Gauss-Legendre Integration | Numerical Methods Lab

Numerical Methods - Gauss-Legendre Integration

Lab 5: Gauss-Legendre Numerical Integration

Lab 5: Gauss-Legendre Numerical Integration Method

Experiment Information

Experiment: Gauss-Legendre Numerical Integration

Course Code: Numerical Methods (SH202)

Description: Complete lab report covering theory, algorithm, Python implementation and analysis of Gauss-Legendre integration method

Prepared by: Important Notes Team

Complete Lab Report PDF

1. Theory of Gauss-Legendre Integration

1.1 Gaussian quadrature fundamentals

1.2 Legendre polynomials and roots

1.3 Weight factors calculation

2. Algorithm

2.1 Step-by-step procedure

2.2 Transformation of integration limits

2.3 Error estimation

3. Implementation

3.1 Python code with numpy

3.2 Function integration

3.3 Tabular data integration

4. Observations

4.1 Data recording table

4.2 Sample calculations

4.3 Accuracy comparison

5. Results

5.1 Integration results

5.2 Error analysis

5.3 Convergence behavior

6. Discussion

6.1 Advantages over other methods

6.2 Limitations and considerations

6.3 Applications in engineering

Digitized lab report prepared by Important Notes Team

Python Implementation for Function Integration

gauss_legendre_function.py

import numpy as np
from math import *

def f(x):
    return e**(-x**2)

def GL(f, a, b, n):
    xi, w = np.polynomial.legendre.leggauss(n)
    
    t = 0.5 * (xi + 1) * (b – a) + a

    I = np.sum(w * f(t) * (b – a) / 2)

    return I

a = float(input(“Enter Lower limit:”))
b = float(input(“Enter Upper limit:”))
n = int(input(“Enter Number of quadrature points:”))

print(“Gauss-Legendre Integration: “, round(GL(f, a, b, n),7))
        

Python Implementation for Tabular Data Integration

gauss_legendre_tabular.py

import numpy as np
from scipy.interpolate import interp1d

def GL_tabular(x_data, y_data, n_points):
    # Create interpolation function
    f = interp1d(x_data, y_data, kind=’cubic’, fill_value=’extrapolate’)
    
    a = min(x_data)
    b = max(x_data)
    
    # Get Gauss-Legendre quadrature points and weights
    xi, w = np.polynomial.legendre.leggauss(n_points)
    
    # Transform points to [a,b] interval
    t = 0.5 * (xi + 1) * (b – a) + a
    
    # Evaluate function at quadrature points
    y = f(t)
    
    # Calculate integral
    I = np.sum(w * y * (b – a) / 2)
    
    return I

# Example usage:
x_data = [0, 0.5, 1.0, 1.5, 2.0]  # Your x values
y_data = [1, 0.8, 0.5, 0.3, 0.1]  # Your y values
n = 5  # Number of quadrature points

result = GL_tabular(x_data, y_data, n)
print(“Integral value:”, result)
        