💡 Problem Formulation: We aim to compute the values of a two-dimensional Hermite series at points defined by the Cartesian product of x and y coordinates. The series coefficients are given as a 3D array in Python. This process is crucial in fields like computational physics and mathematical modeling. Input: arrays x, y, and a 3D coefficients array c. Desired output: an array representing the evaluated Hermite series over the Cartesian grid formed by x and y.
Method 1: Using NumPy and SciPy
This method involves leveraging the power of NumPy for array manipulation along with the SciPy library’s Hermite polynomial functions. Specifically, the numpy.polynomial.hermite.hermval2d function from SciPy can be used to evaluate a 2D Hermite series.
Here’s an example:
import numpy as np from scipy.special import hermval2d # Define the x and y grid points x = np.array([0.1, 0.5, 0.9]) y = np.array([0.2, 0.6]) # 3D array of coefficients shaped (order+1, order+1, number_of_variables) c = np.random.rand(4, 4, 2) # Evaluate the 2D Hermite series on the Cartesian product of x and y result = hermval2d(x, y, c[:,:,0]) print(result)
The output will be an array of values evaluating the 2D Hermite series at all combinations of x and y.
This code snippet creates a grid of points using the provided x and y arrays, then randomly generates a 3D array for the coefficients and uses hermval2d from SciPy to evaluate the series. The output is printed as a two-dimensional array corresponding to each point on the Cartesian grid.
Method 2: Custom Implementation Using Polynomial Theory
A more direct approach involves implementing the Hermite polynomial evaluation manually using the definition of Hermite polynomials. This method will use the recursive definition of Hermite polynomials to compute values without any special external library requirements.
Here’s an example:
import numpy as np
# Define coefficients
coefficients = np.random.rand(3, 3, 2)
# Define the Hermite polynomial evaluation function
def hermite_polynomial(x, y, coeffs):
# Compute the Hermite polynomial for each pair (x, y)
Hx = np.polynomial.hermite.hermval(x, coeffs[:,:,0])
Hy = np.polynomial.hermite.hermval(y, coeffs[:,:,1])
return np.outer(Hx, Hy)
# Cartesian product of x and y arrays
x = np.linspace(-1, 1, 5)
y = np.linspace(-1, 1, 5)
# Evaluate the 2D Hermite series
result = hermite_polynomial(x, y, coefficients)
# Print the result
print(result)The output is a matrix representing the evaluated Hermite polynomials on the Cartesian grid.
In this example, we manually implement the Hermite series evaluation using the recursive Hermite polynomial properties. The code uses NumPy’s outer product function to combine the individually evaluated polynomial series from each axis, offering a simple yet effective solution without any library beyond NumPy.
Method 3: Vectorized Evaluation with NumPy
Vectorized operations are a powerful feature of NumPy, allowing for more efficient and concise code implementations. This method uses vectorized functions to evaluate the 2D Hermite series, improving performance significantly.
Here’s an example:
import numpy as np # Define the x and y grid points x = np.array([0.1, 0.5, 0.9]) y = np.array([0.2, 0.6]) # 3D array of coefficients c = np.random.rand(4, 4, 2) # 2D array of x, y points using meshgrid X, Y = np.meshgrid(x, y) # Evaluate the 2D Hermite series using vectorized operations result = np.sum(c[:,:,0]*hermite_h(X)**c[:,:,1]*hermite_h(Y).reshape(-1,1), axis=(0,1)) print(result)
The output is an array representing the evaluated values over the Cartesian grid.
This snippet uses NumPy’s meshgrid function to compute the Cartesian product grid and then performs a vectorized summation to evaluate the Hermite series. The codes vectorizes the polynomial evaluation across all points on the Cartesian grid, making it extremely efficient for large-scale computations.
Method 4: Utilizing Tensor Products
Tensor products allow for a multidimensional generalization of vector space operations, which can be used for efficiently calculating the Hermite series over a Cartesian grid. This method requires understanding tensor algebra but provides a robust computational tool.
Here’s an example:
import numpy as np # Define the x and y grid points x = np.linspace(-1, 1, 5) y = np.linspace(-1, 1, 5) # 3D array of coefficients c = np.random.rand(3, 3, 2) # Tensor product of 1D Hermite series evaluations on x and y hx = np.polynomial.hermite.hermval(x, c[:,:,0].T) hy = np.polynomial.hermite.hermval(y, c[:,:,1].T) # Evaluate the 2D Hermite series using tensor products result = np.tensordot(hx, hy, axes=0) print(result)
Output will be a multi-dimensional array of the evaluated series over the Cartesian grid.
Here, the tensordot function from NumPy is used to compute the tensor product between the 1D evaluations of the Hermite series on x and y. This effectively evaluates the 2D series across the entire grid. Tensor products provide an elegant way to generalize operations to higher dimensions, which is clearly demonstrated in this approach.
Bonus One-Liner Method 5: Using NumPy’s Polynomial Class
NumPy provides an integrated polynomial class, which can be used for a concise one-liner evaluation of the 2D Hermite series. This method relies on NumPy’s ability to handle polynomial objects.
Here’s an example:
import numpy as np # Coefficients for the Hermite polynomial c = np.random.rand(3, 3, 2) # Cartesian product of x and y arrays with computations in one line result = np.polynomial.hermite.Hermite(c[:,:,0])(np.outer(x, y)) print(result)
The output is an array representing the evaluated Hermite series over the product of input vectors.
This one-liner code uses the Hermite class constructor Hermite() and is an example of how NumPy’s integrated polynomial classes provide quick and elegant solutions. The use of np.outer to compute the Cartesian product before evaluation is a clever use of built-in NumPy functions for a concise operation.
Summary/Discussion
- Method 1: Using NumPy and SciPy. Strengths: Leverages well-tested libraries; simple syntax. Weaknesses: Requires external SciPy installation.
- Method 2: Custom Implementation. Strengths: Fully customizable; no need for external libraries beside NumPy. Weaknesses: More complex and potentially error-prone.
- Method 3: Vectorized Evaluation. Strengths: Efficient for large data; clear and concise. Weaknesses: Might be less readable for those not familiar with vectorized operations.
- Method 4: Tensor Products. Strengths: Robust mathematical foundation; extends easily to higher dimensions. Weaknesses: Requires understanding of tensor algebra.
- Bonus Method 5: NumPy’s Polynomial Class. Strengths: Extremely concise; takes full advantage of NumPy’s features. Weaknesses: Less explicit, which might confuse some users.