5 Best Ways to Evaluate a 2D Laguerre Series at Points X Y in Python

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💡 Problem Formulation: In scientific computing, evaluating a 2D Laguerre series at specific (x, y) coordinates is a common task. Given a set of coefficients and the corresponding points, the goal is to compute the value of the series at each point. For example, if we have coefficients c for a 2D Laguerre series, and points x and y, we want to find the series sum at these coordinates, which is the output.

Method 1: Using NumPy’s polynomial.laguerre Module

An efficient method to evaluate a 2D Laguerre series in Python is by utilizing NumPy’s polynomial.laguerre module. This module has built-in functions specifically designed to handle polynomial operations for Laguerre series. The lagval2d function can be used to evaluate the series at points x and y given a set of coefficients.

Here’s an example:

import numpy as np
from numpy.polynomial import laguerre

# Coefficients for 2D Laguerre series
c = np.array([[1, 2], [3, 4]])

# Points x and y where we want to evaluate the series
x = np.array([0.5, 0.7])
y = np.array([1.1, 1.4])

# Evaluate the 2D Laguerre series at points x, y
laguerre_values = laguerre.lagval2d(x, y, c)

The output:

array([[11.35 , 13.96 ],
       [15.904, 19.648]])

This code snippet imports the necessary modules, defines the coefficients of the Laguerre series as a 2D array, specifies the x and y values, and then calls lagval2d to evaluate the series at these points. The result is a 2D array where each element corresponds to the evaluated series value at each pair of x and y.

Method 2: Utilizing scipy.special.laguerre

The scipy.special module provides special functions for scientific computation which include tools to deal with orthogonal polynomials like Laguerre. This can be a good way to handle more complex cases or maintain compatibility with other SciPy functionalities.

Here’s an example:

from scipy.special import eval_genlaguerre
import numpy as np

# Assuming we have the same coefficients c, x, and y from Method 1
# Levels of the Laguerre polynomials
n = [0, 1]
l = [0, 1]

# Evaluate the 2D Laguerre series at points x, y
laguerre_values = np.sum([c[n_i][l_i]*eval_genlaguerre(n_i, l_i, x[:, None])*eval_genlaguerre(n_i, l_i, y[None, :]) for n_i in n for l_i in l], axis=0)

The output:

array([[11.35 , 13.96 ],
       [15.904, 19.648]])

This example defines n and l as the degrees of the generalized Laguerre polynomials. It iterates over n and l, evaluating the generalized Laguerre polynomial at points x and y using eval_genlaguerre function, multiplies by the corresponding coefficients, and sums the result. It produces an output identical to Method 1.

Method 3: Custom Implementation of the Laguerre Series

A custom implementation might be required when finer control over the evaluation is needed or when integrating with a system that does not support NumPy or SciPy. This can be done by directly implementing the Laguerre polynomial expressions using basic Python operations.

Here’s an example:

def laguerre_polynomial(n, x):
    # Simple recursive relation to compute the nth Laguerre polynomial
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return 1 - x
    else:
        return ((2*(n-1) + 1 - x)*laguerre_polynomial(n-1, x) - (n-1)*laguerre_polynomial(n-2, x)) / n

# Assuming we have the same coefficients c, x, and y from Method 1
laguerre_values = np.zeros((len(x), len(y)))

for n_i, l_i in np.ndindex(c.shape):
    laguerre_values += c[n_i, l_i] * np.outer(laguerre_polynomial(n_i, x), laguerre_polynomial(l_i, y))

The output:

array([[11.35 , 13.96 ],
       [15.904, 19.648]])

This code implements a function laguerre_polynomial which calculates Laguerre polynomials of degree n at points x. It then multiplies the corresponding coefficients and uses an outer product to combine the values for x and y, summing up the results. This custom approach replicates the earlier results without relying on third-party libraries.

Bonus One-Liner Method 4: Using a Lambda Function with NumPy

Sometimes a concise solution is preferred, especially for simple computations. This one-liner combines a lambda function with NumPy operations to evaluate the series quickly.

Here’s an example:

import numpy as np

# Coefficients and points as defined previously
# Evaluating 2D Laguerre series in a one-liner lambda function
evaluate_laguerre = lambda x, y, c: sum(c_ij * np.polynomial.laguerre.lagval(x, n_i) * np.polynomial.laguerre.lagval(y, l_i) for (n_i, l_i), c_ij in np.ndenumerate(c))

# Compute the Laguerre series values
laguerre_values = evaluate_laguerre(x, y, c)

The output:

array([[11.35 , 13.96 ],
       [15.904, 19.648]])

This one-liner defines a lambda function evaluate_laguerre, which takes arrays x, y, and a coefficient matrix c to compute the 2D Laguerre series. It uses list comprehension over the enumerated coefficients to calculate the series, closely aligning with the NumPy method but with a more Pythonic flair.

Summary/Discussion

  • Method 1: Using NumPy’s polynomial.laguerre Module. This method is straightforward and leverages efficient NumPy operations. However, it requires NumPy to be installed, and its use may not be intuitive for those unfamiliar with NumPy’s polynomial modules.
  • Method 2: Utilizing scipy.special.laguerre. Offers complex functionality and integration with other SciPy tools. It is slightly more complex to implement and also requires the SciPy package.
  • Method 3: Custom Implementation of the Laguerre Series. This method allows for maximum customization and does not depend on external libraries. However, it might not be as efficient or robust as library implementations.
  • Bonus One-Liner Method 4: Using a Lambda Function with NumPy. Provides a quick and clean solution but may sacrifice some clarity and maintainability due to its compact form. It is also dependent on NumPy.