Evaluate a 2-D Laguerre series on the Cartesian product of x and y with 3d array of coefficient in Python

To evaluate a 2-D Laguerre series on the Cartesian product of x and y with a 3D array of coefficients, use the numpy.polynomial.laguerre.laggrid2d() method in Python. This method returns the values of the two-dimensional Laguerre series at points in the Cartesian product of x and y.

The coefficient array c should be ordered so that the coefficient of the term of multi-degree i,j is contained in c[i,j]. When c has more than two dimensions, the additional indices enumerate multiple sets of coefficients, allowing for batch evaluation.

Syntax

numpy.polynomial.laguerre.laggrid2d(x, y, c)

Parameters

x, y ? Arrays of point coordinates. If they are lists or tuples, they are converted to ndarrays.

c ? Array of coefficients with shape (M, N, ...). The first two dimensions correspond to the Laguerre polynomial degrees.

Example

Let's evaluate a 2-D Laguerre series using a 3D coefficient array ?

import numpy as np
from numpy.polynomial import laguerre as L

# Create a 3D array of coefficients
c = np.arange(24).reshape(2, 2, 6)
print("Coefficient Array:")
print("Shape:", c.shape)
print("Array:\n", c)

# Define evaluation points
x_points = [1, 2]
y_points = [1, 2]

# Evaluate the 2-D Laguerre series
result = L.laggrid2d(x_points, y_points, c)
print("\nResult shape:", result.shape)
print("Result:\n", result)

Coefficient Array:
Shape: (2, 2, 6)
Array:
 [[[ 0  1  2  3  4  5]
  [ 6  7  8  9 10 11]]

 [[12 13 14 15 16 17]
  [18 19 20 21 22 23]]]

Result shape: (6, 2, 2)
Result:
 [[[ 0. -6.]
  [-12.  0.]]

 [[ 1. -6.]
  [-12.  0.]]

 [[ 2. -6.]
  [-12.  0.]]

 [[ 3. -6.]
  [-12.  0.]]

 [[ 4. -6.]
  [-12.  0.]]

 [[ 5. -6.]
  [-12.  0.]]]

Understanding the Output

The result has shape (6, 2, 2) because:

• The coefficient array has shape (2, 2, 6)
• We evaluate at 2 x-points and 2 y-points
• The output shape follows: c.shape[2:] + x.shape + y.shape = (6,) + (2,) + (2,) = (6, 2, 2)

Each of the 6 coefficient sets produces a (2, 2) result matrix corresponding to the Cartesian product of the evaluation points.

Conclusion

The laggrid2d() method efficiently evaluates 2-D Laguerre series on Cartesian products. When using 3D coefficient arrays, it performs batch evaluation across multiple coefficient sets, making it useful for processing multiple series simultaneously.