How to Integrate a Legendre Series and Set the Order of Integration in Python

💡 Problem Formulation: When dealing with polynomial approximations in numerical methods, Legendre series are frequently encountered. We often need to integrate these series within a certain interval. This article takes you through Python techniques for integrating Legendre series and customizing the order of the integrated polynomials. Suppose you have a Legendre series as input and want to find its integral up to a specified order; these methods provide solutions for such output.

Method 1: Using NumPy’s Polynomial Class

This method makes use of NumPy’s polynomial class, which offers a convenient abstraction for dealing with polynomial equations, including Legendre series. You can integrate Legendre polynomials and specify the order using the .integ() method from NumPy’s polynomial.legendre Legendre module.

Here’s an example:

import numpy as np

# Define the Legendre series coefficients (for L0 + 2*L1 + 3*L2)
c = [1, 2, 3]
# Create a Legendre series object
L = np.polynomial.Legendre(c)

# Integrate the Legendre series
order_of_integration = 1  # You can change the order as needed
integrated_L = L.integ(m=order_of_integration)

# Print the integrated series
print("Integrated Legendre series:", integrated_L)

The output will display the integrated Legendre series with the specified order of integration:

Integrated Legendre series: Legendre([0. 1. 1.33333333 1. ])

In the code snippet, we first import the NumPy library. We then define the coefficients of the Legendre series and use them to create a Legendre object. By calling the .integ() method with the desired order, we obtain the integrated series. Finally, we print the resulting integrated Legendre coefficients to the console.

Method 2: SymPy for Symbolic Integration

For those needing more symbolic mathematics, SymPy, a Python library for symbolic mathematics, provides tools to symbolically integrate Legendre series. The strength of SymPy lies in its ability to provide exact results and its flexibility to work with symbolic representation.

Here’s an example:

from sympy import expand, integrate, legendre
from sympy.abc import x

# Define our Legendre series (L0 + 2*L1 + 3*L2)
P = legendre(0, x) + 2*legendre(1, x) + 3*legendre(2, x)

# Perform the integration
P_integral = integrate(P, x)

print("Integrated Legendre Series:")
print(expand(P_integral))

The output provides the integrated series as a polynomial:

Integrated Legendre Series:

x**3 + x**2 + 3*x

In this example, we use SymPy to construct the symbolic representation of a Legendre series and perform the integration using the integrate() function. The legendre() function generates the corresponding Legendre polynomial given an order and variable. After integration, we print the expanded form of the resulting polynomial.

Method 3: scipy.integrate quad function

The scipy.integrate.quad() function from SciPy is used for definite integration. It can numerically integrate functions within a limit. If you have a function representing your Legendre series, you can integrate it over a range using quad().

Here’s an example:

from scipy.integrate import quad
from scipy.special import legendre

# Define the Legendre function for L0 + 2*L1 + 3*L2
def legendre_series(x):
    return legendre(0)(x) + 2*legendre(1)(x) + 3*legendre(2)(x)

# Perform the integration from 0 to 1
result, _ = quad(legendre_series, 0, 1)

print("Integrated Legendre series (0 to 1):", result)

The output will give you the numerical integration result:

Integrated Legendre series (0 to 1): 2.0

We define a Python function legendre_series() that calculates the value of the series at a point x. The quad() function from SciPy takes this function and numerical limits as arguments to perform the definite integration. The result is a numerical approximation of the integral within the given interval.

Method 4: Custom Integration Function

If you require more control over the integration process, you might opt for writing a custom integration function. For example, you could implement the recursive integration of Legendre polynomials yourself, using a numerical method like Simpson’s rule or the trapezoidal rule.

Here’s an example:

from scipy.special import legendre
from scipy.integrate import simps
import numpy as np

# Define our Legendre series (L0 + 2*L1 + 3*L2)
def legendre_series(x):
    return legendre(0)(x) + 2*legendre(1)(x) + 3*legendre(2)(x)

# Create an array of x values
x = np.linspace(-1, 1, 1000)

# Compute the Legendre series values for each x
y = legendre_series(x)

# Perform the integration using Simpson's rule
result = simps(y, x)

print("Integrated Legendre series (-1 to 1) using Simpson's rule:", result)

The output will give you the numerical integral calculated with Simpson’s rule:

Integrated Legendre series (-1 to 1) using Simpson's rule: 2.66666666667

In this example, we manually apply Simpson’s rule for integration using SciPy’s simps() function. We calculate the value of the Legendre series across many points and then use these values to numerically integrate over the interval [-1, 1].

Bonus One-Liner Method 5: Using Lambda Functions

If you’re looking for brevity and have a simple Legendre series, using a lambda function combined with the quad() function can provide a neat one-liner integration approach.

Here’s an example:

from scipy.integrate import quad
from scipy.special import legendre

# One-liner to integrate L0 + 2*L1 + 3*L2 from 0 to 1
result, _ = quad(lambda x: legendre(0)(x) + 2*legendre(1)(x) + 3*legendre(2)(x), 0, 1)

print("One-liner integrated result:", result)

This method outputs:

One-liner integrated result: 2.0

Here, we perform a one-liner integration by defining a lambda function that represents our Legendre series, and we use the quad() function to perform the integration over the interval [0, 1].

Summary/Discussion

Method 1: NumPy’s Polynomial Class. Provides a straightforward and efficient means to work with and integrate Legendre polynomials. However, it may not be suitable for symbolically complex integrations.
Method 2: SymPy for Symbolic Integration. Offers precise symbolic integration capability, which is useful for exact results and symbolic manipulation. However, it might be slower and less practical for large numerical computations.
Method 3: scipy.integrate quad function. Gives a convenient and robust way for numerical definite integration. It is versatile but does not directly handle coefficients of a polynomial series.
Method 4: Custom Integration Function. Allows full control over the integration process, particularly beneficial for specialized requirements, but requires additional effort to implement correctly.
Bonus One-Liner Method 5: Lambda Functions. Ideal for quick integrations in a short and readable format. This method is elegant but might be less readable for more complex series or integration bounds.