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Module « scipy.special »

Fonction chebyt - module scipy.special

Signature de la fonction chebyt

def chebyt(n, monic=False) 

Description

help(scipy.special.chebyt)

Chebyshev polynomial of the first kind.

Defined to be the solution of

.. math::
    (1 - x^2)\frac{d^2}{dx^2}T_n - x\frac{d}{dx}T_n + n^2T_n = 0;

:math:`T_n` is a polynomial of degree :math:`n`.

Parameters
----------
n : int
    Degree of the polynomial.
monic : bool, optional
    If `True`, scale the leading coefficient to be 1. Default is
    `False`.

Returns
-------
T : orthopoly1d
    Chebyshev polynomial of the first kind.

See Also
--------
chebyu : Chebyshev polynomial of the second kind.

Notes
-----
The polynomials :math:`T_n` are orthogonal over :math:`[-1, 1]`
with weight function :math:`(1 - x^2)^{-1/2}`.

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples
--------
Chebyshev polynomials of the first kind of order :math:`n` can
be obtained as the determinant of specific :math:`n \times n`
matrices. As an example we can check how the points obtained from
the determinant of the following :math:`3 \times 3` matrix
lay exactly on :math:`T_3`:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.linalg import det
>>> from scipy.special import chebyt
>>> x = np.arange(-1.0, 1.0, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-2.0, 2.0)
>>> ax.set_title(r'Chebyshev polynomial $T_3$')
>>> ax.plot(x, chebyt(3)(x), label=rf'$T_3$')
>>> for p in np.arange(-1.0, 1.0, 0.1):
...     ax.plot(p,
...             det(np.array([[p, 1, 0], [1, 2*p, 1], [0, 1, 2*p]])),
...             'rx')
>>> plt.legend(loc='best')
>>> plt.show()

They are also related to the Jacobi Polynomials
:math:`P_n^{(-0.5, -0.5)}` through the relation:

.. math::
    P_n^{(-0.5, -0.5)}(x) = \frac{1}{4^n} \binom{2n}{n} T_n(x)

Let's verify it for :math:`n = 3`:

>>> from scipy.special import binom
>>> from scipy.special import jacobi
>>> x = np.arange(-1.0, 1.0, 0.01)
>>> np.allclose(jacobi(3, -0.5, -0.5)(x),
...             1/64 * binom(6, 3) * chebyt(3)(x))
True

We can plot the Chebyshev polynomials :math:`T_n` for some values
of :math:`n`:

>>> x = np.arange(-1.5, 1.5, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-4.0, 4.0)
>>> ax.set_title(r'Chebyshev polynomials $T_n$')
>>> for n in np.arange(2,5):
...     ax.plot(x, chebyt(n)(x), label=rf'$T_n={n}$')
>>> plt.legend(loc='best')
>>> plt.show()

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