Evaluate a Hermite_e series at points x in Python

To evaluate a Hermite_e series at points x, use the hermite_e.hermeval() method in Python NumPy. This function computes the value of a Hermite_e polynomial series at specified points using the provided coefficients.

Syntax

numpy.polynomial.hermite_e.hermeval(x, c, tensor=True)

Parameters

The hermeval() function accepts the following parameters ?

• x: Array of points at which to evaluate the series. Can be a scalar, list, tuple, or ndarray
• c: Array of coefficients ordered so that coefficients for terms of degree n are in c[n]
• tensor: Boolean flag (default True) controlling how multidimensional coefficients are handled

Basic Example

Let's evaluate a simple Hermite_e series with coefficients [1, 2, 3] at point x = 1 ?

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

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

# Evaluate at x = 1
result = H.hermeval(1, c)
print("H(1) =", result)

Coefficients: [1 2 3]
H(1) = 3.0

Evaluating at Multiple Points

You can evaluate the series at multiple points by passing an array of x values ?

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

# Coefficients for the series
c = np.array([1, 2, 3])

# Evaluate at multiple points
x_values = np.array([0, 1, 2, 3])
results = H.hermeval(x_values, c)

print("x values:", x_values)
print("H(x) values:", results)

x values: [0 1 2 3]
H(x) values: [ 1.  3. 13. 37.]

How It Works

The Hermite_e polynomial series is evaluated using the formula ?

H(x) = c? + c?·He?(x) + c?·He?(x) + ... + c?·He?(x)

Where He?(x) are the probabilists' Hermite polynomials (Hermite_e polynomials).

Working with Multidimensional Coefficients

The function can handle multidimensional coefficient arrays for evaluating multiple series simultaneously ?

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

# Multidimensional coefficients (2 series)
c = np.array([[1, 2], [3, 4], [5, 6]])
print("Coefficient matrix shape:", c.shape)

# Evaluate both series at x = 1
results = H.hermeval(1, c)
print("Results for both series:", results)

Coefficient matrix shape: (3, 2)
Results for both series: [ 6. 10.]

Key Points

• Coefficients are ordered by degree: c[0] for degree 0, c[1] for degree 1, etc.
• The function automatically handles scalar and array inputs for x
• Multidimensional coefficient arrays allow evaluation of multiple series
• The tensor parameter controls broadcasting behavior for multidimensional inputs

Conclusion

The hermite_e.hermeval() function provides an efficient way to evaluate Hermite_e polynomial series at specified points. It supports both single and multiple point evaluation with flexible coefficient handling for complex mathematical computations.