Convert a polynomial to Legendre series in Python

To convert a polynomial to a Legendre series, use the legendre.poly2leg() method in Python NumPy. This function converts an array representing the coefficients of a polynomial (ordered from lowest degree to highest) to an array of the coefficients of the equivalent Legendre series.

Syntax

numpy.polynomial.legendre.poly2leg(pol)

Parameters:

• pol − 1-D array containing the polynomial coefficients ordered from lowest to highest degree

Returns: 1-D array containing the coefficients of the equivalent Legendre series.

Example

Let's convert a polynomial with coefficients [1, 2, 3, 4, 5] to its Legendre series representation ?

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

# Create polynomial coefficients array
coefficients = np.array([1, 2, 3, 4, 5])

# Display the array
print("Polynomial coefficients:", coefficients)
print("Array dimensions:", coefficients.ndim)
print("Array shape:", coefficients.shape)
print("Data type:", coefficients.dtype)

# Convert polynomial to Legendre series
legendre_coeffs = L.poly2leg(coefficients)
print("\nLegendre series coefficients:", legendre_coeffs)

Polynomial coefficients: [1 2 3 4 5]
Array dimensions: 1
Array shape: (5,)
Data type: int64

Legendre series coefficients: [3.         4.4        4.85714286 1.6        1.14285714]

How It Works

The polynomial 1 + 2x + 3x² + 4x³ + 5x? is converted to its equivalent representation using Legendre polynomials. Each coefficient in the result corresponds to the weight of the respective Legendre polynomial in the series expansion.

Simple Example

Here's a simpler example with a quadratic polynomial ?

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

# Simple quadratic: 1 + x + x²
poly_coeffs = [1, 1, 1]
legendre_coeffs = L.poly2leg(poly_coeffs)

print("Original polynomial coefficients:", poly_coeffs)
print("Legendre series coefficients:", legendre_coeffs)

Original polynomial coefficients: [1, 1, 1]
Legendre series coefficients: [1.66666667 1.         0.33333333]

Conclusion

Use numpy.polynomial.legendre.poly2leg() to convert polynomial coefficients to Legendre series representation. This is useful in numerical analysis and scientific computing where Legendre polynomials provide better numerical properties for certain calculations.