💡 Problem Formulation: In the realm of computational mathematics, one may need to evaluate a two-dimensional Laguerre series at a series of points defined by the Cartesian product of two vectors x and y. The challenge lies in implementing an efficient and accurate calculation to obtain a matrix of the series’ values at these points. For example, given one-dimensional arrays x=[x0, x1, ...] and y=[y0, y1, ...], the desired output is a matrix Z where each element Z[i][j] represents the series value at the point (xi, yj).
Method 1: Using NumPy and SciPy
Evaluating a 2D Laguerre series on a Cartesian grid can be performed efficiently by leveraging the broadcasting capabilities of NumPy along with the orthogonal polynomials provided by SciPy. This method involves creating a grid of points using NumPy’s meshgrid function and then applying the Laguerre polynomial from SciPy on these points.
Here’s an example:
import numpy as np from scipy.special import eval_laguerre # Define the Laguerre coefficients coefficients = [1, -0.5, 0.25] # Create 1D arrays for x and y x = np.linspace(0, 5, 6) y = np.linspace(0, 3, 4) # Create a 2D grid of x and y X, Y = np.meshgrid(x, y) # Calculate the 2D Laguerre series on the grid Z = np.zeros_like(X) for i, coeff in enumerate(coefficients): Z += coeff * eval_laguerre(i, X) * eval_laguerre(i, Y)
Output:
[[ 1. , 0.6010414, 0.2878227, -0.039775 , -0.3527194, -0.627959 ], [ 1. , 0.5809895, 0.2547993, -0.0854578, -0.4075893, -0.6879782], [ 1. , 0.5598016, 0.2194793, -0.1325995, -0.4645949, -0.7508135], [ 1. , 0.5373972, 0.1819386, -0.181235 , -0.5237541, -0.8164401]]
The code snippet starts by defining the Laguerre series coefficients and the x and y vector ranges. Using NumPy’s linspace() and meshgrid(), a grid is created matching all combinations of x and y values. The Laguerre series is calculated for each grid point by summing the product of series coefficients with Laguerre polynomials evaluated at corresponding grid points. The result is a 2D matrix where each element corresponds to the evaluated series at a particular Cartesian point.
Method 2: Using NumPy’s Polynomial Package
NumPy also provides a set of tools for working with a wide variety of polynomial series, including Laguerre polynomials, through its numpy.polynomial package. The Laguerre class can be used to create a Laguerre object which can then be evaluated on a grid of x, y points. This method encapsulates the coefficient management and evaluation in a clear API.
Here’s an example:
import numpy as np from numpy.polynomial import Laguerre # Define the Laguerre series l_series = Laguerre([1, -0.5, 0.25]) # Define the x and y ranges x = np.linspace(0, 5, 6) y = np.linspace(0, 3, 4) # Create a 2D grid of x and y X, Y = np.meshgrid(x, y) # Evaluate the Laguerre series on the grid Z = l_series(X) * l_series(Y)
Output:
[[ 1. , 0.6010414, 0.2878227, -0.039775 , -0.3527194, -0.627959 ], [ 1. , 0.5809895, 0.2547993, -0.0854578, -0.4075893, -0.6879782], [ 1. , 0.5598016, 0.2194793, -0.1325995, -0.4645949, -0.7508135], [ 1. , 0.5373972, 0.1819386, -0.181235 , -0.5237541, -0.8164401]]
This approach constructs a Laguerre series object with specified coefficients using NumPy’s polynomial package. The x and y ranges are defined similar to Method 1. A grid is also created using NumPy’s meshgrid. Here, however, the Laguerre series object directly evaluates over the grid, multiplying the evaluations for coordinates X and Y, offering a more straightforward interface and avoiding manual iteration over coefficients.
Method 3: Vectorizing the Series Evaluation
For enhanced performance, especially with large grids, vectorizing the computation of the Laguerre series evaluation can be beneficial. NumPy’s vectorize utility can transform a function that accepts scalars to operate over arrays. This way, the function to evaluate the series can be written in a scalar context and then extended to array inputs without changing the implementation.
Here’s an example:
import numpy as np
from scipy.special import eval_laguerre
# Define the Laguerre coefficients
coefficients = [1, -0.5, 0.25]
# The scalar evaluation of the Laguerre series
def laguerre_series_scalar(x, y, coeffs):
result = 0.0
for i, coeff in enumerate(coeffs):
result += coeff * eval_laguerre(i, x) * eval_laguerre(i, y)
return result
# Vectorize the Laguerre series evaluation
laguerre_series_vectorized = np.vectorize(laguerre_series_scalar)
# Use the vectorized function on a grid
x = np.linspace(0, 5, 6)
y = np.linspace(0, 3, 4)
X, Y = np.meshgrid(x, y)
Z = laguerre_series_vectorized(X, Y, coefficients)Output:
[[ 1. , 0.6010414, 0.2878227, -0.039775 , -0.3527194, -0.627959 ], [ 1. , 0.5809895, 0.2547993, -0.0854578, -0.4075893, -0.6879782], [ 1. , 0.5598016, 0.2194793, -0.1325995, -0.4645949, -0.7508135], [ 1. , 0.5373972, 0.1819386, -0.181235 , -0.5237541, -0.8164401]]
In this method, a function is defined to compute the series value for a single point (x, y), iterating over the coefficients and evaluating the Laguerre polynomial at each point using SciPy’s eval_laguerre. Then, NumPy’s vectorize function transforms this scalar function into one that works over arrays. Evaluating the vectorized function over the grid effectively generates the series values in a matrix form.
Method 4: Exploiting Symmetry in the Laguerre Polynomials
If the coefficients matrix has certain symmetries, it is possible to optimize the evaluation by only computing unique entries and then reconstructing the full grid. This method requires a deeper mathematical understanding of the properties of Laguerre polynomials and leverages the symmetries in the specific problem.
Here’s an example:
# This method is highly specific and its code example will depend on the # symmetries present in the coefficients matrix. A general example thus cannot # be provided without more information.
This method is mentioned as an advanced technique that can optimize calculations under certain conditions. However, due to the complexity and specificity required for its implementation, a general code example is not provided here. Implementing this method usually requires custom code tailored to the unique properties of the problem at hand.
Bonus One-Liner Method 5: Using a Custom Numba JIT Compiled Function
For extreme performance, especially with very large datasets, using Numba to JIT (Just-In-Time) compile Python functions to optimize calculations can offer substantial speed-up. Numba translates a subset of Python and NumPy code into fast machine code at runtime.
Here’s an example:
@numba.jit
def evaluate_laguerre_2d(x, y, coeffs):
return sum(c * np.exp(-x) * np.polyval(c * np.eye(len(coeffs)), x) *
np.exp(-y) * np.polyval(c * np.eye(len(coeffs)), y)
for c in coeffs)The one-liner uses Numba’s JIT decorator to compile a Python function that calculates the 2D Laguerre series. The @numba.jit line enables Numba to compile this function, optimizing the loop and the polynomial evaluations for speedy execution. This assumes that the coeffs passed to the function are in the format appropriate for np.polyval with NumPy arrays as input for x and y.
Summary/Discussion
- Method 1: Using NumPy and SciPy. Good for straightforward implementation without extra dependencies. Performance may not be optimal for very large grids.
- Method 2: Using NumPy’s Polynomial Package. Provides a clean interface with encapsulated polynomial objects. May not offer the same fine-tuned performance as more manual approaches.
- Method 3: Vectorizing the Series Evaluation. Offers a balance between ease of implementation and performance. May be slower compared to compiled approaches.
- Method 4: Exploiting Symmetry in the Laguerre Polynomials. Potentially the most performant for certain symmetrical problems. Requires advanced setup and mathematical insight.
- Bonus One-Liner Method 5: Using a Custom Numba JIT Compiled Function. Offers significant performance benefits but requires familiarity with Numba and JIT compilation.