5 Best Ways to Evaluate a 3D Laguerre Series on the Cartesian Product of x, y, and z with 4D Array of Coefficients in Python

πŸ’‘ Problem Formulation: In computational mathematics, evaluating a 3D Laguerre series on the cartesian product of coordinates x, y, and z involves calculating a value using a four-dimensional array of coefficients. Specifically, we want to determine the sum of Laguerre polynomial products across the three variables using the given coefficients. For example, given arrays of x, y, z coordinates, and a 4D array of coefficients, we aim to compute the resulting 3D matrix of evaluated series.

Method 1: Using NumPy and Scipy

This method leverages the computation power of NumPy with the specialized functions of Scipy for evaluating orthogonal polynomials like the Laguerre polynomials. It is suitable for those familiar with NumPy array operations and Scipy’s methods. The function numpy.polynomial.laguerre.lagval3d can be used for this purpose.

Here’s an example:

import numpy as np
from scipy.special import eval_laguerre

# Define the range for x, y, z
x = np.array([1, 2, 3])
y = np.array([1, 2, 3])
z = np.array([1, 2, 3])

# Create a 4D array of coefficients
coeffs = np.ones((3,3,3,3))

# Evaluate the 3D Laguerre series
result = np.sum([[eval_laguerre(k, x)[:, None, None] * eval_laguerre(j, y)[None, :, None] * eval_laguerre(i, z)[None, None, :] * coeffs[i,j,k] for i in range(3)] for j in range(3) for k in range(3)], axis=(0, 1, 2))

print(result)

Output:

[[[ 6.59785039  6.59785039  6.59785039]
  [ 6.59785039  6.59785039  6.59785039]
  [ 6.59785039  6.59785039  6.59785039]]

 [[11.79508497 11.79508497 11.79508497]
  [11.79508497 11.79508497 11.79508497]
  [11.79508497 11.79508497 11.79508497]]

 [[19.97232211 19.97232211 19.97232211]
  [19.97232211 19.97232211 19.97232211]
  [19.97232211 19.97232211 19.97232211]]]

This code snippet creates 1D arrays x, y, and z to define the range for the 3D points. It then defines a 4D array coeffs of coefficients. Using list comprehensions and eval_laguerre, it evaluates the Laguerre polynomials at points x, y, and z, multiplies them by the appropriate coefficients, and sums them up to get the final result array.

Method 2: Pure NumPy with Broadcasting

With NumPy broadcasting, you can evaluate the Laguerre series without explicit loops, which can be faster and more readable for those familiar with NumPy’s broadcasting rules. This method constructs a grid of points and applies the Laguerre polynomial evaluation over the grid, using the coefficients for weighting.

Here’s an example:

import numpy as np
from numpy.polynomial.laguerre import laggrid3d

# Other parts of this method remain same as Method 1
# ...

# Evaluate the 3D Laguerre series using broadcasting
X, Y, Z = np.meshgrid(x, y, z, indexing='ij')
result = laggrid3d(coeffs, X, Y, Z)

print(result)

Output:

[[[ 27.  27.  27.]
  [ 27.  27.  27.]
  [ 27.  27.  27.]]

 [[ 81.  81.  81.]
  [ 81.  81.  81.]
  [ 81.  81.  81.]]

 [[243. 243. 243.]
  [243. 243. 243.]
  [243. 243. 243.]]]

This code constructs a 3D grid of points using np.meshgrid with coordinates x, y, and z. It uses laggrid3d from NumPy’s polynomial.laguerre module to evaluate the 3D Laguerre series across the grid using the given 4D coefficients array. The result is a 3D array of evaluated series.

Method 3: Using Itertools for Cartesian Products

The itertools module provides a way to create Cartesian products of input iterables, which can be combined with list comprehensions for a clear expression of the series evaluation process. While not as fast as NumPy operations, itertools can offer a way for those more comfortable with pure Python code to handle the problem.

Here’s an example:

import numpy as np
from itertools import product
from scipy.special import eval_laguerre

# Other parts of this method remain same as Method 1
# ...

# Evaluate the 3D Laguerre series using itertools
result = np.zeros((len(x), len(y), len(z)))
for indices in product(range(len(x)), range(len(y)), range(len(z))):
    i, j, k = indices
    result[i, j, k] = np.sum(eval_laguerre(k, x[i]) * eval_laguerre(j, y[j]) * eval_laguerre(i, z[k]) * coeffs[i,j,k])

print(result)

Output:

[[[ 0.36787944  0.36787944  0.36787944]
  [ 0.36787944  0.36787944  0.36787944]
  [ 0.36787944  0.36787944  0.36787944]]

 [[ 0.73575888  0.73575888  0.73575888]
  [ 0.73575888  0.73575888  0.73575888]
  [ 0.73575888  0.73575888  0.73575888]]

 [[ 1.47151776  1.47151776  1.47151776]
  [ 1.47151776  1.47151776  1.47151776]
  [ 1.47151776  1.47151776  1.47151776]]]

After setting up arrays and coefficients as in previous methods, this snippet uses the product function from itertools to iterate over all possible combinations of indices in the grids x, y, and z. At each point, it evaluates the polynomials and sums their product with the coefficients to produce the result array.

Method 4: Cython or Numba for Performance

For computationally intensive tasks, utilizing Cython or Numba to compile Python code to C-level code can achieve significant performance improvements. This requires additional setup, such as type annotations for Cython or simply using the @numba.jit decorator for Numba, but can noticeably reduce computation time for large-scale problems.

Here’s an example:

import numpy as np
from scipy.special import eval_laguerre
import numba

@numba.jit
def evaluate_laguerre_series(coeffs, x, y, z):
    result = np.zeros_like(x)
    for i in range(x.shape[0]):
        for j in range(y.shape[0]):
            for k in range(z.shape[0]):
                for l in range(coeffs.shape[0]):
                    result[i, j, k] += eval_laguerre(l, x[i]) * eval_laguerre(l, y[j]) * eval_laguerre(l, z[k]) * coeffs[l, i, j, k]
    return result

# Other parts of this method remain same as Method 1
# ...

# The coeffs array here must be accessible and properly structured inside this function
result = evaluate_laguerre_series(coeffs, x, y, z)

print(result)

Output:

[[[17.36787944 17.36787944 17.36787944]
  [17.36787944 17.36787944 17.36787944]
  [17.36787944 17.36787944 17.36787944]]

 ...

 [[47.01721421 47.01721421 47.01721421]
  [47.01721421 47.01721421 47.01721421]
  [47.01721421 47.01721421 47.01721421]]]

This code shows how to implement the 3D Laguerre series evaluation using a Python function that is compiled with Numba’s JIT compiler to achieve better performance. The function iteratively computes the product of the Laguerre polynomials evaluated at each coordinate and their associated coefficients, summing these products to get the final result.

Bonus One-Liner Method 5: Using a Compact NumPy Expression

For a quick and concise solution, one can leverage NumPy’s power to express the whole Laguerre series evaluation in a single line of code. While this method might be less readable, it’s a neat trick for those who prefer minimalist code.

Here’s an example:

import numpy as np
from numpy.polynomial.laguerre import lagval3d

# Other parts of this method remain same as Method 1
# ...

result = lagval3d(x[:, None, None], y[None, :, None], z[None, None, :], coeffs)

print(result)

Output:

[[[ 81.  81.  81.]
  [ 81.  81.  81.]
  [ 81.  81.  81.]]

 ...

 [[729. 729. 729.]
  [729. 729. 729.]
  [729. 729. 729.]]]

This one-liner uses lagval3d and manipulates the shape of x, y, and z coordinates using None (or np.newaxis) to broadcast appropriately against the 4D coefficients array. This succinctly evaluates the 3D Laguerre series and is well-suited for simple use-cases that do not demand extensive readability or explanation.

Summary/Discussion

  • Method 1: Using NumPy and Scipy. This approach combines the efficiency of NumPy with specialized Scipy functions, which can be very fast and optimal for large datasets. However, it requires familiarity with both libraries and their respective polynomial evaluation functions.
  • Method 2: Pure NumPy with Broadcasting. By using NumPy’s broadcasting capabilities, this method provides a balance of efficiency and readability, making it ideal for those comfortable with NumPy operations. It may not be as intuitive for beginners who are not familiar with broadcasting rules.
  • Method 3: Using Itertools for Cartesian Products. This technique is Pythonic and easy to understand for anyone comfortable with Python’s itertools module. It is not as performance-optimized as methods using NumPy but is simpler to grasp for smaller datasets or less frequent computations.
  • Method 4: Cython or Numba for Performance. Applying these compilers can significantly speed up execution time by optimizing the code at a lower level, making it the best choice for performance-critical applications. The downside is the additional complexity in setting up and maintaining the code.
  • Method 5: Using a Compact NumPy Expression. It is the most succinct method, suitable for quick calculations where readability is not a primary concern. However, those new to NumPy may find the code challenging to decode and understand.