Evaluate a 2-D polynomial at points (x, y) in Python

To evaluate a 2-D polynomial at points (x, y), use the numpy.polynomial.polynomial.polyval2d() method. This function evaluates a two-dimensional polynomial at specified coordinate points and returns the computed values.

Syntax

numpy.polynomial.polynomial.polyval2d(x, y, c)

Parameters

The function accepts three parameters ?

• x, y ? Coordinates where the polynomial is evaluated. Must have the same shape.
• c ? Array of coefficients where c[i,j] contains the coefficient for the term of multidegree i,j.

Understanding Coefficient Array

The coefficient array c represents a 2-D polynomial where each element c[i,j] corresponds to the coefficient of x^i * y^j. For a 2×2 coefficient array, the polynomial takes the form ?

# For c = [[c00, c01], [c10, c11]]
# Polynomial: c00 + c01*y + c10*x + c11*x*y

Example

import numpy as np
from numpy.polynomial.polynomial import polyval2d

# Create a 2x2 coefficient array
c = np.arange(4).reshape(2, 2)
print("Coefficient Array:")
print(c)

print("\nArray Properties:")
print("Dimensions:", c.ndim)
print("Datatype:", c.dtype)
print("Shape:", c.shape)

# Evaluate polynomial at points (1,1) and (2,2)
x_points = [1, 2]
y_points = [1, 2]
result = polyval2d(x_points, y_points, c)

print("\nEvaluating at points (1,1) and (2,2):")
print("Result:", result)

Coefficient Array:
[[0 1]
 [2 3]]

Array Properties:
Dimensions: 2
Datatype: int64
Shape: (2, 2)

Evaluating at points (1,1) and (2,2):
Result: [ 6. 18.]

How It Works

For the coefficient array [[0, 1], [2, 3]], the polynomial is 0 + 1*y + 2*x + 3*x*y. At point (1,1): 0 + 1*1 + 2*1 + 3*1*1 = 6. At point (2,2): 0 + 1*2 + 2*2 + 3*2*2 = 18.

Different Input Shapes

import numpy as np
from numpy.polynomial.polynomial import polyval2d

# Coefficient array for polynomial: 1 + 2*y + 3*x + 4*x*y
c = np.array([[1, 2], [3, 4]])

# Single point evaluation
result1 = polyval2d(1, 2, c)
print("Single point (1,2):", result1)

# Multiple points with arrays
x_vals = np.array([0, 1, 2])
y_vals = np.array([0, 1, 2])
result2 = polyval2d(x_vals, y_vals, c)
print("Multiple points:", result2)

Single point (1,2): 12.0
Multiple points: [ 1. 10. 25.]

Conclusion

The polyval2d() function efficiently evaluates 2-D polynomials at specified coordinate points. The coefficient array structure directly maps to polynomial terms, making it intuitive for mathematical computations involving two-dimensional polynomial expressions.