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Module « scipy.integrate »

Fonction trapezoid - module scipy.integrate

Signature de la fonction trapezoid

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

Description

help(scipy.integrate.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
:math:`\int y(x) dx`.
When `x` is specified, this integrates along the parametric curve,
computing :math:`\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. 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.

See Also
--------
cumulative_trapezoid, simpson, romb

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
--------
Use the trapezoidal rule on evenly spaced points:

>>> import numpy as np
>>> from scipy import integrate
>>> integrate.trapezoid([1, 2, 3])
4.0

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

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

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

>>> integrate.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 :math:`\int_0^1 x^2 = 1/3` using:

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

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))
3.141571941375841

``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]])
>>> integrate.trapezoid(a, axis=0)
array([1.5, 2.5, 3.5])
>>> integrate.trapezoid(a, axis=1)
array([2.,  8.])

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