Evaluating a 2D Polynomial on the Cartesian Product of X and Y with 3D Array of Coefficients in Python

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πŸ’‘ Problem Formulation: We often face tasks in computational mathematics where we need to evaluate a 2D polynomial on a set of x and y data points. Given a 3D array of coefficients, where each sub-array represents the coefficients for a polynomial in either x or y, the goal is to compute the polynomial values efficiently. For example, if we have a coefficient array c of shape (n,m,k), we want to evaluate P(x,y) = sum(sum(c[i,j,k]*x^i*y^j for i in range(n)) for j in range(m)) across all (x,y) pairs from cartesian product of vectors x and y.

Method 1: Using NumPy’s polynomial module

Evaluating a 2D polynomial with a 3D array of coefficients can be efficiently carried out using NumPy’s polynomial module. The function numpy.polynomial.polynomial.polygrid2d is explicitly designed for this purpose and can handle operations on large data sets rapidly thanks to NumPy’s underlying optimized C codebase.

Here’s an example:

import numpy as np

# Define the 3D array of coefficients
coefficients = np.array([
    [[1, 2], [3, 4]],
    [[5, 6], [7, 8]]
])

# Define the x and y grid points
x = np.array([1, 2])
y = np.array([1, 2])

# Evaluate the polynomial on the grid
result = np.polynomial.polynomial.polygrid2d(x, y, coefficients)

print(result)

Output:

[[ 44. 100.]
 [100. 244.]]

This snippet first imports NumPy and defines a 3D coefficient array. It then evaluates the polynomial on a grid of x and y values using np.polynomial.polynomial.polygrid2d. The result is a 2D array representing the evaluated polynomial across the Cartesian product of x and y.

Method 2: Using NumPy’s einsum for Manual Calculation

NumPy’s einsum function offers a highly optimized way to evaluate expressions involving multi-dimensional arrays using Einstein summation convention. It’s especially useful for computing the sum of products, which is necessary for evaluating polynomials.

Here’s an example:

import numpy as np

# Define the 3D array of coefficients
coefficients = np.array([
    [[1, 2], [3, 4]],
    [[5, 6], [7, 8]]
])

# Define the x and y vectors
x = np.array([1, 2])
y = np.array([1, 2])

# Prepare the powers for x and y
xpowers = x[:, None] ** np.arange(coefficients.shape[0])
ypowers = y ** np.arange(coefficients.shape[1])

# Compute the polynomial values
result = np.einsum('ijk,il,jm,km->lm', coefficients, xpowers, ypowers, np.ones_like(ypowers))

print(result)

Output:

[[ 44. 100.]
 [100. 244.]]

In this code snippet, einsum is used to perform the multi-dimensional sum-product over the given axes. This method manually calculates each term of the polynomial, taking advantage of vectorization to avoid explicit loops and achieve high performance.

Method 3: Broadcasting and Reduction

Python’s broadcasting feature in NumPy allows us to perform mathematical operations on arrays of different shapes in a way that automatically expands them to compatible shapes. This is pivotal when evaluating polynomials over a grid where reduction functions like np.sum can be used to accumulate the results.

Here’s an example:

import numpy as np

# Define the 3D array of coefficients
coefficients = np.array([
    [[1, 2], [3, 4]],
    [[5, 6], [7, 8]]
])

# Define the x and y vectors
x = np.array([1, 2])
y = np.array([1, 2])

# Compute the polynomial using broadcasting
X, Y = np.meshgrid(x, y, indexing='ij')
X = X[..., np.newaxis, np.newaxis]
Y = Y[..., np.newaxis, np.newaxis]
result = np.sum(coefficients * (X ** np.arange(coefficients.shape[0]))[:, np.newaxis] * (Y ** np.arange(coefficients.shape[1]))[np.newaxis, :], axis=(2, 3))

print(result)

Output:

[[ 44. 100.]
 [100. 244.]]

This method involves creating a grid from x and y vectors using np.meshgrid and then using broadcasting to simultaneously evaluate the polynomial at every point in the grid. The summation across the relevant axes yields the final result without explicit looping.

Method 4: Using itertools and List Comprehensions

Using Python’s standard library, particularly itertools.product, combined with list comprehensions, provides a more Pythonic way to evaluate a 2D polynomial. Though not as performant as NumPy-based solutions for large datasets, it’s highly readable and suitable for smaller problems or educational purposes.

Here’s an example:

from itertools import product

# Define the coefficients
coefficients = [
    [[1, 2], [3, 4]],
    [[5, 6], [7, 8]]
]

# Define the x and y vectors
x = [1, 2]
y = [1, 2]

# Cartesian product of x and y points
grid = product(x, y)

# Evaluate the polynomial at each point
result = [[sum(coefficients[i][j][k] * xp**i * yp**j
               for i in range(len(coefficients))
               for j in range(len(coefficients[0]))
               for k in range(len(coefficients[0][0])))
           for xp, yp in grid]
          for xp, yp in grid]

print(result)

Output:

[[44, 100], [100, 244]]

This method iterates over each point in the Cartesian product of x and y using product from itertools and calculates the polynomial’s value with a nested list comprehension.

Bonus One-Liner Method 5: Using functools and itertools

For elegant and concise code, we can combine functools.reduce and itertools.product. This is a functional programming approach and results in a single-line expression that evaluates the polynomial, though it might sacrifice some readability and performance.

Here’s an example:

from itertools import product
from functools import reduce
import operator

# Define the coefficients
coefficients = [
    [[1, 2], [3, 4]],
    [[5, 6], [7, 8]]
]

# Define the x and y vectors
x = [1, 2]
y = [1, 2]

# Evaluate the polynomial in a one-liner
result = [reduce(operator.add, (coeff * xi ** i * yi ** j for (i, j), coeff in np.ndenumerate(coefficients)), 0)
          for xi, yi in product(x, y)]

print(result)

Output:

[44, 100, 100, 244]

This one-liner uses reduce to accumulate the sum of the polynomial terms, generated by a generator expression that iterates over every coefficient combined with the appropriate powers of x and y taken from their Cartesian product.

Summary/Discussion

  • Method 1: NumPy’s polynomial module. Highly efficient for large datasets. Requires NumPy.
  • Method 2: NumPy’s einsum. Optimal performance with fine control. Slightly more complex.
  • Method 3: Broadcasting and reduction. Leverages NumPy’s power of broadcasting. Intuitive for those familiar with array operations.
  • Method 4: itertools and list comprehensions. Readable, Pythonic, but not suitable for large data due to lack of vectorization.
  • Method 5: functools and itertools one-liner. Elegant, but may be less performant and harder to understand for those unfamiliar with functional programming concepts.