This program implements the Gaussian Elimination algorithm, a fundamental method in linear algebra for solving systems of linear equations, determining matrix inverses, and calculating determinants.
The project is inspired by the teachings of Stefan Harmeling in the Linear Algebra lecture at TU Dortmund. It translates mathematical concepts into a practical and interactive tool with a graphical user interface (GUI) built in WPF (Windows Presentation Foundation).
- Matrix Input: Allows users to input a matrix and a right-hand side vector via a GUI.
- Gaussian Elimination: Automatically performs forward elimination and backward substitution to solve the system of equations.
- Detailed Logs: Displays step-by-step logs of the calculations, highlighting pivot operations, row eliminations, and the final solution.
- Results Display: Clearly shows the original matrix, the transformed matrix, the right-hand side vector, and the solution in a structured format inspired by LaTeX-style formatting.
- Error Handling: Validates user input and provides feedback if the system is unsolvable or invalid data is provided.
The program implements the Gaussian Elimination algorithm in two main phases:
- Goal: Transform the input matrix into an upper triangular matrix.
- For each pivot column:
- Ensure the pivot element is non-zero (row swaps if necessary).
- Eliminate all entries below the pivot by subtracting suitable multiples of the pivot row.
- Goal: Solve for the unknowns starting from the last row.
- Starting from the last equation, substitute the known values into the equations above to solve for all variables.
Example: For the system Ax = b, where:
A = [[2, 1], [1, 3]],
b = [8, 13]
The program performs elimination to solve for x.
-
Input Matrix and Vector:
- Enter the matrix and right-hand side vector in the input fields provided in the GUI.
- Alternatively, use the "Fill with Random Values" button to generate a random solvable system.
-
Start the Calculation:
- Click the "Solve" button to begin the Gaussian Elimination process.
- The program will validate your input, perform the calculations, and display the results.
-
Review the Results:
- The original matrix, the transformed matrix, the solution vector, and detailed logs are displayed in the result window.
-
Close the Result Window:
- Use the "OK" button to close the result window.
While the program has been extensively tested, there may still be edge cases or unexpected errors. For example:
- Extremely large matrices or floating-point precision issues might lead to inaccuracies.
- Non-invertible matrices (singular matrices) will cause the program to throw exceptions.
If you encounter any bugs or errors, I would greatly appreciate it if you could report them.
Feel free to reach out to me via my website: Leon Burghardt. Your feedback will help improve the tool for everyone!
This program was developed by Leon Burghardt in 2024. All rights reserved.
Thank you for using this program! I hope it serves as a helpful tool for understanding and applying Gaussian Elimination in your studies or work.