Evaluate a 2D Legendre series at points (x, y) with 1D array of coefficient in Python

To evaluate a 2D Legendre series at points (x, y), use the polynomial.legendre.legval2d() method in NumPy. This method returns the values of the two-dimensional Legendre series at points formed from pairs of corresponding values from x and y arrays.

Syntax

numpy.polynomial.legendre.legval2d(x, y, c)

Parameters

The function takes the following parameters:

• x, y: The two dimensional series is evaluated at points (x, y). Both must have the same shape. If x or y is a list or tuple, it is converted to an ndarray.
• c: Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in c[i,j].

Example

Let's create a 1D array of coefficients and evaluate a 2D Legendre series at specific points ?

import numpy as np
from numpy.polynomial import legendre as L

# 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:")
print(c.dtype)

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

# Evaluate 2D Legendre series at points (1,1) and (2,2)
result = L.legval2d([1, 2], [1, 2], c)
print("\nResult:")
print(result)

Our Array:
[3 5]

Dimensions of our Array:
1

Datatype of our Array:
int64

Shape of our Array:
(2,)

Result:
[21. 34.]

How It Works

The legval2d() function evaluates the 2D Legendre polynomial using the coefficient array. For a 1D coefficient array [3, 5], it treats this as coefficients for terms of different degrees in the Legendre series expansion.

Multiple Points Example

You can evaluate the series at multiple coordinate pairs simultaneously ?

import numpy as np
from numpy.polynomial import legendre as L

# Create coefficient array
coefficients = np.array([2, 4, 1])

# Evaluate at multiple points
x_points = [0, 1, 2]
y_points = [0, 1, 2]

result = L.legval2d(x_points, y_points, coefficients)
print("Coefficients:", coefficients)
print("X points:", x_points)
print("Y points:", y_points)
print("Results:", result)

Coefficients: [2 4 1]
X points: [0, 1, 2]
Y points: [0, 1, 2]
Results: [ 6. 13. 28.]

Conclusion

The legval2d() function efficiently evaluates 2D Legendre series at specified coordinate points. It accepts 1D coefficient arrays and coordinate pairs, returning the computed polynomial values at each point.