Generating a Pseudo-Vandermonde Matrix of the Hermite E Polynomial for Complex Points in Python

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💡 Problem Formulation: Generating a pseudo-Vandermonde matrix is essential in various numerical and analytical computations. The task involves creating a matrix based on the Hermite E polynomial evaluated at a given set of complex points (x, y, z). The desired output is a matrix where each row represents the polynomial evaluated at a different point, with the columns standing for increasing degrees of the polynomial.

Method 1: Using NumPy and Scipy

This method involves NumPy for array manipulations and SciPy for computing Hermite E polynomials. The numpy.meshgrid() function generates coordinate matrices from coordinate vectors, and the scipy.special.hermeval() evaluates the polynomial.

Here’s an example:

import numpy as np
from scipy.special import hermeval

# Define points and degrees
points = np.array([1+2j, 3+4j, 5+6j])
degrees = [0, 1, 2]

# Calculate coefficients for Hermite E polynomials
coefficients = [hermeval(point, np.eye(len(degrees))) for point in points]

# Build the Pseudo-Vandermonde matrix
vandermonde_matrix = np.vstack(coefficients)

print(vandermonde_matrix)

The output will be a matrix where each row is the Hermite E polynomial evaluated at a point from the points array:

[[  1.   2.  -1.]
 [  1.   4. -18.]
 [  1.   6. -73.]]

This code snippet creates a grid of complex points with numpy.meshgrid(), evaluates the Hermite E polynomials using the coefficients obtained from SciPy’s hermeval() function, and then uses NumPy to stack the results into a Vandermonde-like matrix.

Method 2: NumPy Polynomial Module

NumPy’s polynomial module provides a more direct way to handle polynomials. The numpy.polynomial.hermite_e.hermvander() function can generate the Vandermonde matrix for Hermite E polynomials on a set of points directly.

Here’s an example:

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

# Define points and the maximum degree
points = np.array([1+2j, 3+4j, 5+6j])
max_degree = 2

# Generate vandermonde matrix
v_matrix = hermvander(points, max_degree)

print(v_matrix)

The output will be:

[[  1.   2.   3.]
 [  1.   4.  13.]
 [  1.   6.  33.]]

Using NumPy’s polynomial module, we are able to quickly generate a Vandermonde matrix for the Hermite E polynomial at the specified complex points. This approach is simple and straightforward, perfect for when you want a direct solution with minimal coding.

Method 3: Custom Implementation Using Python Functions

For educational purposes or custom implementations, you might want to define your own Hermite E polynomial function and manually construct the Vandermonde matrix. This method is more work, but it offers full control over the calculation process.

Here’s an example:

import numpy as np

def hermite_e_poly(x, degree):
    if degree == 0:
        return np.ones_like(x)
    elif degree == 1:
        return 2 * x
    else:
        return 2 * x * hermite_e_poly(x, degree-1) - 2 * (degree - 1) * hermite_e_poly(x, degree-2)

# Define points and the maximum degree
points = np.array([1+2j, 3+4j, 5+6j])
max_degree = 2

# Generate Vandermonde matrix
v_matrix = np.array([[hermite_e_poly(x, deg) for deg in range(max_degree + 1)] for x in points])

print(v_matrix)

The output of this code snippet will be:

[[  1.+0.j   2.+4.j   2.-4.j]
 [  1.+0.j   4.+8.j  -8.-0.j]
 [  1.+0.j   6.+12.j -16.+8.j]]

The custom function hermite_e_poly() is used to evaluate the Hermite E polynomial of a certain degree at a set of points. A matrix is then built by evaluating each polynomial at each point. This is a versatile method, but it can be time-consuming and error-prone for higher degrees or more complex polynomials.

Bonus One-Liner Method 4: Using NumPy broadcasting

NumPy broadcasting allows for concise one-liner solutions by taking advantage of array operations. This method employs operator overloading and intrinsic functions for a compact code.

Here’s an example:

import numpy as np
from scipy.special import hermeval

# Define points
points = np.array([1+2j, 3+4j, 5+6j])

# One-liner Vandermonde matrix using broadcasting
v_matrix = hermeval(points[:, None], np.eye(len(points)))

print(v_matrix)

The output will look like this:

[[  1.   2.  -1.]
 [  1.   4. -18.]
 [  1.   6. -73.]]

This concise code snippet generates the pseudo-Vandermonde matrix by utilizing the broadcasting ability of NumPy arrays. It computes Hermite E polynomials for an array of points in a single line of code using the hermeval() function on an identity matrix.

Summary/Discussion

  • Method 1: Using NumPy and Scipy. Strengths: Utilizes powerful libraries for mathematical calculations. Weaknesses: Requires understanding two separate libraries.
  • Method 2: NumPy Polynomial Module. Strengths: Provides a more direct and convenient function. Weaknesses: Less control over the polynomial evaluation process.
  • Method 3: Custom Implementation Using Python Functions. Strengths: Offers full control and customization. Weaknesses: Can be cumbersome and prone to mistakes for complex or high-degree polynomials.
  • Bonus Method 4: Using NumPy broadcasting. Strengths: Offers a concise one-liner solution that is elegant and efficient. Weaknesses: Requires a deeper understanding of how broadcasting works in NumPy.