scipy.special.eval_chebyt#

scipy.special.eval_chebyt(n, x, out=None) = <wrapped_ufunc 'eval_chebyt'>#

Evaluate Chebyshev polynomial of the first kind at a point.

The Chebyshev polynomials of the first kind can be defined via the Gauss hypergeometric function \({}_2F_1\) as

\[T_n(x) = {}_2F_1(n, -n; 1/2; (1 - x)/2).\]

When \(n\) is an integer the result is a polynomial of degree \(n\). See 22.5.47 in [AS] (or equivalently [DLMF]) for details.

Parameters:

narray_like: Degree of the polynomial. If not an integer, the result is determined via the relation to the Gauss hypergeometric function.
xarray_like: Points at which to evaluate the Chebyshev polynomial
outndarray, optional: Optional output array for the function values

Returns:

Tscalar or ndarray: Values of the Chebyshev polynomial

See also

roots_chebyt: roots and quadrature weights of Chebyshev polynomials of the first kind
chebyu: Chebychev polynomial object
eval_chebyu: evaluate Chebyshev polynomials of the second kind
hyp2f1: Gauss hypergeometric function
numpy.polynomial.chebyshev.Chebyshev: Chebyshev series

Notes

This routine is numerically stable for x in [-1, 1] at least up to order 10000.

Array API Standard Support

eval_chebyt has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	✅
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable.

With PyTorch, noninteger values of the degree n are cast to integers before evaluation.

See Support for the array API standard for more information.

References

[AS]

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

[DLMF]

NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/18.5.E11_2

Examples

>>> import numpy as np
>>> from scipy.special import eval_chebyt
>>> import matplotlib.pyplot as plt

Evaluate the order 3 polynomial at a few values of x.

>>> eval_chebyt(3, [0, 0.25, 0.5])
array([-0.    , -0.6875, -1.    ])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(-1, 1, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_chebyt(n, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Chebyshev Polynomials of the First Kind")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$T_n(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

../../_images/scipy-special-eval_chebyt-1.png