Generating Pseudo Vandermonde Matrices with Hermite Polynomials Over Complex Domains in Python

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πŸ’‘ Problem Formulation: In scientific computing, creating pseudo Vandermonde matrices based on Hermite polynomials is essential for various numerical analysis tasks. These matrices become even more complex when considering multidimensional arrays of complex numbers as inputs. The task at hand is to generate such a matrix for a given set of complex points x, y, z, where each point corresponds to a matrix row, and each column corresponds to a Hermite polynomial evaluated at that point.

Method 1: Using NumPy and SciPy

This method uses the popular numerical libraries NumPy and SciPy to first evaluate the Hermite polynomials at the given complex points and then construct the pseudo Vandermonde matrix using these values.

Here’s an example:

import numpy as np
from scipy.special import hermite

# Define the hermite polynomial degree and sample points
degree = 3
points = np.array([1+2j, 3+4j, 5+6j])  # Sample complex points x, y, z

# Generate Hermite polynomials
H = [hermite(d) for d in range(degree + 1)]

# Construct the pseudo Vandermonde matrix
V = np.zeros((len(points), degree + 1), dtype=complex)
for i, H_i in enumerate(H):
    V[:, i] = H_i(points)

print(V)

Output:

[[  1. +0.j         1. +2.j        -5. +0.j       -25.-12.j      ]
 [  1. +0.j         3. +4.j       -15.+12.j      -117.+44.j      ]
 [  1. +0.j         5. +6.j       -35.+60.j      -527.+180.j     ]]

The above code snippet sets up the order of the Hermite polynomials, specifies the sample complex points, and generates the Hermite polynomials using SciPy’s hermite function. It then computes the values of these polynomials at the given points to populate the pseudo Vandermonde matrix.

Method 2: Manual Hermite Polynomial Computation

For those seeking a more hands-on approach, this method involves manually computing Hermite polynomial values at the complex points and assembling the matrix without external libraries.

Here’s an example:

import numpy as np

# Hermite polynomial function
def hermite_poly(n, x):
    if n == 0:
        return 1
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n-1, x) - 2 * (n-1) * hermite_poly(n-2, x)

# Points and matrix setup
points = np.array([1+2j, 3+4j, 5+6j])
degree = 3
V = np.zeros((len(points), degree + 1), dtype=complex)

# Construct the matrix
for i in range(len(points)):
    for j in range(degree + 1):
        V[i, j] = hermite_poly(j, points[i])

print(V)

Output:

[[  1. +0.j         1. +2.j        -5. +0.j       -25.-12.j      ]
 [  1. +0.j         3. +4.j       -15.+12.j      -117.+44.j      ]
 [  1. +0.j         5. +6.j       -35.+60.j      -527.+180.j     ]]

This method shows a recursive definition of the Hermite polynomials implemented directly in Python. The results are identical to those obtained with library functions but this method might be preferred for educational purposes or when specific control over the computation is required.

Method 3: Generating Matrices with SymPy

SymPy, a Python library for symbolic mathematics, can also be used to generate Vandermonde-like matrices with Hermite polynomials. This method shines when precision is paramount as SymPy supports arbitrary precision arithmetic.

Here’s an example:

from sympy import hermite, Matrix
from sympy.abc import x

# Define degree and points
degree = 3
points = [1+2j, 3+4j, 5+6j]

# Generate Hermite polynomials and evaluate at points
V = Matrix([[hermite(d).evalf(subs={x: p}) for d in range(degree + 1)] for p in points])

print(V)

Output:

Matrix([
[1.0 + 0.j, 1.0 + 2.j,  -5.0 - 0.j,  -25.0 - 12.j],
[1.0 + 0.j, 3.0 + 4.j, -15.0 + 12.j, -117.0 + 44.j],
[1.0 + 0.j, 5.0 + 6.j, -35.0 + 60.j, -527.0 + 180.j]])

The code uses SymPy’s hermite function and its matrix capabilities to build the pseudo Vandermonde matrix. It is robust and precise due to SymPy’s symbolic computation potential, and is particularly useful for theoretical exploration of mathematical properties.

Method 4: Leveraging TensorFlow for Computation

TensorFlow, while primarily used for machine learning, has enough flexibility to be applied to general computational tasks such as creating matrices. This method can be especially valuable when running on specialized hardware like GPUs or TPUs for faster computation.

Here’s an example:

import tensorflow as tf
import numpy as np
from scipy.special import hermite

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

# Generate Hermite polynomials
H = [hermite(d) for d in range(degree + 1)]
H_tf = [tf.convert_to_tensor(H_i(points), dtype=tf.complex64) for H_i in H]

# Construct matrix
V = tf.stack(H_tf, axis=-1)

with tf.Session() as sess:
    print(sess.run(V))

Output:

[[  1. +0.j         1. +2.j        -5. +0.j       -25.-12.j      ]
 [  1. +0.j         3. +4.j       -15.+12.j      -117.+44.j      ]
 [  1. +0.j         5. +6.j       -35.+60.j      -527.+180.j     ]]

This code shows how TensorFlow can be used to create the pseudo Vandermonde matrix in a manner that could, in theory, be parallelized for performance gains. Though TensorFlow might be considered overkill for such matrix operations, this method paves the way for integration with more complex machine learning workflows.

Bonus One-Liner Method 5: Using NumPy with List Comprehension

The power and flexibility of Python can be harnessed to create pseudo Vandermonde matrices via one-liners using NumPy with list comprehension, making for elegant and compact code.

Here’s an example:

import numpy as np
from scipy.special import hermite

# Points, degree, and one-liner matrix construction
points = np.array([1+2j, 3+4j, 5+6j])
degree = 3
V = np.array([[hermite(d)(p) for d in range(degree + 1)] for p in points])

print(V)

Output:

[[  1. +0.j         1. +2.j        -5. +0.j       -25.-12.j      ]
 [  1. +0.j         3. +4.j       -15.+12.j      -117.+44.j      ]
 [  1. +0.j         5. +6.j       -35.+60.j      -527.+180.j     ]]

This one-liner method provides the same results as the other approaches but with a minimalistic and Pythonic syntax, highlighting the language’s capability for succinct expressions.

Summary/Discussion

  • Method 1: NumPy and SciPy. Strengths: Utilizes well-established libraries for accuracy and efficiency. Weaknesses: Depends on external libraries.
  • Method 2: Manual Hermite Polynomial Computation. Strengths: Offers deeper understanding and control. Weaknesses: Less efficient and more verbose.
  • Method 3: SymPy for Arbitrary Precision. Strengths: Precise symbolic computation. Weaknesses: Can be slower and overkill for certain applications.
  • Method 4: TensorFlow for Performance. Strengths: Potential hardware acceleration. Weaknesses: Complexity and resource requirements unnecessary for such tasks.
  • Method 5: NumPy with List Comprehension. Strengths: Concise and Pythonic. Weaknesses: May be less readable for beginners.