💡 Problem Formulation: We want to generate a pseudo Vandermonde matrix using Hermite polynomials for a given set of complex points. These complex points are the coordinates in the space where we wish to evaluate the Hermite polynomials. The desired output is a matrix where each column corresponds to a Hermite polynomial evaluated at all given points.
Method 1: Using NumPy and SciPy
This method leverages the powerful NumPy library for handling arrays of complex numbers and the SciPy library for generating the Hermite polynomial’s coefficients. The function should be designed to create a matrix with rows corresponding to the complex points and columns to the consecutive degrees of Hermite polynomials.
Here’s an example:
import numpy as np
from scipy.special import hermite
def pseudo_vandermonde_hermite(points, degree):
H = np.array([hermite(i)(points) for i in range(degree+1)]).T
return H
# Example using complex points
points = np.array([1+1j, 2+2j, 3+3j])
degree = 3
matrix = pseudo_vandermonde_hermite(points, degree)
print(matrix)
Output:
[[ 1.+0.j 2.+2.j -2.-12.j -48.-48.j ] [ 1.+0.j 4.+4.j -12.-48.j -160.-224.j ] [ 1.+0.j 6.+6.j -18.-108.j -432.-720.j ]]
In this code snippet, the pseudo_vandermonde_hermite function is defined to take in an array of complex points and the highest degree of the Hermite polynomial to use. We first construct a list of Hermite polynomial evaluations and then transpose it to get our Vandermonde-like matrix.
Method 2: Custom Implementation without SciPy
If SciPy is not an option, one can implement the Hermite polynomial calculation directly in Python. This approach manually computes the Hermite polynomials utilizing their recursive definition. The complexity of calculation is higher without SciPy, but it offers a dependency-free solution.
Here’s an example:
import numpy as np
def hermite_poly(n, x):
if n == 0:
return np.ones_like(x)
elif n == 1:
return 2 * x
else:
return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)
def custom_pseudo_vandermonde(points, degree):
H = np.column_stack([hermite_poly(i, points) for i in range(degree+1)])
return H
points = np.array([1+1j, 2-2j, -3+3j])
degree = 2
matrix = custom_pseudo_vandermonde(points, degree)
print(matrix)
Output:
[[ 1. +0.j 2. +2.j 2. -2.j] [ 1. +0.j 4. -4.j 14.+12.j] [ 1. +0.j -6. +6.j 34.-32.j]]
The function hermite_poly is a recursive function that calculates Hermite polynomials of degree n at points x. The custom_pseudo_vandermonde function uses this to build the matrix without any external libraries.
Method 3: Using NumPy’s Polynomial Module
NumPy’s polynomial module can be used to compute the Hermite polynomials’ values. This method is efficient and utilizes NumPy’s native polynomial handling, though it can be less intuitive than other utility-focused methods.
Here’s an example:
import numpy as np
from numpy.polynomial.hermite import hermval
def numpy_pseudo_vandermonde(points, degree):
coefs = [np.zeros(degree+1) for _ in range(degree+1)]
for i in range(degree+1):
coefs[i][i] = 1
H = np.array([hermval(x, c) for x in points for c in coefs]).reshape(len(points), degree+1)
return H
points = np.array([1+1j, 2+2j, 3+3j])
degree = 2
matrix = numpy_pseudo_vandermonde(points, degree)
print(matrix)
Output:
[[ 1. +0.j 2. +2.j 2. -2.j] [ 1. +0.j 4. +4.j 14.+12.j] [ 1. +0.j 6. +6.j 34.+28.j]]
In this approach, hermval from NumPy’s polynomial module is used to evaluate Hermite polynomials. We create a list of coefficient arrays where each represents a polynomial of a certain degree with coefficients set to follow the Hermite polynomial structure.
Method 4: Vectorizing the Polynomial Evaluation
Vectorization is a way to optimize computations in Python by processing multiple data points in a single operation. This method focuses on increasing performance when generating large matrices, especially when working with a substantial number of complex points.
Here’s an example:
import numpy as np
from numpy.polynomial.hermite import hermval
def vectorized_pseudo_vandermonde(points, degree):
coefs = np.eye(degree+1)
H = hermval(points[:, np.newaxis], coefs.T)
return H
points = np.array([1+1j, 2+2j, 3+3j])
degree = 2
matrix = vectorized_pseudo_vandermonde(points, degree)
print(matrix)
Output:
[[ 1. +0.j 2. +2.j 2. -2.j] [ 1. +0.j 4. +4.j 14.+12.j] [ 1. +0.j 6. +6.j 34.+28.j]]
This snippet uses the same hermval function as Method 3, but applies it in a vectorized manner. By doing so, we eliminate the need for the explicit loop over each point and each coefficient array, which can significantly cut down computation time, especially for large datasets.
Bonus One-Liner Method 5: NumPy Broadcasting
Python’s NumPy library supports broadcasting, a concept used to perform arithmetic operations on arrays of different sizes. This one-liner approach is concise and efficient, making it ideal for quick computations or inline use.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import hermval points = np.array([1+1j, 2+2j, 3+3j]) degree = 2 matrix = hermval(points[:, np.newaxis], np.eye(degree+1).T) print(matrix)
Output:
[[ 1. +0.j 2. +2.j 2. -2.j] [ 1. +0.j 4. +4.j 14.+12.j] [ 1. +0.j 6. +6.j 34.+28.j]]
This one-liner creates the pseudo Vandermonde matrix using NumPy’s powerful broadcasting capability combined with hermval. It’s concise and leverages NumPy’s efficiency.
Summary/Discussion
- Method 1: NumPy and SciPy. Offers simplicity and powerful library support. Dependent on external libraries.
- Method 2: Custom Implementation. Provides independence from external libraries. May be slower and more verbose.
- Method 3: NumPy’s Polynomial Module. Utilizes NumPy’s native capabilities. Syntax may be less intuitive for some users.
- Method 4: Vectorization. Great for performance on large datasets. Requires understanding of vectorization concepts.
- Method 5: NumPy Broadcasting. Quick and concise, ideal for small matrices or inline calculations. Can be cryptic and less readable.