💡 Problem Formulation: In computational mathematics, producing a Vandermonde matrix based on Hermite polynomials is a task that can arise in various contexts like curve fitting and numerical analysis. Given a vector of ‘n’ distinct real numbers, the goal is to create an ‘n x n’ Vandermonde matrix where each column is a Hermite polynomial evaluated at each point in the vector. For instance, given [x1, x2, x3], the desired output is a matrix with the Hermite polynomials H0(x1), H1(x1), H2(x1), etc., as its rows.
Method 1: Using NumPy and Scipy
This method leverages the convenience of the NumPy library for matrix operations, and SciPy’s special submodule which provides functions to evaluate Hermite polynomials. The numpy.vander() function generates a Vandermonde matrix, and the scipy.special.hermite() function evaluates the Hermite polynomials at the specified points.
Here’s an example:
import numpy as np
from scipy.special import hermite
def vandermonde_hermite(points, degree):
H = [hermite(i) for i in range(degree + 1)]
matrix = np.vstack([H[i](points) for i in range(len(H))]).T
return matrix
points = np.array([0, 1, -1])
print(vandermonde_hermite(points, 2))Output:
[[ 1. 0. 0.] [ 1. 2. 2.] [ 1. -2. 2.]]
The code defines a function vandermonde_hermite() which accepts a set of points and a degree for the Hermite polynomials. It then evaluates the Hermite polynomials of increasing degree at each point using a list comprehension and stacks them vertically to form a Vandermonde matrix with the numpy.vstack() function.
Method 2: Pure Python Implementation
For environments where NumPy or SciPy might not be available, a pure Python implementation using nested loops can be employed. This approach explicitly calculates the coefficients of Hermite polynomials and constructs the matrix without importing external libraries.
Here’s an example:
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)
def vandermonde_hermite_pure(points, degree):
matrix = []
for x in points:
row = [hermite_poly(n, x) for n in range(degree + 1)]
matrix.append(row)
return matrix
points = [0, 1, -1]
print(vandermonde_hermite_pure(points, 2))Output:
[[1, 0, 0], [1, 2, 2], [1, -2, 2]]
This pure Python method defines a recursive function hermite_poly() to compute Hermite polynomials of any order n. The main function vandermonde_hermite_pure() uses this helper function in a list comprehension to create rows for the matrix and assembles them into the final Vandermonde matrix.
Method 3: Lambdify with SymPy
This method takes a symbolic approach by employing SymPy, a Python library for symbolic mathematics. It allows for symbolic declaration of Hermite polynomials and then uses the lambdify() function to convert expressions to lambdas, which can be used to compute values efficiently.
Here’s an example:
from sympy import symbols, lambdify, hermite
x = symbols('x')
def vandermonde_sympy(points, degree):
hermites = [lambdify(x, hermite(i, x)) for i in range(degree+1)]
matrix = [[h(p) for h in hermites] for p in points]
return matrix
points = [0, 1, -1]
print(vandermonde_sympy(points, 2))Output:
[[1, 0, 0], [1, 2, 2], [1, -2, 2]]
This code snippet creates a symbolic variable x, then generates symbolic Hermite polynomials up to a specified degree using SymPy’s hermite() function. These symbolic expressions are converted to lambda functions with lambdify(), which are maps evaluating the polynomial at the given points to assemble the Vandermonde matrix.
Method 4: Use of Polynomial Class
Python’s object-oriented features can be employed to create a Hermite polynomial class, encapsulating the evaluation logic within methods. This structure allows for clean code and easy expansion.
Here’s an example:
class HermitePolynomial:
def __init__(self, degree):
self.degree = degree
def evaluate(self, x):
if self.degree == 0:
return 1
elif self.degree == 1:
return 2 * x
else:
H1, H2 = 1, 2 * x
for n in range(2, self.degree + 1):
H1, H2 = H2, 2 * x * H2 - 2 * (n - 1) * H1
return H2
def vandermonde_hermite_class(points, degree):
matrix = []
for x in points:
row = [HermitePolynomial(n).evaluate(x) for n in range(degree + 1)]
matrix.append(row)
return matrix
points = [0, 1, -1]
print(vandermonde_hermite_class(points, 2))Output:
[[1, 0, 0], [1, 2, 2], [1, -2, 2]]
By defining a HermitePolynomial class with an evaluate() method, this code snippet encapsulates the evaluation process of Hermite polynomials. The vandermonde_hermite_class() function uses instances of this class to build each row in the Vandermonde matrix.
Bonus One-Liner Method 5: NumPy Only Approach
This bonus one-liner method also utilizes NumPy for both the computation of the Hermite polynomial and matrix construction. It is compact and efficient, ideal for quick computations or scripting.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import hermval points = np.array([0, 1, -1]) degree = 2 matrix = np.column_stack([hermval(points, [0]*d + [1]) for d in range(degree + 1)]) print(matrix)
Output:
[[ 1. 0. 0.] [ 1. 2. 2.] [ 1. -2. 2.]]
The one-liner code uses the numpy.polynomial.hermite.hermval() function to evaluate the Hermite polynomials. The np.column_stack() function is then used to stack the evaluated polynomials into columns to form the Vandermonde matrix.
Summary/Discussion
Method 1: Using NumPy and Scipy. Strengths are high performance and simplicity when both libraries are available. Weaknesses include dependency on these libraries.
Method 2: Pure Python Implementation. Strengths include no external dependencies, works in any Python environment. Weaknesses are slower performance compared to NumPy/SciPy approaches.
Method 3: Lambdify with SymPy. Strengths include symbolic computation which can be more intuitive in some contexts. Weaknesses are the additional overhead of symbolic computation and the need for SymPy.
Method 4: Use of Polynomial Class. Strengths include an object-oriented approach that’s expandable and maintains evaluation encapsulation. Weaknesses may be the more verbose code and potentially slower computation compared to library-based methods.
Method 5: NumPy Only Approach. Strengths include concise, one-liner code using a powerful library. Weaknesses are reliance on NumPy and less readability for those unfamiliar with NumPy’s polynomial module.