Generate a Vandermonde matrix of the Hermite_e polynomial with complex array of points in Python

To generate a Vandermonde matrix of the Hermite_e polynomial, use the hermite_e.hermvander() function in Python NumPy. The method returns the pseudo-Vandermonde matrix where each row corresponds to a point in the input array, and each column corresponds to a polynomial degree.

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 will be the same as the converted input array.

Parameters

x: Array of points. The dtype is converted to float64 or complex128 depending on whether any elements are complex. If x is scalar, it is converted to a 1-D array.

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

Syntax

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

Example

Let's create a Vandermonde matrix using complex array points ?

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

# Create a complex array
x = np.array([-2.+2.j, -1.+2.j, 0.+2.j, 1.+2.j, 2.+2.j])

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

# Check the dimensions
print("\nDimensions of our Array...")
print(x.ndim)

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

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

# Generate Vandermonde matrix of Hermite_e polynomial with degree 2
print("\nVandermonde Matrix (degree 2)...")
result = H.hermvander(x, 2)
print(result)

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

Dimensions of our Array...
1

Datatype of our Array object...
complex128

Shape of our Array object...
(5,)

Vandermonde Matrix (degree 2)...
[[ 1.+0.j -2.+2.j -1.-8.j]
 [ 1.+0.j -1.+2.j -4.-4.j]
 [ 1.+0.j  0.+2.j -5.+0.j]
 [ 1.+0.j  1.+2.j -4.+4.j]
 [ 1.+0.j  2.+2.j -1.+8.j]]

How It Works

The Vandermonde matrix contains evaluations of Hermite_e polynomials of different degrees at each point:

• Column 0: H_0(x) = 1 (constant polynomial)
• Column 1: H_1(x) = x (linear polynomial)
• Column 2: H_2(x) = x² - 1 (quadratic polynomial)

Each row corresponds to evaluating these polynomials at one complex point from the input array.

Conclusion

Use hermite_e.hermvander() to generate Vandermonde matrices for Hermite_e polynomials with complex arrays. The function efficiently evaluates multiple polynomial degrees simultaneously, making it useful for polynomial fitting and numerical analysis tasks.