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Module « scipy.interpolate »

Fonction approximate_taylor_polynomial - module scipy.interpolate

Signature de la fonction approximate_taylor_polynomial

def approximate_taylor_polynomial(f, x, degree, scale, order=None) 

Description

help(scipy.interpolate.approximate_taylor_polynomial)

Estimate the Taylor polynomial of f at x by polynomial fitting.

Parameters
----------
f : callable
    The function whose Taylor polynomial is sought. Should accept
    a vector of `x` values.
x : scalar
    The point at which the polynomial is to be evaluated.
degree : int
    The degree of the Taylor polynomial
scale : scalar
    The width of the interval to use to evaluate the Taylor polynomial.
    Function values spread over a range this wide are used to fit the
    polynomial. Must be chosen carefully.
order : int or None, optional
    The order of the polynomial to be used in the fitting; `f` will be
    evaluated ``order+1`` times. If None, use `degree`.

Returns
-------
p : poly1d instance
    The Taylor polynomial (translated to the origin, so that
    for example p(0)=f(x)).

Notes
-----
The appropriate choice of "scale" is a trade-off; too large and the
function differs from its Taylor polynomial too much to get a good
answer, too small and round-off errors overwhelm the higher-order terms.
The algorithm used becomes numerically unstable around order 30 even
under ideal circumstances.

Choosing order somewhat larger than degree may improve the higher-order
terms.

Examples
--------
We can calculate Taylor approximation polynomials of sin function with
various degrees:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.interpolate import approximate_taylor_polynomial
>>> x = np.linspace(-10.0, 10.0, num=100)
>>> plt.plot(x, np.sin(x), label="sin curve")
>>> for degree in np.arange(1, 15, step=2):
...     sin_taylor = approximate_taylor_polynomial(np.sin, 0, degree, 1,
...                                                order=degree + 2)
...     plt.plot(x, sin_taylor(x), label=f"degree={degree}")
>>> plt.legend(bbox_to_anchor=(1.05, 1), loc='upper left',
...            borderaxespad=0.0, shadow=True)
>>> plt.tight_layout()
>>> plt.axis([-10, 10, -10, 10])
>>> plt.show()

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