Evaluating a 3D Legendre Series on the Cartesian Product of X, Y, and Z Using 4D Array of Coefficients in Python

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💡 Problem Formulation: We aim to evaluate a 3D Legendre series on a grid defined by the Cartesian product of X, Y, and Z coordinates using a 4D array that represents the polynomial coefficients. Imagine a situation where we have arrays x, y, z, each representing a dimensional space, and a 4D array coefficients[l,m,n], where l, m, n are the degrees of the corresponding Legendre polynomials. Our task is to compute the series sum for each combination of (x, y, z).

Method 1: Using NumPy’s Polynomials

This method uses the numpy.polynomial.legendre module, which provides a convenient interface for working with Legendre series. A nested loop approach is used to iterate over the Cartesian product of x, y, z, and the legval function is employed to compute the series sum for each combination.

Here’s an example:

import numpy as np
import itertools

# Sample x, y, z arrays and 4D coefficients
x = np.linspace(-1, 1, 10)
y = np.linspace(-1, 1, 10)
z = np.linspace(-1, 1, 10)
coefficients = np.random.rand(5, 5, 5)

# Evaluate the Legendre series
results = np.zeros((len(x), len(y), len(z)))
for i, j, k in itertools.product(range(len(x)), range(len(y)), range(len(z))):
    coef_2d = coefficients[:, :, k]
    results[i, j, k] = np.polynomial.legendre.legval2d(x[i], y[j], coef_2d)

Output:

A 3D array where each entry represents the evaluated series at the respective (x, y, z) coordinate.

This code snippet iterates over each point in the Cartesian product using itertools.product, calculates the corresponding 2D coefficient matrix for each z value, and then evaluates the 2D Legendre polynomial with legval2d. This approach can handle arbitrary polynomial degrees but may be slow for large arrays due to the loop.

Method 2: Vectorized Computation Using NumPy

Here we employ a fully vectorized approach using NumPy’s broadcasting capabilities. This method is significantly faster for large datasets as it avoids the use of Python loops, instead relying on NumPy’s optimized C-based computation.

Here’s an example:

# Assume x, y, z, and coefficients are defined as before
X, Y, Z = np.meshgrid(x, y, z, indexing='ij')

def evaluate_legendre(X, Y, Z, coefficients):
    results = np.zeros_like(X)
    for i in range(coefficients.shape[0]):
        for j in range(coefficients.shape[1]):
            for k in range(coefficients.shape[2]):
                results += coefficients[i, j, k] * np.polynomial.legendre.legval(X, i) * np.polynomial.legendre.legval(Y, j) * np.polynomial.legendre.legval(Z, k)
    return results

result_vectorized = evaluate_legendre(X, Y, Z, coefficients)

Output:

A 3D array analogous to Method 1, containing evaluated series values at corresponding coordinates.

The above code defines a function evaluate_legendre to handle the series evaluation. The function calculates the Legendre polynomial values for each degree and dimension using broadcasting, yielding a grid of values. These are then scaled by the coefficients and summed to get the final result. This method boasts significant performance improvements for large datasets by leveraging vectorized operations.

Method 3: Employing SciPy’s Special Functions

This method takes advantage of specialized functions available in SciPy’s scipy.special module, which include implementations for Legendre polynomials. While similar to Method 1, it can provide performance improvements by using more optimized special function evaluations.

Here’s an example:

from scipy.special import lpmv

# Assume x, y, z, and coefficients are defined as before
results_scipy = np.zeros((len(x), len(y), len(z)))

# Evaluate the Legendre series
for l in range(coefficients.shape[0]):
    for m in range(coefficients.shape[1]):
        for n in range(coefficients.shape[2]):
            results_scipy += coefficients[l, m, n] * \
                             np.outer(np.outer(lpmv(0, l, x), lpmv(0, m, y)), lpmv(0, n, z))

Output:

A result array structured similarly to the outputs in previous methods.

By using scipy.special.lpmv, we compute Legendre polynomial values across each dimension and combine them using outer products to match the coefficient indices. This results in an array of evaluated series, where each element corresponds to a specific set of (x, y, z) coordinates. SciPy’s specialized functions usually offer more efficient computational strategies, which can be advantageous for performance.

Method 4: Exploiting SymPy’s Symbolic Computation

Method 4 explores symbolic computation through SymPy, a Python library for symbolic mathematics. This method is ideal when exact expressions are required or for symbolic manipulation and analysis before numerical evaluation.

Here’s an example:

from sympy import symbols, legendre, lambdify
import numpy as np

x, y, z = symbols('x y z')

# Assume coef is a sympy matrix
coef = sympy.Matrix(coefficients)

# Form the Legendre polynomial series symbolically
leg_series = sum(legendre(i, x) * legendre(j, y) * legendre(k, z) * coef[i, j, k] 
                 for i in range(coef.shape[0]) 
                 for j in range(coef.shape[1]) 
                 for k in range(coef.shape[2]))

# Lambdify the series for numerical evaluation
leg_series_num = lambdify((x, y, z), leg_series, "numpy")

# Evaluate the series over a grid
X, Y, Z = np.meshgrid(np.linspace(-1, 1, 10), np.linspace(-1, 1, 10), np.linspace(-1, 1, 10))
result_sympy = leg_series_num(X, Y, Z)

Output:

A 3D grid of evaluated series, similar to previous methods.

The code defines symbolic variables representing the coordinates, constructs the series symbolically, and then converts this expression into a numerical function using lambdify. This function is then evaluated over our grid. While generally slower than purely numerical approaches, this method provides exact arithmetic and symbolic manipulation capabilities.

Bonus One-Liner Method 5: Using Numba for JIT Compilation

Numba is a just-in-time compiler for Python that can dramatically speed up the computation of functions that are heavy on numerical loops, such as the evaluation of a 3D Legendre series.

Here’s an example:

from numba import jit
import numpy as np

# Assume x, y, z, and coefficients are defined as before

@jit
def evaluate_legendre_jit(X, Y, Z, coefficients):
    # Corresponding code to evaluate the series
    # Similar to that in Method 2, just decorated with @jit for JIT compilation

result_numba = evaluate_legendre_jit(X, Y, Z, coefficients)

Output:

Evaluation results as in previous methods, but usually faster due to JIT acceleration.

The @jit decorator is added to our function from Method 2, signaling to Numba to compile this function into fast machine code. This can lead to significant speed-ups especially in CPU-bound numerical calculations.

Summary/Discussion

  • Method 1: NumPy’s Polynomials. Straightforward implementation. May be slow for large data due to Python loops.
  • Method 2: Vectorized Computation. Faster for large datasets. Fully utilizes NumPy’s optimized operations.
  • Method 3: SciPy’s Special Functions. Makes use of optimized special functions. A balance between simplicity and performance.
  • Method 4: Symbolic Computation with SymPy. Provides exact results and symbolic manipulation. Less efficient for large numerical evaluations.
  • Method 5: JIT with Numba. Offers the best performance for large loop-driven computations. Requires compatibility with Numba’s limitations and types.