Evaluate a 2-D polynomial on the Cartesian product of x and y with 1d array of coefficient in Python

To evaluate a 2-D polynomial on the Cartesian product of x and y, use the numpy.polynomial.polynomial.polygrid2d() method in Python. This function returns the values of the two-dimensional polynomial at points in the Cartesian product of x and y arrays.

Syntax

numpy.polynomial.polynomial.polygrid2d(x, y, c)

Parameters

The function accepts three parameters:

• x, y: One-dimensional arrays of coordinates. If x or y is a list or tuple, it is first converted to an ndarray
• c: Array of coefficients ordered so that coefficients for terms of degree i,j are contained in c[i,j]. If c has fewer than two dimensions, ones are implicitly appended to make it 2-D

Example

Let's create a 1D array of coefficients and evaluate a 2-D polynomial ?

import numpy as np
from numpy.polynomial.polynomial import polygrid2d

# Create a 1d array of coefficients
c = np.array([3, 5])

# Display the array
print("Our Array...")
print(c)

# Check the dimensions
print("\nDimensions of our Array...")
print(c.ndim)

# Get the datatype
print("\nDatatype of our Array object...")
print(c.dtype)

# Get the shape
print("\nShape of our Array object...")
print(c.shape)

# Evaluate 2-D polynomial on Cartesian product
print("\nResult...")
print(polygrid2d([1,2], [1,2], c))

The output of the above code is ?

Our Array...
[3 5]

Dimensions of our Array...
1

Datatype of our Array object...
int64

Shape of our Array object...
(2,)

Result...
[21. 34.]

How It Works

The function evaluates the polynomial with coefficients c = [3, 5] at the Cartesian product of points [1,2] and [1,2]. Since c is 1-dimensional, it's treated as a 2-D array with shape (2,1), representing the polynomial 3 + 5*x.

import numpy as np
from numpy.polynomial.polynomial import polygrid2d

# Example with 2D coefficient array
c_2d = np.array([[1, 2], [3, 4]])  # Represents: 1 + 2*y + 3*x + 4*x*y

# Evaluate at specific points
x_vals = [0, 1]
y_vals = [0, 1] 

result = polygrid2d(x_vals, y_vals, c_2d)
print("2D polynomial result:")
print(result)

The output shows the evaluation at all combinations ?

2D polynomial result:
[[ 1.  3.]
 [ 4. 10.]]

Conclusion

The polygrid2d() function efficiently evaluates 2-D polynomials on Cartesian products of coordinate arrays. It automatically handles coefficient array dimensions and returns results for all point combinations.