Evaluating 3D Legendre Series Over Cartesian Products in Python

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💡 Problem Formulation: This article addresses the computational task of evaluating a 3-dimensional (3D) Legendre series over the Cartesian product coordinates, \( x, y, \) and \( z. \) Specifically, it explores methods to compute values of a 3D polynomial where each term is a product of Legendre polynomials in \( x, y, \), and \( z. \) For example, given arrays of \( x, y, \) and \( z \) values, the goal is to find the corresponding values of the Legendre series at these points.

Method 1: Using NumPy and SciPy

Evaluating a 3D Legendre series using a combination of NumPy and SciPy is a common approach in Python. NumPy offers efficient array operations, while SciPy provides the necessary special functions, specifically the Legendre polynomials. One can use NumPy’s broadcasting to evaluate the Legendre polynomials for each dimension and then combine them.

Here’s an example:

import numpy as np
from scipy.special import eval_legendre

# Define the degree of Legendre polynomial and grid points
l, m, n = 2, 3, 4
x = np.linspace(-1, 1, 100)
y = np.linspace(-1, 1, 100)
z = np.linspace(-1, 1, 100)

# Evaluate Legendre polynomials
P_l = eval_legendre(l, x[:, np.newaxis, np.newaxis])
P_m = eval_legendre(m, y[np.newaxis, :, np.newaxis])
P_n = eval_legendre(n, z[np.newaxis, np.newaxis, :])

# Compute the 3D Legendre series
result = P_l * P_m * P_n

The output is a 3D array of the evaluated Legendre series at the Cartesian product points.

This method utilizes the power of array broadcasting, where arrays of different shapes during arithmetic operations behave as if they have compatible shapes. This feature is fundamental to NumPy and allows the concise computation of the Legendre series over the grid without explicit loops.

Method 2: Using NumPy’s Polynomial Package

NumPy’s polynomial package contains classes for dealing with polynomial series, including Legendre series. The Legendre class can be used to initialize the polynomial and evaluate it over a meshgrid created by the Cartesian product of \( x, y, \) and \( z \).

Here’s an example:

import numpy as np
from numpy.polynomial.legendre import Legendre

# Define Legendre coefficients for a 3D series: c[l, m, n] for term P_l(x)*P_m(y)*P_n(z)
coeffs = np.zeros((3, 3, 3))
coeffs[2, 2, 2] = 1  # For example, only the term with l=m=n=2 has a non-zero coefficient

# Create Legendre polynomial
P = Legendre.fromroots([])  # creates a Legendre polynomial P(x) = x

# Grid of points
x, y, z = np.mgrid[-1:1:100j, -1:1:100j, -1:1:100j]

# Evaluate the 3D series
result = sum(coeffs[l, m, n] * P(x) * P(y) * P(z) for l in range(coeffs.shape[0])
                                                       for m in range(coeffs.shape[1])
                                                       for n in range(coeffs.shape[2]))

The output is a 3D array containing the evaluated values of the Legendre series across the grid.

In this approach, the Legendre polynomial’s coefficient array defines the terms of our 3D series. We use generator expressions within the sum function to iterate over all combinations of coefficients, thus calculating the full series.

Method 3: Using mpmath for Arbitrary Precision

For applications requiring high precision, the Python library mpmath can be used to evaluate Legendre polynomials with arbitrary precision. Mpmath is designed to work with floating-point arithmetic of arbitrary precision and provides extensive library functions.

Here’s an example:

import mpmath
import numpy as np

# Set arbitrary precision (number of digits)
mpmath.mp.dps = 50

# Define grid points
x = np.linspace(-1, 1, 10)
y = np.linspace(-1, 1, 10)
z = np.linspace(-1, 1, 10)

# Convert to mpmath objects
x, y, z = map(mpmath.mp.matrix, (x, y, z))

# Define the degrees of the Legendre polynomials
l, m, n = 2, 3, 4

# Evaluate the 3D Legendre series with arbitrary precision
result = mpmath.legendre(l, x) * mpmath.legendre(m, y) * mpmath.legendre(n, z)

The output is a 3D matrix of mpmath floating-point numbers with the evaluated values of the Legendre series.

This snippet demonstrates how to set the precision with mpmath and accomplish high-precision calculations. mpmath’s matrix types are used instead of NumPy arrays to work seamlessly with high precision.

Method 4: Custom Implementation using Orthogonal Polynomial Properties

If specific properties of the Legendre polynomials are known, a custom implementation can be developed to evaluate the series more efficiently. This approach might be beneficial in situations where performance is critical and the algorithms can be highly customized and optimized for a specific problem structure.

Here’s an example:

import numpy as np

# Recursive definition of Legendre Polynomials
def legendre_poly(n, x):
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return x
    else:
        return ((2 * n - 1) * x * legendre_poly(n - 1, x) - (n - 1) * legendre_poly(n - 2, x)) / n

# Grid points
x, y, z = np.linspace(-1, 1, 100), np.linspace(-1, 1, 100), np.linspace(-1, 1, 100)

# Compute the 3D Legendre series using the recursive definition for each dimension
result = legendre_poly(2, x[:, np.newaxis, np.newaxis]) * \
         legendre_poly(3, y[np.newaxis, :, np.newaxis]) * \
         legendre_poly(4, z[np.newaxis, np.newaxis, :])

The output is a 3D array with the Legendre series values evaluated over the Cartesian product space.

This code defines a recursive function for computing Legendre polynomials and uses it similarly to the method shown in Method 1, applying vectorized calculations over a 3D grid, exploiting the recursive relationship directly.

Bonus One-Liner Method 5: using itertools and NumPy

Python’s itertools product can generate Cartesian products combined with NumPy’s vectorized evaluation for a very concise one-liner approach.

Here’s an example:

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

# Degrees and grid points
degrees = (2, 3, 4)
grid = [np.linspace(-1, 1, 100)] * 3

# One-liner to evaluate the Legendre series
P = np.prod([eval_legendre(d, g) for d, g in zip(degrees, grid)], axis=0)

The output is a 3D array with the proper evaluations of the position-wise products of Legendre polynomials.

This one-liner leverages list comprehension, itertools.product, and NumPy’s prod function to succinctly perform the task at hand, reducing code verbosity while maintaining performance benefits from vectorized operations.

Summary/Discussion

  • Method 1: NumPy and SciPy. Best for general use. Efficient and utilizes well-known libraries. May not suit highly specialized needs.
  • Method 2: NumPy’s Polynomial Package. Convenient for handling polynomial coefficients directly. It might have performance overheads for large-scale problems.
  • Method 3: mpmath for Arbitrary Precision. Essential for high-precision requirements. Slower due to arbitrary-precision arithmetic.
  • Method 4: Custom Implementation. Highly efficient when the problem structure is well understood. Requires more in-depth mathematical knowledge and may be less flexible.
  • Method 5: One-Liner with itertools and NumPy. Offers brevity and readability. Good for simple or prototype implementations but might obscure complexities for more intricate tasks.