Evaluate a Legendre series at points x in Python

To evaluate a Legendre series at specific points x in Python, use the polynomial.legendre.legval() method from NumPy. This function computes the value of a Legendre polynomial series at given points using the provided coefficients.

Syntax

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

Parameters

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

c: Array of coefficients ordered so that coefficients for terms of degree n are in c[n]. For multidimensional arrays, remaining indices enumerate multiple polynomials.

tensor: If True (default), the coefficient array shape is extended for broadcasting. If False, x is broadcast over columns of c.

Basic Example

Let's evaluate a simple Legendre series with coefficients [1, 2, 3] at point x = 1 ?

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

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

print("Coefficients:", c)
print("Shape:", c.shape)
print("Dimensions:", c.ndim)

# Evaluate Legendre series at x = 1
result = L.legval(1, c)
print("Result at x=1:", result)

Coefficients: [1 2 3]
Shape: (3,)
Dimensions: 1
Result at x=1: 6.0

Multiple Points Evaluation

You can evaluate the same series at multiple points simultaneously ?

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

c = np.array([1, 2, 3])
x_points = np.array([0, 0.5, 1, 1.5])

result = L.legval(x_points, c)
print("X points:", x_points)
print("Results:", result)

X points: [0.  0.5 1.  1.5]
Results: [1.   2.75 6.   11.75]

How It Works

For coefficients [1, 2, 3], the Legendre series is: 1×P?(x) + 2×P?(x) + 3×P?(x), where P?(x)=1, P?(x)=x, and P?(x)=(3x²-1)/2.

Multidimensional Coefficients

You can work with multidimensional coefficient arrays to evaluate multiple series ?

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

# Two different series with coefficients in columns
c = np.array([[1, 2], [2, 3], [3, 1]])  # 3x2 array
x = np.array([0, 1])

result = L.legval(x, c)
print("Coefficient array shape:", c.shape)
print("Results shape:", result.shape)
print("Results:\n", result)

Coefficient array shape: (3, 2)
Results shape: (2, 2)
Results:
[[-0.5  0.5]
 [ 6.   6. ]]

Conclusion

The legval() function efficiently evaluates Legendre polynomial series at specified points. Use it for single or multiple point evaluations, and leverage multidimensional coefficients for evaluating multiple series simultaneously.