Or estimate 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.trapezoid(np.cos(theta), x=np.sin(theta))
array(3.14157194)

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

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