Integral Approximation Project

Overview

This project focuses on the numerical approximation of integrals using Chebyshev polynomials. It implements and compares the Trapezoidal and Simpson's methods for integral approximation.

Features

• Implements numerical integration using Trapezoidal and Simpson's methods.
• Utilizes Chebyshev polynomials of the first and second kind.
• Includes error measurement and optimization techniques.

Usage

1. Add the project folders to the MATLAB search path.
2. Use functions (trapezoidal, simpson) for approximate integral calculations of Chebyshev polynomials. In the project we provide chebyshev_combination function that represents polynomials in the form of: $$w_n(x) = \sum_{k=0}^n a_kT_k(x)U_k(x)$$ where $T_k$ is a Chebyshev polynomial of the first kind and $U_k$ - of the second kind. To use chebyshev_combination with trapezoidal (similarly simpson) use the following code (a - beggining of the integration interval, b - end of the integration interval, N - number of subintervals in the composite trapezoidal method, coefficients - $a_k$ coefficients for the Chebyshev polynomial):

    result = trapezoidal(a, b, N, @chebyshev_combination, coefficients)

3. Use functions (trapezoidal_general, simpson_general) for approximate integral calculations of any function $y = f(x)$. For example

    result = trapezoidal_general(a, b, N, @my_function)

4. Explore the GUI application examplesGUI.mlapp for visual demonstration and comparison of aprroximate results with true integrals.
5. Explore the GUI application errorsGUI.mlapp for error visualisations.

Examples

• Several examples (chebyshev_example_1, chebyshev_example_2, etc.) are provided to demonstrate the effectiveness of the methods in different scenarios. See the report for full description of the examples.

Error Analysis

• The project includes a detailed analysis of the errors associated with each method, including heat maps and relative error comparisons.
• One of the takeaways is that increasing the number of subintervals (hyperparameter N) for both methods decreases the relative error of the approximation.

Figure 1: Graph of the dependence of the relative error expressed in (%) on the number of subintervals in integration by the trapezoidal method

Authors

• Hubert Kowalski
• Adam Kaniasty

Date

• 25.11.2023