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Module « scipy.special »

Fonction chebyu - module scipy.special

Signature de la fonction chebyu

def chebyu(n, monic=False) 

Description

help(scipy.special.chebyu)

Chebyshev polynomial of the second kind.

Defined to be the solution of

.. math::
    (1 - x^2)\frac{d^2}{dx^2}U_n - 3x\frac{d}{dx}U_n
      + n(n + 2)U_n = 0;

:math:`U_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
-------
U : orthopoly1d
    Chebyshev polynomial of the second kind.

See Also
--------
chebyt : Chebyshev polynomial of the first kind.

Notes
-----
The polynomials :math:`U_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 second 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:`U_3`:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.linalg import det
>>> from scipy.special import chebyu
>>> 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 $U_3$')
>>> ax.plot(x, chebyu(3)(x), label=rf'$U_3$')
>>> for p in np.arange(-1.0, 1.0, 0.1):
...     ax.plot(p,
...             det(np.array([[2*p, 1, 0], [1, 2*p, 1], [0, 1, 2*p]])),
...             'rx')
>>> plt.legend(loc='best')
>>> plt.show()

They satisfy the recurrence relation:

.. math::
    U_{2n-1}(x) = 2 T_n(x)U_{n-1}(x)

where the :math:`T_n` are the Chebyshev polynomial of the first kind.
Let's verify it for :math:`n = 2`:

>>> from scipy.special import chebyt
>>> x = np.arange(-1.0, 1.0, 0.01)
>>> np.allclose(chebyu(3)(x), 2 * chebyt(2)(x) * chebyu(1)(x))
True

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

>>> x = np.arange(-1.0, 1.0, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-1.5, 1.5)
>>> ax.set_title(r'Chebyshev polynomials $U_n$')
>>> for n in np.arange(1,5):
...     ax.plot(x, chebyu(n)(x), label=rf'$U_n={n}$')
>>> plt.legend(loc='best')
>>> plt.show()

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