Update README and final touches

README.md

@@ -1,7 +1,7 @@
 # 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.
+This project focuses on the numerical approximation of integrals using [Chebyshev polynomials](https://en.wikipedia.org/wiki/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.
@@ -10,14 +10,28 @@ This project focuses on the numerical approximation of integrals using Chebyshev
 
 ## Usage
 1. Add the project folders to the MATLAB search path.
-2. Use provided functions (`trapezoidal`, `simpson`, `chebyshev_combination`) for integral calculations.
-3. Explore the GUI application (`examplesGUI.mlapp`) for visual demonstration and error analysis.
+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):
+```matlab
+    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
+```matlab
+    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.
+- Several examples (`chebyshev_example_1`, `chebyshev_example_2`, etc.) are provided to demonstrate the effectiveness of the methods in different scenarios. See the [report](Integral_Approximation_Report_ENG.pdf) 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.
+
+<div align="center">
+  <img src="img/trapezoidal_error.png" alt="Error of trapezoidal method" width="500">
+  <p>Figure 1: Graph of the dependence of the relative error expressed in (%) on the number of subintervals in integration by the trapezoidal method</p>
+</div>
 
 ## Authors
 - Hubert Kowalski

scripts/errors/kaniastyKowalskiError.m

@@ -23,7 +23,7 @@
    errors = zeros(1, maxN); % Inicjalizacja wektora błędów
    for N = 1:maxN % Pętla obliczająca błąd dla każdej liczby przedziałów
        approxIntegral = method(a, b, N, @chebyshev_combination, coefficients); % Obliczanie przybliżonej całki
-       unpenalizedError = abs(trueIntegral - approxIntegral) / trueIntegral; % Obliczanie błędu bez kary
+       unpenalizedError = abs((trueIntegral - approxIntegral) / trueIntegral); % Obliczanie błędu bez kary
        penalty = log(1+N); % Obliczanie kary kwadratowej
        errors(N) = unpenalizedError + penalty; % Sumowanie błędu i kary
    end

scripts/tests/plotN.m

@@ -41,8 +41,8 @@
             simpsonIntegral = simpson(a, b, i, @chebyshev_combination, currentCoefficients);
 
             % Obliczenie błędów względnych
-            errors_s(j, i) = abs(trueIntegral - simpsonIntegral) / trueIntegral;
-            errors_t(j, i) = abs(trueIntegral - trapezoidalIntegral) / trueIntegral;
+            errors_s(j, i) = abs((trueIntegral - simpsonIntegral) / trueIntegral);
+            errors_t(j, i) = abs((trueIntegral - trapezoidalIntegral) / trueIntegral);
         end
     end
 end

scripts/tests/runTestsTable.m

@@ -1,3 +1,8 @@
+% Skrypt testujący
+% Skrypt generuje tabelę zawierającą prawdziwe wartości całki oraz całki
+% wyznaczone numerycznie za pomocą metody Simpsona i trapezów dla 12
+% przykładowych funkcji.
+
 examples = cell(12,1);
 
 examples{1} = struct('a', -2, 'b', 10, 'N', 50, 'Func', @chebyshev_example_1);
@@ -10,7 +15,7 @@
 examples{8} = struct('a', -1, 'b', 1, 'N', 50, 'Func', @chebyshev_example_8);
 examples{9} = struct('a', -5, 'b', 5, 'N', 50, 'Func', @chebyshev_example_9);
 examples{10} = struct('a', -0.7, 'b', 0, 'N', 50, 'Func', @chebyshev_example_10); 
-examples{11} = struct('a', -1, 'b', -0.5, 'N', 50, 'Func', @example_11);s
+examples{11} = struct('a', -1, 'b', -0.5, 'N', 50, 'Func', @example_11);
 examples{12} = struct('a', -10, 'b', 10, 'N', 100, 'Func', @example_12);