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bundles / scipy latest / scipy / integrate / _quadrature / trapezoid

function

`scipy.integrate._quadrature:trapezoid`

source: /scipy/integrate/_quadrature.py :21

Signature

def trapezoid ( y , x = None , dx = 1.0 , axis = -1 )

Summary

Integrate along the given axis using the composite trapezoidal rule.

Extended Summary

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 $\int y (x) d x$ . When x is specified, this integrates along the parametric curve, computing $\int_{t} y (t) d t = \int_{t} y (t) \frac{d x}{d t}_{x = x (t)} d t$ .

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. The default is the last axis.

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.

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.

Array API Standard Support

trapezoid has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ✅                   
JAX                   ⛔                     ⛔                   
Dask                  ✅                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

Use the trapezoidal rule on evenly spaced points:

import numpy as np
from scipy import integrate

✓

integrate.trapezoid([1, 2, 3])

✗

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

integrate.trapezoid([1, 2, 3], x=[4, 6, 8])
integrate.trapezoid([1, 2, 3], dx=2)

✗

Using a decreasing ``x`` corresponds to integrating in reverse:

integrate.trapezoid([1, 2, 3], x=[8, 6, 4])

✗

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

x = np.linspace(0, 1, num=50)
y = x**2

✓

integrate.trapezoid(y, x)

✗

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)

✓

integrate.trapezoid(np.cos(theta), x=np.sin(theta))

✗

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

a = np.arange(6).reshape(2, 3)
a
integrate.trapezoid(a, axis=0)

✓

integrate.trapezoid(a, axis=1)

`scipy.integrate._quadrature:trapezoid`

Signature

Summary

Extended Summary

Parameters

Returns

Notes

Examples

See also

Aliases

Referenced by

This package