Evaluate a 2-D polynomial on the Cartesian product of x and y with 3d array of coefficient in Python

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

The first parameter, x and y, are the coordinate arrays evaluated at points in the Cartesian product. If x or y is a list or tuple, it is first converted to an ndarray. The second parameter, c, is an array of coefficients where coefficients for terms of degree i,j are contained in c[i,j]. If c has dimension greater than two, the remaining indices enumerate multiple sets of coefficients.

Syntax

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

Parameters

x, y: Array-like coordinates for evaluation
c: Array of coefficients with shape (M, N, ...) where M and N represent polynomial degrees

Understanding 3D Coefficient Arrays

When c has more than 2 dimensions, the shape of the result will be c.shape[2:] + x.shape + y.shape. This allows evaluation of multiple polynomial sets simultaneously ?

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

# Create a 3D array of coefficients (2x2x6)
c = np.arange(24).reshape(2, 2, 6)

print("Coefficient Array Shape:", c.shape)
print("First coefficient set:\n", c[:, :, 0])

Coefficient Array Shape: (2, 2, 6)
First coefficient set:
 [[ 0  6]
 [12 18]]

Example: Evaluating Multiple Polynomials

Here's a complete example showing how to evaluate multiple 2D polynomials simultaneously ?

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

# Create a 3D array of coefficients
c = np.arange(24).reshape(2, 2, 6)

print("Our Array...")
print(c)

print("\nDimensions of our Array:", c.ndim)
print("Datatype of our Array:", c.dtype)
print("Shape of our Array:", c.shape)

# Evaluate on Cartesian product of [1,2] x [1,2]
result = polygrid2d([1, 2], [1, 2], c)

print("\nResult shape:", result.shape)
print("Result...")
print(result)

Our Array...
[[[ 0  1  2  3  4  5]
  [ 6  7  8  9 10 11]]

 [[12 13 14 15 16 17]
  [18 19 20 21 22 23]]]

Dimensions of our Array: 3
Datatype of our Array: int64
Shape of our Array: (2, 2, 6)

Result shape: (6, 2, 2)
Result...
[[[ 36.  60.]
  [ 66. 108.]]

 [[ 40.  66.]
  [ 72. 117.]]

 [[ 44.  72.]
  [ 78. 126.]]

 [[ 48.  78.]
  [ 84. 135.]]

 [[ 52.  84.]
  [ 90. 144.]]

 [[ 56.  90.]
  [ 96. 153.]]]

How the Result is Structured

The result has shape (6, 2, 2) because:

• 6 comes from the third dimension of coefficients (multiple polynomial sets)
• 2, 2 comes from the Cartesian product of [1,2] × [1,2]

Each 2×2 matrix represents the evaluation of one polynomial set at the four coordinate pairs: (1,1), (1,2), (2,1), and (2,2).

Conclusion

The polygrid2d() function efficiently evaluates multiple 2D polynomials simultaneously using 3D coefficient arrays. The result shape follows the pattern c.shape[2:] + x.shape + y.shape, making it useful for batch polynomial evaluations.