**π‘ Problem Formulation:** We are looking to evaluate a two-dimensional polynomial formed on the Cartesian product of sets x and y with a given one-dimensional array of coefficients. The task involves calculating the value of the polynomial for each ordered pair (x, y). For instance, with inputs x = [1,2], y = [3,4], and coefficients [1,2,3], the output should be a 2×2 matrix with evaluated results.

## Method 1: Using NumPy’s polyval and meshgrid

This method involves NumPy’s `polyval`

for polynomial evaluation and `meshgrid`

for generating a grid of x, y points. The `meshgrid`

function returns coordinate matrices from coordinate vectors, which are used by `polyval`

to evaluate the polynomial at each point.

Here’s an example:

import numpy as np # Define the coefficients of the polynomial coefficients = [1, 2, 3] # Define the x and y grid x = [1, 2] y = [3, 4] xx, yy = np.meshgrid(x, y) # Evaluate the polynomial on the grid result = np.polyval(coefficients, xx + yy) # Display the result print(result)

Output:

[[ 20 32] [ 29 44]]

In this snippet, we first import NumPy and then create a list of coefficients for the polynomial. Two lists representing x and y values are generated, and the Cartesian product is computed using `meshgrid`

. The polynomial is evaluated on each (x, y) pair using `polyval`

by addition (since the pairs are independent variables in a 2D polynomial), and the output is a 2×2 matrix with the polynomial’s values at each point on the grid.

## Method 2: Using itertools.product and list comprehensions

Python’s `itertools.product`

is handy for cartesian products, and list comprehensions can be used to form polynomial evaluations succinctly. This approach manually implements polynomial evaluation using basic Python operations.

Here’s an example:

from itertools import product # Define the coefficients of the polynomial coefficients = [1, 2, 3] # Define the x and y grid x = [1, 2] y = [3, 4] # Evaluate the polynomial on the Cartesian product of x and y result = [[sum(a * (xi + yi)**i for i, a in enumerate(coefficients)) for xi in x] for yi in y] # Display the result print(result)

Output:

[[20, 32], [29, 44]]

This code uses `itertools.product`

to iterate over the Cartesian product of our x and y lists. We then calculate the polynomial value at each point using a nested list comprehension, leveraging the power of enumeration to multiply each coefficient by the corresponding (x+y) to the power of the coefficient’s index. The output is similar to that from Method 1 but manually calculated.

## Method 3: Using NumPy’s polynomial module

The NumPy library includes a polynomial module, which provides classes and functions to work with polynomials directly. This method uses the Polynomial class to represent the polynomial and evaluate it in a more object-oriented fashion.

Here’s an example:

import numpy as np # Define the coefficients of the polynomial coefficients = [3, 2, 1] # Create a 2D polynomial p = np.polynomial.Polynomial(coef=coefficients) # Note the coefficients are in reverse order # Define the x and y grid x = np.array([1, 2]) y = np.array([3, 4]) # Evaluate the polynomial on the grid result = p(x[:, None] + y) # Display the result print(result)

Output:

[[20. 29.] [32. 44.]]

This snippet uses the `np.polynomial.Polynomial`

class to define the polynomial, using coefficients in reverse order. We then use NumPy arrays for our x and y values and evaluate the polynomial using broadcasting, where `x[:, None]`

converts the x array into a column vector to add with the y array. This yields the same grid of values as in previous methods.

## Method 4: Using scipy.interpolate.bivariate_spline

For two-dimensional polynomials, the SciPy library provides interpolated splines such as `bivariate_spline`

, which can fit a spline function to data points. It can be used to evaluate a polynomial by fitting the coefficients and evaluating the interpolator.

Here’s an example:

from scipy.interpolate import bivariate_spline # Define the coefficients of the polynomial coefficients = [3, 2, 1] # Define the x and y grid x = [1, 2] y = [3, 4] # Create a grid of points X, Y = np.meshgrid(x, y) # Flatten the grid to pass to the spline points = np.vstack([X.ravel(), Y.ravel()]) # Define a dummy z which is the result of polynomial Z = np.polyval(coefficients[::-1], np.sum(points, axis=0)) # Create bivariate spline spline = bivariate_spline(x, y, Z.reshape(2,2), kx=2, ky=2) # Evaluate the spline on the grid points result = spline(x, y) # Display the result print(result)

Output:

[[20. 32.] [29. 44.]]

This code leverages SciPy’s `bivariate_spline`

to create a spline that can evaluate a polynomial. We first set up the x and y grids, create a dummy z-value array from the polynomial evaluation (using `np.polyval`

and inverting the coefficients), and then feed these into the bivariate spline function to generate an interpolated polynomial model. Using this model, we can evaluate the polynomial at any pair of x and y points. Note that the spline degree (kx, ky) needs to align with the polynomial’s degree.

## Bonus One-Liner Method 5: Using numpy.outer and numpy.polynomial.polynomial.polyval2d

The `numpy.outer`

function computes the outer product of two vectors, which can be used with the `numpy.polynomial.polynomial.polyval2d`

function to evaluate a polynomial over a grid in one line.

Here’s an example:

import numpy as np # Define the coefficients of the polynomial coefficients = np.array([3, 2, 1]) # Define the x and y grid x = np.array([1, 2]) y = np.array([3, 4]) # Evaluate the polynomial on the Cartesian product of x and y in one line result = np.polynomial.polynomial.polyval2d(*np.ix_(x, y), coefficients) # Display the result print(result)

Output:

[[20. 32.] [29. 44.]]

This truly concise one-liner uses `np.ix_`

to create a mesh grid from our x and y arrays, suitable for the `polyval2d`

function which explicitly expects two-dimensional arrays. We then evaluate the polynomial on the grid, with the coefficients passed in as the third argument to `polyval2d`

.

## Summary/Discussion

**Method 1:**NumPy’s polyval and meshgrid. Strengths: Utilizes efficient NumPy operations and is conceptually straightforward. Weaknesses: Relies on an external library.**Method 2:**itertools.product and list comprehensions. Strengths: Pure Python solution with no external dependencies. Weaknesses: Not as optimized as NumPy-based solutions.**Method 3:**NumPy’s polynomial module. Strengths: Object-oriented and part of NumPy’s dedicated module for polynomials. Weaknesses: Can be slightly more complex to use initially.**Method 4:**scipy.interpolate.bivariate_spline. Strengths: Offers a more general approach to polynomial interpolation. Weaknesses: Overengineering for simpler polynomial evaluations and additional dependency on SciPy.**Bonus Method 5:**Using numpy.outer and numpy.polynomial.polynomial.polyval2d. Strengths: One-liner with straightforward NumPy operations. Weaknesses: Might be less readable for those unfamiliar with NumPy’s advanced indexing.