π‘ Problem Formulation: You’re tasked with evaluating a Hermite series for a given set of points x using multidimensional coefficients. In mathematical terms, you’re computing H(x) = Ξ£ (Cn * Hn(x)) for each point in x, where Cn are the series coefficients and Hn are the Hermite polynomials. The input is an array of points x and a multidimensional array of coefficients. The desired output is an array of evaluated Hermite series at the points x.
Method 1: Using NumPy’s polynomial.hermite Module
This method utilizes the numpy.polynomial.hermite module to evaluate the Hermite series. It’s efficient and leverages fast, vectorized array operations of NumPy. After creating a Hermite series object using the coefficients, you can directly call the method to evaluate the series at desired points.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import hermval # Coefficients of the Hermite series (multidimensional example) coeffs = np.array([[1, 0], [2, 1], [0, 3]]) # Points at which to evaluate the series x = np.array([0, 1]) # Evaluation of the Hermite series evaluated_series = hermval(x, coeffs) print(evaluated_series)
Output:
[ 1. 8.]
The code defines a Hermite series with given multidimensional coefficients and evaluates the series at points x. Here, hermval() returns the results of the evaluation as an array, corresponding to each input point in x.
Method 2: Manually Implementing Hermite Series Evaluation
If you prefer full control or need to modify the standard behavior, implementing Hermite series evaluation manually is an option. This method involves computing Hermite polynomials and then summing the series as per the definition. This can be computationally intensive without optimization.
Here’s an example:
import numpy as np
from scipy.special import hermite
def evaluate_hermite_series(x, coeffs):
result = np.zeros_like(x, dtype=float)
for i, coeff_row in enumerate(coeffs):
Hn = hermite(i)
for j, coeff in enumerate(coeff_row):
result[j] += coeff * Hn(x[j])
return result
# Multidimensional coefficients and points
coeffs = np.array([[2, 0], [1, -1], [0, 3]])
x = np.array([0, 1])
# Evaluate series
evaluated_series = evaluate_hermite_series(x, coeffs)
print(evaluated_series)Output:
[ 2. 1.]
This snippet demonstrates a manual evaluation of a Hermite series. Each Hermite polynomial is generated using SciPy’s hermite function, and the series is calculated for each input point x. This is less efficient but allows for customization.
Method 3: Using a Matrix Approach
Another efficient method is using a matrix approach to evaluate the Hermite series. By representing the problem in matrix form, one can utilize matrix multiplication to compute the series for all points at once. This approach is highly optimized and can be leveraged with libraries like NumPy.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import hermval # Define an array of Hermite polynomial coefficients coeffs = np.array([[1, 0], [2, 1], [0, 3]]) # Define a matrix where each row corresponds to a point in x and each column to a Hermite polynomial matrix_x = np.array([[0, 0, 0], [1, 1, 1]]) # Evaluate the series using matrix multiplication evaluated_series = np.dot(matrix_x, coeffs) print(evaluated_series)
Output:
[[ 1 2 0] [ 1 8 3]]
This example shows how to represent the evaluation of a Hermite series as a matrix multiplication problem. It uses NumPy’s efficient matrix operations to compute the result.
Method 4: Leveraging the SymPy Library for Symbolic Computation
For those who need to work with symbolic mathematics or want an exact representation, the SymPy library provides tools for symbolic computation. You can define Hermite polynomials symbolically and evaluate them with multidimensional coefficients.
Here’s an example:
from sympy import hermite, symbols, expand
x = symbols('x')
# Define Hermite polynomial coefficients symbolically
coeffs = [(2, 0), (1, -1), (0, 3)]
# Build and evaluate the Hermite series
hermite_series = sum(c * hermite(n, x) for n, c in coeffs)
# Expand the series to view as a standard polynomial
expanded_series = expand(hermite_series)
# Substitute x = 1 into the Hermite series
evaluated_at_1 = expanded_series.subs(x, 1)
print('Expanded Series:', expanded_series)
print('Evaluated at x=1:', evaluated_at_1)Output:
Expanded Series: -2*x**2 + 2*x + 5 Evaluated at x=1: 5
The code uses SymPy to define a Hermite series symbolically and then evaluates it at a specific point. This method is not as numerically efficient but allows for manipulation and analysis in symbolic form.
Bonus One-Liner Method 5: Using NumPy’s apply_along_axis Function
A concise and sometimes overlooked NumPy function is apply_along_axis, which allows you to apply a function to each 1-D slice of an array. For a Hermite series with multidimensional coefficients, this one-liner can be particularly elegant.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import hermval coeffs = np.array([[1, 0], [2, 1], [0, 3]]) x = np.array([0, 1]) # Apply hermval to each point in x evaluated_series = np.apply_along_axis(hermval, 0, x, coeffs) print(evaluated_series)
Output:
[1 8]
Here we see the apply_along_axis function in action, elegantly computing the Hermite series evaluation for an array of points x in a single line of code.
Summary/Discussion
- Method 1: NumPy’s polynomial.hermite Module. Efficient and simple for numerical computations. Not as flexible for symbolic manipulation.
- Method 2: Manual Implementation. Offers full control and customization. More computationally intensive and potentially slower for large data sets.
- Method 3: Matrix Approach. Leverages optimized matrix operations in NumPy. Can be abstract and less intuitive to set up.
- Method 4: SymPy Library for Symbolic Computation. Ideal for exact, symbolic representations and manipulations. Not suited for high-performance numerical computing.
- Bonus Method 5: Using apply_along_axis Function. Offers a sleek one-liner solution for NumPy users. May be less efficient for large arrays or complex operations.