Evaluate a Hermite series at points x in Python

To evaluate a Hermite series at points x, use the hermite.hermval() method in Python NumPy. This function evaluates a Hermite polynomial series at given points using the coefficients provided.

Syntax

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

Parameters

The function accepts three parameters ?

• x: Points at which to evaluate the Hermite series. Can be a scalar, list, or array.
• c: Array of coefficients ordered so that coefficients for terms of degree n are in c[n].
• tensor: If True (default), evaluates each coefficient column for every element of x. If False, broadcasts x over coefficient columns.

Basic Example

Let's start with a simple example evaluating a Hermite series at a single point ?

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

# Create coefficient array [1, 2, 3] representing: 1 + 2*H_1(x) + 3*H_2(x)
c = np.array([1, 2, 3])

print("Coefficient array:", c)
print("Array dimensions:", c.ndim)
print("Array shape:", c.shape)
print("Array datatype:", c.dtype)

# Evaluate at x = 1
result = H.hermval(1, c)
print("\nHermite series value at x=1:", result)

Coefficient array: [1 2 3]
Array dimensions: 1
Array shape: (3,)
Array datatype: int64

Hermite series value at x=1: 11.0

Evaluating at Multiple Points

You can evaluate the Hermite series at multiple points simultaneously ?

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

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

# Evaluate at multiple points
x_points = np.array([0, 1, 2, -1])
results = H.hermval(x_points, c)

print("Coefficients:", c)
print("Evaluation points:", x_points)
print("Results:", results)

# Show individual evaluations
for i, (x, result) in enumerate(zip(x_points, results)):
    print(f"At x={x}: {result}")

Coefficients: [1 2 3]
Evaluation points: [ 0  1  2 -1]
Results: [ 1. 11. 37. -1.]

At x=0: 1.0
At x=1: 11.0
At x=2: 37.0
At x=-1: -1.0

Understanding the Calculation

The Hermite series is calculated as: c[0] + c[1]*H?(x) + c[2]*H?(x) + ... where H?(x) are Hermite polynomials ?

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

# For coefficients [1, 2, 3], the series is: 1 + 2*H_1(x) + 3*H_2(x)
c = np.array([1, 2, 3])

# Let's verify the calculation manually at x=1
# H_0(x) = 1, H_1(x) = 2x, H_2(x) = 4x² - 2
x = 1
manual_calc = 1*1 + 2*(2*x) + 3*(4*x**2 - 2)
hermval_result = H.hermval(x, c)

print(f"Manual calculation at x=1: {manual_calc}")
print(f"hermval() result at x=1: {hermval_result}")
print(f"Results match: {manual_calc == hermval_result}")

Manual calculation at x=1: 11.0
hermval() result at x=1: 11.0
Results match: True

Conclusion

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