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

To evaluate a 3-D Laguerre series on the Cartesian product of x, y and z, use the polynomial.laguerre.laggrid3d() method in Python. The method returns the values of the three-dimensional Laguerre series at points in the Cartesian product of x, y and z.

If the coefficient array c has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape.

Parameters

The method accepts the following parameters:

• x, y, z ? Three-dimensional series is evaluated at the points in the Cartesian product of x, y, and z. If x, y, or z is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar.
• c ? Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in c[i,j]. If c has dimension greater than two, the remaining indices enumerate multiple sets of coefficients.

Example

Let's create a 2D array of coefficients and evaluate the 3-D Laguerre series ?

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

# Create a 2d array of coefficients
c = np.arange(4).reshape(2,2)

# Display the array
print("Our Array...\n", c)

# Check the Dimensions
print("\nDimensions of our Array...\n", c.ndim)

# Get the Datatype
print("\nDatatype of our Array object...\n", c.dtype)

# Get the Shape
print("\nShape of our Array object...\n", c.shape)

# Evaluate 3-D Laguerre series on Cartesian product
result = L.laggrid3d([1,2], [1,2], [1,2], c)
print("\nResult...\n", result)

Our Array...
 [[0 1]
 [2 3]]

Dimensions of our Array...
2

Datatype of our Array object...
int64

Shape of our Array object...
(2, 2)

Result...
 [[ 0.  2.]
 [-1. -1.]]

How It Works

The laggrid3d() method evaluates the Laguerre series using the coefficient array c. Since our coefficient array is 2D with shape (2,2), it gets implicitly expanded to 3D. The evaluation points [1,2] for each dimension create a 2×2×2 Cartesian product, and the method computes the Laguerre series values at these points.

Conclusion

The laggrid3d() method provides an efficient way to evaluate 3-D Laguerre series on Cartesian products. It automatically handles dimension expansion and returns the evaluated series values at the specified coordinate points.