Generate a Vandermonde matrix of the Hermite_e polynomial in Python

To generate a Vandermonde matrix of the Hermite_e polynomial, use the hermite_e.hermevander() function in Python NumPy. This method returns a pseudo-Vandermonde matrix where each row corresponds to evaluating Hermite_e polynomials of increasing degrees at a given point.

The shape of the returned matrix is x.shape + (deg + 1,), where the last index represents the degree of the corresponding Hermite_e polynomial. The dtype will match the converted input array x.

Parameters

The function accepts two main parameters:

• x ? Array of points where polynomials are evaluated. Converted to float64 or complex128 depending on element types
• deg ? Degree of the resulting matrix, determining the number of polynomial columns

Basic Example

Here's how to generate a Hermite_e Vandermonde matrix ?

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

# Create an array of points
x = np.array([0, 1, -1, 2])

# Display the array
print("Our Array...")
print(x)

# Check array properties
print("\nDimensions:", x.ndim)
print("Datatype:", x.dtype)
print("Shape:", x.shape)

# Generate Vandermonde matrix of degree 2
result = H.hermevander(x, 2)
print("\nVandermonde Matrix:")
print(result)

Our Array...
[ 0  1 -1  2]

Dimensions: 1
Datatype: int64
Shape: (4,)

Vandermonde Matrix:
[[ 1.  0. -1.]
 [ 1.  1.  0.]
 [ 1. -1.  0.]
 [ 1.  2.  3.]]

Understanding the Output

Each column represents a Hermite_e polynomial degree:

• Column 0 ? Hermite_e polynomial of degree 0 (always 1)
• Column 1 ? Hermite_e polynomial of degree 1 (equals x)
• Column 2 ? Hermite_e polynomial of degree 2 (equals x² - 1)

Different Degrees

You can specify different degrees to get more polynomial evaluations ?

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

x = np.array([0, 1, 2])

# Degree 1 matrix
print("Degree 1 matrix:")
print(H.hermevander(x, 1))

print("\nDegree 3 matrix:")
print(H.hermevander(x, 3))

Degree 1 matrix:
[[1. 0.]
 [1. 1.]
 [1. 2.]]

Degree 3 matrix:
[[ 1.  0. -1. -0.]
 [ 1.  1.  0. -2.]
 [ 1.  2.  3.  2.]]

Conclusion

The hermite_e.hermevander() function efficiently generates Vandermonde matrices for Hermite_e polynomials. Each row evaluates polynomials of degrees 0 to deg at corresponding x values, making it useful for polynomial fitting and numerical computations.