scipy.special.eval_legendre#

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

Evaluate Legendre polynomial at a point.

The Legendre polynomials can be defined via the Gauss hypergeometric function \({}_2F_1\) as

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

When \(n\) is an integer the result is a polynomial of degree \(n\). See 22.5.49 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 Legendre polynomial
outndarray, optional: Optional output array for the function values

Returns:

Pscalar or ndarray: Values of the Legendre polynomial

See also

roots_legendre: roots and quadrature weights of Legendre polynomials
legendre: Legendre polynomial object
hyp2f1: Gauss hypergeometric function
numpy.polynomial.legendre.Legendre: Legendre series

Notes

Array API Standard Support

eval_legendre 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/15.9.E7

Examples

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

Evaluate the zero-order Legendre polynomial at x = 0

>>> eval_legendre(0, 0)
1.0

Evaluate the first-order Legendre polynomial between -1 and 1

>>> x = np.linspace(-1, 1, 5)  # Domain of Legendre polynomials
>>> eval_legendre(1, x)
array([-1. , -0.5,  0. ,  0.5,  1. ])

Evaluate Legendre polynomials of order 0 through 4 at x = 0

>>> eval_legendre(range(5), 0)
array([ 1.   ,  0.   , -0.5  ,  0.   ,  0.375])

Plot Legendre 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_legendre(n, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Legendre Polynomials")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$P_n(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

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