scipy.special.eval_chebyc#

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

Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a point.

These polynomials are defined as

\[C_n(x) = 2 T_n(x/2)\]

where \(T_n\) is a Chebyshev polynomial of the first kind. See 22.5.11 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 eval_chebyt.
xarray_like: Points at which to evaluate the Chebyshev polynomial
outndarray, optional: Optional output array for the function values

Returns:

Cscalar or ndarray: Values of the Chebyshev polynomial

See also

roots_chebyc: roots and quadrature weights of Chebyshev polynomials of the first kind on [-2, 2]
chebyc: Chebyshev polynomial object
numpy.polynomial.chebyshev.Chebyshev: Chebyshev series
eval_chebyt: evaluate Chebycshev polynomials of the first kind

Notes

Array API Standard Support

eval_chebyc 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. eval_chebyc does not currently support out for the PyTorch backend.

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/18.1.E3

Examples

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

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

>>> eval_chebyc(3, [0, 0.25, 0.5])
array([-0.      , -0.734375, -1.375   ])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(-2, 2, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_chebyc(n, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Chebyshev Polynomials of the First Kind on [-2, 2]")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$C_n(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

../../_images/scipy-special-eval_chebyc-1_00_00.png

They are a scaled version of the Chebyshev polynomials of the first kind.

>>> x = np.linspace(-2, 2, 6)
>>> eval_chebyc(3, x)
array([-2.   ,  1.872,  1.136, -1.136, -1.872,  2.   ])
>>> 2 * eval_chebyt(3, x / 2)
array([-2.   ,  1.872,  1.136, -1.136, -1.872,  2.   ])