Generate a Vandermonde matrix of the Hermite polynomial in Python

To generate a Vandermonde matrix of the Hermite polynomial, use the hermite.hermvander() function in Python NumPy. This method returns a pseudo-Vandermonde matrix where each row corresponds to a point and each column represents increasing degrees of Hermite polynomials.

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

Parameters

x: Array of points where the dtype is converted to float64 or complex128 depending on whether any elements are complex. Scalar values are converted to 1-D arrays.

deg: The degree of the resulting matrix, determining the number of polynomial columns.

Syntax

numpy.polynomial.hermite.hermvander(x, deg)

Example

Let's create a Vandermonde matrix for Hermite polynomials up to degree 2 ?

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

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

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

# Check the Dimensions
print("\nDimensions of our Array...\n", x.ndim)

# Get the Datatype
print("\nDatatype of our Array object...\n", x.dtype)

# Get the Shape
print("\nShape of our Array object...\n", x.shape)

# To generate a Vandermonde matrix of the Hermite polynomial, use hermite.hermvander()
print("\nResult...\n", H.hermvander(x, 2))

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

Dimensions of our Array...
1

Datatype of our Array object...
int64

Shape of our Array object...
(4,)

Result...
 [[ 1.  0. -2.]
 [ 1.  2.  2.]
 [ 1. -2.  2.]
 [ 1.  4. 14.]]

How It Works

The Vandermonde matrix contains Hermite polynomial values where:

• Column 0: H?(x) = 1 (degree 0)
• Column 1: H?(x) = 2x (degree 1)
• Column 2: H?(x) = 4x² - 2 (degree 2)

For each input point, the function evaluates all Hermite polynomials up to the specified degree.

Conclusion

The hermite.hermvander() function generates a Vandermonde matrix for Hermite polynomials, useful in polynomial fitting and numerical analysis. Each row represents evaluations at a specific point, while columns represent increasing polynomial degrees.