Evaluate a 3D Laguerre series at points (x,y,z) with 4D array of coefficient using NumPy in Python

A 3D Laguerre series can be evaluated using NumPy in Python by following these steps:

• Import the necessary libraries:

import numpy as np
from scipy.special import genlaguerre

• Define the 3D Laguerre series function:

def laguerre_3d(x, y, z, coeffs):
    n, m, l = coeffs.shape
    result = np.zeros_like(x)
    for i in range(n):
        for j in range(m):
            for k in range(l):
                result += coeffs[i, j, k] * genlaguerre(i, 0)(x) * genlaguerre(j, 0)(y) * genlaguerre(k, 0)(z)
    return result

This function takes in the points (x, y, z) and the 4D array of coefficients (coeffs) and returns the evaluated Laguerre series.

• Define the points at which to evaluate the Laguerre series:

x = np.linspace(0, 1, 10)
y = np.linspace(0, 1, 10)
z = np.linspace(0, 1, 10)
xx, yy, zz = np.meshgrid(x, y, z, indexing='ij')

Here, we create a meshgrid of points using NumPy's meshgrid function.

• Define the coefficients of the Laguerre series:

coeffs = np.random.rand(5, 5, 5)

Here, we create a 4D array of random coefficients.

• Evaluate the Laguerre series at the points:

result = laguerre_3d(xx, yy, zz, coeffs)

Here, we call the laguerre_3d function with the meshgrid of points and the coefficients.

• Visualize the result:

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(xx, yy, zz, c=result)
plt.show()

Here, we create a 3D scatter plot of the points with the color of each point determined by the evaluated Laguerre series.

Alternative method:

Another way to evaluate a 3D Laguerre series using NumPy is to use the numpy.polynomial.laguerre.Laguerre class. Here's an example:

from numpy.polynomial.laguerre import Laguerre

def laguerre_3d(x, y, z, coeffs):
    n, m, l = coeffs.shape
    result = np.zeros_like(x)
    for i in range(n):
        for j in range(m):
            for k in range(l):
                result += coeffs[i, j, k] * Laguerre.basis(i)(x) * Laguerre.basis(j)(y) * Laguerre.basis(k)(z)
    return result

Here, we use the Laguerre.basis method to generate the Laguerre polynomials and evaluate the series. The rest of the code is the same as the previous method.