Evaluate a Legendre series at list of points x in Python

To evaluate a Legendre series at specific points, use the polynomial.legendre.legval() method in NumPy. This function allows you to compute Legendre polynomial values efficiently at single points or arrays of points.

Syntax

numpy.polynomial.legendre.legval(x, c, tensor=True)

Parameters

x: Points at which to evaluate the Legendre series. Can be a scalar, list, or array.

c: Array of coefficients where c[n] contains the coefficient for the degree-n term.

tensor: If True (default), evaluates every coefficient column for every element of x. If False, broadcasts x over coefficient columns.

Example

Let's evaluate a Legendre series with coefficients [1, 2, 3] at multiple points:

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

# Create an array of coefficients
c = np.array([1, 2, 3])

# Display the coefficient array
print("Coefficients:", c)
print("Dimensions:", c.ndim)
print("Shape:", c.shape)

# Define evaluation points
x = [5, 10, 15]

# Evaluate the Legendre series at points x
result = L.legval(x, c)
print("\nEvaluating at points", x)
print("Result:", result)

Coefficients: [1 2 3]
Dimensions: 1
Shape: (3,)

Evaluating at points [5, 10, 15]
Result: [ 122.  469.5 1042. ]

How It Works

The Legendre series is evaluated using the formula:

P(x) = c[0] × L?(x) + c[1] × L?(x) + c[2] × L?(x) + ...

Where L?(x) = 1, L?(x) = x, L?(x) = (3x² - 1)/2, etc.

Single Point Evaluation

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

# Coefficients for polynomial: 1 + 2x + 3L?(x)
coefficients = np.array([1, 2, 3])

# Evaluate at a single point
point = 2.0
result = L.legval(point, coefficients)
print(f"L({point}) = {result}")

L(2.0) = 25.0

Conclusion

Use legval() to efficiently evaluate Legendre series at single or multiple points. The function handles the complex Legendre polynomial calculations automatically, making it ideal for numerical analysis and scientific computing applications.