scipy.special.eval_hermite

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

Evaluate physicist’s Hermite polynomial at a point.

Defined by

\[H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2};\]

\(H_n\) is a polynomial of degree \(n\). See 22.11.7 in [AS] or [DLMF] for details.

Parameters:

narray_like

Degree of the polynomial

xarray_like

Points at which to evaluate the Hermite polynomial

outndarray, optional

Optional output array for the function values

Returns:

Hscalar or ndarray

Values of the Hermite polynomial

See also

roots_hermite

roots and quadrature weights of physicist’s Hermite polynomials

hermite

physicist’s Hermite polynomial object

numpy.polynomial.hermite.Hermite

Physicist’s Hermite series

eval_hermitenorm

evaluate Probabilist’s Hermite polynomials

Notes

Array API Standard Support

eval_hermite 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.T1

Examples

Try it in your browser!

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

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

>>> eval_hermite(3, [0, 0.25, 0.5])
array([-0.   , -2.875, -5.   ])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(-1.6, 1.6, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_hermite(n, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Physicist's Hermite Polynomials")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$H_n(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

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