GaussianQuadrature

Overview

This set of Python scripts provides tools for performing numerical integration using various methods including Gauss-Legendre and Gauss-Chebyshev quadrature, the trapezoidal rule, and Simpson's rule. The GaussQuad.py file contains classes and functions for calculating basis polynomials and applying Gaussian Quadrature, while comparisons.py includes methods for comparing the accuracy of different numerical integration techniques.

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src/GaussQuad.py

Description

This file defines classes and methods for computing basis polynomials and performing Gauss Quadrature integration. It supports both Legendre and Chebyshev polynomials.

Key Classes and Functions

• BasisPolynomials: Class for calculating basis polynomials (Legendre, Chebyshev) and their derivatives.
• GaussQuadrature: Class for performing Gauss Quadrature integration.
• Gaussian_Quad: Function for executing Gaussian quadrature and returning the integral value, nodes, and weights.

Usage

1. Import the GaussQuadrature class or Gaussian_Quad function.
2. Create a function to integrate.
3. Use Gaussian_Quad to perform the integration, specifying the number of nodes, interval, and method ('legendre' or 'chebyshev').

CSV File Format

When using GaussQuadrature.generate_and_save(), the script generates a CSV file containing two columns:

• Nodes: The computed nodes for the quadrature.
• Weights: The corresponding weights for each node.

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src/Comparisons.py

Description

This file contains functions to compare the performance of various numerical integration methods.

Key Functions

• trapezoidal_rule: Implements the trapezoidal rule for integration.
• simpsons_rule: Implements Simpson's rule for integration.
• compare_methods: Compares the convergence and error of Gauss-Legendre, Gauss-Chebyshev, trapezoidal rule, and Simpson's rule.

Usage

1. Import the required functions.
2. Define the function to integrate and, if necessary, a modified version for use with Chebyshev quadrature.
3. Call compare_methods with the standard and modified functions, the exact integral value for comparison, the interval of integration, and the maximum power of 2 for the number of intervals or nodes.

Visual Output

The compare_methods function generates a plot showing the convergence and error of each method.

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Requirements

• Python 3.x
• Libraries: numpy, pandas, matplotlib, scipy, math

Installation

Ensure that Python 3.x is installed and the required libraries are available. If not, they can be installed via pip:

pip install numpy pandas matplotlib scipy