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bundles / scipy latest / scipy / special / _orthogonal / chebyu

function

scipy.special._orthogonal:chebyu

source: /scipy/special/_orthogonal.py :1861

Signature

def   chebyu ( n ,  monic = False )

Summary

Chebyshev polynomial of the second kind.

Extended Summary

Defined to be the solution of

is a polynomial of degree .

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.

Notes

The polynomials are orthogonal over with weight function .

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()
✓
fig-18a3c10d1ed5c1c2.png
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))
✓
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()
✓
fig-140c8d77c0362d5a.png

See also

chebyt

Chebyshev polynomial of the first kind.

Aliases

  • scipy.special.chebyu