Evaluate a Hermite series at array of points x in Python

To evaluate a Hermite series at points x, use the hermite.hermval() method in Python NumPy. This function allows you to compute the value of a Hermite polynomial at specified points using an array of coefficients.

Syntax

numpy.polynomial.hermite.hermval(x, c, tensor=True)

Parameters

The function accepts three parameters ?

• x ? Array of points where the Hermite series will be evaluated. If x is a list or tuple, it is converted to an ndarray
• c ? Array of coefficients ordered so that coefficients for terms of degree n are contained in c[n]
• tensor ? If True (default), evaluates every column of coefficients for every element of x. If False, broadcasts x over columns of c

Example

Let's evaluate a Hermite series with coefficients [1, 2, 3] at various points ?

import numpy as np
from numpy.polynomial import hermite as H

# Create an array of coefficients
c = np.array([1, 2, 3])

# Display the coefficient array
print("Coefficients:", c)
print("Dimensions:", c.ndim)
print("Shape:", c.shape)

# Evaluate at a 2D array of points
x = np.array([[1, 2], [3, 4]])
print("\nPoints x:")
print(x)

# Evaluate the Hermite series
result = H.hermval(x, c)
print("\nHermite series evaluation:")
print(result)

Coefficients: [1 2 3]
Dimensions: 1
Shape: (3,)

Points x:
[[1 2]
 [3 4]]

Hermite series evaluation:
[[ 11.  51.]
 [115. 203.]]

How It Works

The Hermite series is evaluated using the formula: c[0] + c[1]*H?(x) + c[2]*H?(x) + ..., where H?(x) are the Hermite polynomials. For our example with coefficients [1, 2, 3], the series becomes: 1 + 2*H?(x) + 3*H?(x).

Using Different Point Arrays

You can evaluate the series at single points or 1D arrays as well ?

import numpy as np
from numpy.polynomial import hermite as H

c = np.array([1, 2, 3])

# Evaluate at a single point
single_point = H.hermval(2, c)
print("At x=2:", single_point)

# Evaluate at a 1D array
x_array = np.array([0, 1, 2])
result_1d = H.hermval(x_array, c)
print("At points [0, 1, 2]:", result_1d)

At x=2: 51.0
At points [0, 1, 2]: [ 1. 11. 51.]

Conclusion

The hermite.hermval() function efficiently evaluates Hermite series at specified points using coefficient arrays. It supports both scalar and array inputs, making it versatile for mathematical computations involving Hermite polynomials.