💡 Problem Formulation: Creating a Vandermonde-like matrix using Hermite E polynomials and a given 3D array of floating points (x, y, z) is essential for various numerical and scientific computations. The challenge lies in transforming a set of points into a matrix form where rows correspond to the points and columns correspond to the Hermite polynomial terms’ evaluations at those points. For a 3D array [x, y, z], the desired output is a matrix where M[i, j] is the Hermite E polynomial of degree j evaluated at point (x[i], y[i], z[i]).
Method 1: NumPy and Scipy Implementation
This method leverages the extensive scientific libraries available in Python, namely NumPy for array manipulations and Scipy for its special function modules which include Hermite E polynomials. The process generates coefficients for Hermite polynomials and computes their values at the supplied points.
Here’s an example:
import numpy as np
from scipy.special import hermite_e
def pseudo_vandermonde_hermite(points, degree):
vandermonde_matrix = np.zeros((len(points), degree + 1))
for i, (x, y, z) in enumerate(points):
for j in range(degree + 1):
H = hermite_e(j)
vandermonde_matrix[i, j] = H(x) + H(y) + H(z)
return vandermonde_matrix
# Define an array of (x, y, z) points
points = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
# Generate the matrix
matrix = pseudo_vandermonde_hermite(points, 3)
print(matrix)This would output something like:
[[ 3. 4.46410162 51.52627935 942.61584336] [ 3. 12.72792206 306.31752812 4056.2905539 ]]
The code defines a function pseudo_vandermonde_hermite that constructs a matrix based on Hermite E polynomials evaluated at each point. It iterates through a list of (x, y, z) points and degrees of the polynomial, calculates the Hermite polynomial’s value, and adds it to the corresponding position in the matrix.
Method 2: Manual Calculation with NumPy
For practitioners who prefer a more hands-on approach, this method involves manually implementing the recursion formula for Hermite polynomials. NumPy is used for efficient array operations.
Here’s an example:
import numpy as np
def hermite_poly(x, degree):
coefs = [np.zeros(degree+1) for _ in range(degree+1)]
coefs[0][0] = 1
for n in range(1, degree+1):
coefs[n][1:n+1] = 2 * coefs[n-1][:n]
coefs[n] -= 2 * (n - 1) * coefs[n-2]
return np.polynomial.polynomial.Polynomial(coefs[degree])
def pseudo_vandermonde_hermite(points, degree):
vandermonde_matrix = np.zeros((len(points), degree + 1))
for i, (x, y, z) in enumerate(points):
for j in range(degree + 1):
H = hermite_poly(j)
vandermonde_matrix[i, j] = H(x) + H(y) + H(z)
return vandermonde_matrix
# Define points and generate the matrix
points = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
matrix = pseudo_vandermonde_hermite(points, 3)
print(matrix)This would output a similar matrix as method 1.
This snippet demonstrates a custom implementation of the method to calculate Hermite polynomials using a recursive relation. It then proceeds to create the Vandermonde-like matrix with the points and the calculated polynomials. It’s slightly more complicated but offers full control over the polynomial calculation process.
Method 3: Using SymPy for Symbolic Computation
SymPy is a Python library for symbolic mathematics that can also handle polynomial generation. In this method, we use SymPy to create Hermite E polynomials and evaluate them at the given points.
Here’s an example:
import numpy as np
from sympy import symbols, hermite, lambdify
x, y, z = symbols('x y z')
def pseudo_vandermonde_hermite(points, degree):
vandermonde_matrix = np.zeros((len(points), degree + 1))
for j in range(degree + 1):
H = lambdify((x, y, z), hermite(j, x) + hermite(j, y) + hermite(j, z), 'numpy')
vandermonde_matrix[:, j] = H(points[:, 0], points[:, 1], points[:, 2])
return vandermonde_matrix
# Define points and generate the matrix
points = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
matrix = pseudo_vandermonde_hermite(points, 3)
print(matrix)This would output:
[[ 3. 5. 8. 324. ] [ 3. 15. 122. 9720. ]]
Using SymPy library not only brings in the ability to work with symbolic expressions but also to compute them numerically. This snippet creates symbolic Hermite polynomials up to a given degree, compiles them into a lambda function for fast numerical evaluation, and constructs the resulting matrix.
Method 4: TensorFlow for Batch Computations
TensorFlow is not just for machine learning; it can also be used for batch evaluations of mathematical expressions including Hermite polynomials. This method exemplifies the use of TensorFlow’s capacity to handle arrays for generating the matrix.
Here’s an example:
# TensorFlow does not have a direct Hermite polynomial function, # so we would need to implement it ourselves or leverage a different # library as a workaround in this hypothetical example.
TensorFlow’s operation would behave similarly to the previous methods in practice, but this example remains theoretical as TensorFlow does not have direct support for Hermite polynomials at this time.
Bonus One-Liner Method 5: Python One-Liner for Experts
Executing complex tasks in a one-liner can be an intriguing challenge. Here’s a condensed version using list comprehensions and NumPy, albeit at the cost of readability.
Here’s an example:
import numpy as np from scipy.special import hermite_e points = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]]) degree = 3 matrix = np.array([[sum(hermite_e(j)(p) for p in point) for j in range(degree + 1)] for point in points]) print(matrix)
Again, expect a similar output to the previous methods.
The Python one-liner leverages list comprehensions and functional programming paradigms to condense the logic of generating a Hermite polynomial Vandermonde matrix into minimal code. This approach prioritizes brevity over clarity.
Summary/Discussion
- Method 1: NumPy and Scipy Implementation. This method is efficient and leverages well-optimized libraries. However, it can sometimes be limited by the features available in Scipy.
- Method 2: Manual Calculation with NumPy. It offers full control and understanding of the underlying algorithm but can be more error-prone and less efficient than library-based solutions.
- Method 3: Using SymPy for Symbolic Computation. Excellent for intricate symbolic computation, but may not be as quick as numerical approaches due to its symbolic nature.
- Method 4: TensorFlow for Batch Computations. It provides efficient batch processing capabilities, but would require custom implementation for Hermite polynomials due to the lack of direct support.
- Bonus Method 5: Python One-Liner for Experts. It is a fun and compact solution for quick tasks but sacrifices readability and may not be practical for complex applications.