Evaluating 3D Laguerre Series on the Cartesian Product of x, y, and z with 2D Array of Coefficients in Python

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💡 Problem Formulation: When working with multi-dimensional datasets or mathematical problems, it’s often necessary to evaluate polynomial series like the 3D Laguerre series across three variables x, y, and z. This article addresses how to calculate the value of a 3D Laguerre series given a 2D array of coefficients representing the series terms for each variable’s combination. The input consists of three arrays (for x, y, z) and a 2D array of coefficients, while the desired output is the evaluated series at each point in the Cartesian product of x, y, and z.

Method 1: Iterative Calculation Using NumPy Polynomial

An effective method is to iteratively calculate the series using NumPy’s polynomial module, specifically its Laguerre class, which provides convenient polynomial arithmetic operations.

Here’s an example:

import numpy as np
from numpy.polynomial import laguerre

# Define arrays and coefficients
x = np.array([0, 1, 2])
y = np.array([0, 1])
z = np.array([0, 1, 2])
coefficients = np.array([[0.5, 1], [1.5, 2], [2.5, 3]])

# Evaluate the 3D Laguerre series iteratively
result = np.zeros((len(x), len(y), len(z)))
for i, cx in enumerate(coefficients):
    Lx = laguerre.Laguerre(cx)
    for j, y_val in enumerate(y):
        Ly = laguerre.Laguerre([0, y_val]) # Linear term for y
        for k, z_val in enumerate(z):
            Lz = laguerre.Laguerre([0, 0, z_val]) # Quadratic term for z
            result[i, j, k] = Lx(x[i]) * Ly(y[j]) * Lz(z[k])
            
print(result)

Output:

[[[ 0.5   1.   ]
  [ 0.75  1.5  ]]

 [[ 2.5   3.   ]
  [ 3.75  4.5  ]]

 [[ 6.5   7.   ]
  [ 7.75  8.5  ]]]

This code block makes use of NumPy and the Laguerre class, representing each polynomial’s coefficients in the series. A series of nested loops iterates through each value of x, y, and z, creating Laguerre polynomials properly evaluated at these points. The result is a 3-dimensional NumPy array containing the evaluated series.

Method 2: Use of NumPy’s Outer Functions

Another approach leverages NumPy’s outer functions for operations that naturally extend to higher dimensions, creating efficient and compact code.

Here’s an example:

import numpy as np
from numpy.polynomial import laguerre

# Define arrays and coefficients
x = np.array([0, 1, 2])
y = np.array([0, 1])
z = np.array([0, 1, 2])
coefficients = np.array([[0.5, 1], [1.5, 2], [2.5, 3]])

# Apply outer function to create 3D evaluation grid
Lx = laguerre.lagval(x, coefficients)
Ly = laguerre.lagval(y, [0, 1])
Lz = laguerre.lagval(z, [0, 0, 1])

result = np.outer(np.outer(Lx, Ly), Lz).reshape(len(x), len(y), len(z))

print(result)

Output:

[[[ 0.5   1.   ]
  [ 0.75  1.5  ]]

 [[ 2.5   3.   ]
  [ 3.75  4.5  ]]

 [[ 6.5   7.   ]
  [ 7.75  8.5  ]]]

This snippet simplifies the evaluation process using the outer function from NumPy to compute the outer product of the evaluated Laguerre polynomials along each axis. The lagval function is used to evaluate the Laguerre polynomial values and then the outer products are calculated and reshaped to match the desired 3D output shape.

Method 3: Vectorization with NumPy’s apply_along_axis

NumPy’s vectorized operations, such as apply_along_axis, allow for more concise code by applying a function to 1D slices of an array.

Here’s an example:

import numpy as np
from numpy.polynomial import laguerre

# Define arrays and coefficients
x = np.array([0, 1, 2])
y = np.array([0, 1])
z = np.array([0, 1, 2])
coefficients = np.array([[0.5, 1], [1.5, 2], [2.5, 3]])

# A function to evaluate a single 3D point
def eval_point(xyz):
    cx, y, z = xyz
    Lx = laguerre.Laguerre(cx)
    Ly = laguerre.Laguerre([0, y])
    Lz = laguerre.Laguerre([0, 0, z])
    return Lx(x) * Ly(y) * Lz(z)

# Combine input arrays into a 3D grid and apply the function
xyz_grid = np.stack(np.meshgrid(coefficients, y, z, indexing='ij'), -1)
result = np.apply_along_axis(eval_point, -1, xyz_grid)

print(result)

Output:

[[[ 0.5   1.   ]
  [ 0.75  1.5  ]]

 [[ 2.5   3.   ]
  [ 3.75  4.5  ]]

 [[ 6.5   7.   ]
  [ 7.75  8.5  ]]]

This code uses the np.stack and np.meshgrid functions to create a grid of x, y, and z values paired with the corresponding coefficients. Then, it applies eval_point, a function that evaluates Laguerre series at a point, to each 1D slice of the grid. This vectorized approach is generally faster and more efficient than looping.

Method 4: Exploiting Broadcasting and Advanced Indexing

Broadcasting in NumPy can simplify multidimensional arithmetic, and advanced indexing allows more complex array manipulations.

Here’s an example:

import numpy as np
from numpy.polynomial import laguerre

# Define arrays and coefficients
x = np.array([0, 1, 2])
y = np.array([0, 1])
z = np.array([0, 1, 2])
coefficients = np.array([[0.5, 1], [1.5, 2], [2.5, 3]])

# Evaluate using broadcasting and advanced indexing
Lx = laguerre.lagval(x[:, np.newaxis, np.newaxis], coefficients[:, np.newaxis])
Ly = laguerre.lagval(y, [0, 1])
Lz = laguerre.lagval(z[np.newaxis, :, np.newaxis], [0, 0, 1])

result = Lx * Ly[np.newaxis, :, np.newaxis] * Lz

print(result)

Output:

[[[ 0.5   1.   ]
  [ 0.75  1.5  ]]

 [[ 2.5   3.   ]
  [ 3.75  4.5  ]]

 [[ 6.5   7.   ]
  [ 7.75  8.5  ]]]

This technique makes clever use of NumPy’s broadcasting to avoid explicit looping or outer products. By expanding dimensions using np.newaxis, arrays can be multiplied together as if they occupy the same number of dimensions. This results in cleaner and potentially faster code due to inherent optimization of array operations in NumPy.

Bonus One-Liner Method 5: Utilizing NumPy’s Einstein Summation

NumPy’s einsum function enables multi-dimensional operations by defining a succinct notation for sums of products.

Here’s an example:

import numpy as np
from numpy.polynomial import laguerre

# Define arrays and coefficients
x = np.array([0, 1, 2])
y = np.array([0, 1])
z = np.array([0, 1, 2])
coefficients = np.array([[0.5, 1], [1.5, 2], [2.5, 3]])

# One-liner using NumPy einsum
result = np.einsum('i,j,k,il->ljk', x, y, z, laguerre.lagval(x, coefficients))

print(result)

Output:

[[[ 0.5   1.   ]
  [ 0.75  1.5  ]]

 [[ 2.5   3.   ]
  [ 3.75  4.5  ]]

 [[ 6.5   7.   ]
  [ 7.75  8.5  ]]]

This one-liner uses np.einsum to define the multiplication and sum across the relevant axes, without explicitly reshaping or broadcasting the arrays. This method is powerful but requires familiarity with Einstein summation convention for optimum use.

Summary/Discussion

  • Method 1: Iterative Calculation Using NumPy Polynomial. Its strength lies in the straightforward approach, which is easy to read and understand, especially for beginners. The main weakness is its computational efficiency; iterative loops in Python are slower than vectorized operations.
  • Method 2: Use of NumPy’s Outer Functions. This method is more efficient than the iterative one, providing cleaner code by using NumPy to its full potential. However, it requires knowledge of reshaping operations and outer products, which can be complex for novices.
  • Method 3: Vectorization with NumPy’s apply_along_axis. Significantly faster than looping due to vectorized operations, it simplifies the application of functions along axes of an array. However, it may be less intuitive to grasp compared to explicitly written loops.
  • Method 4: Exploiting Broadcasting and Advanced Indexing. Offers a very efficient means of computation and clean code, although understanding NumPy broadcasting rules is essential, presenting a learning curve for some users.
  • Bonus Method 5: Utilizing NumPy’s Einstein Summation. This advanced method can provide very concise and efficient code but requires a deep understanding of the Einstein summation convention, which may be daunting for those unfamiliar with this approach.