scipy.special.eval_sh_chebyt#

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

Evaluate shifted Chebyshev polynomial of the first kind at a point.

These polynomials are defined as

\[T_n^*(x) = T_n(2x - 1)\]

where \(T_n\) is a Chebyshev polynomial of the first kind. See 22.5.14 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 shifted Chebyshev polynomial
outndarray, optional: Optional output array for the function values

Returns:

Tscalar or ndarray: Values of the shifted Chebyshev polynomial

See also

roots_sh_chebyt: roots and quadrature weights of shifted Chebyshev polynomials of the first kind
sh_chebyt: shifted Chebyshev polynomial object
eval_chebyt: evaluate Chebyshev polynomials of the first kind
numpy.polynomial.chebyshev.Chebyshev: Chebyshev series

Notes

Array API Standard Support

eval_sh_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/18.7.E7

Examples

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

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

>>> eval_sh_chebyt(3, [0, 0.125, 0.25, 0.375, 0.5])
array([-1.    ,  0.5625,  1.    ,  0.6875, -0.    ])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(0, 1, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_sh_chebyt(n, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Shifted 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_sh_chebyt-1.png