Gaussian Elimination Algorithm

This work presents a computational implementation of the Gaussian Elimination method for solving systems of linear equations. The system accepts user-defined equations in symbolic string format, converts them into an augmented matrix representation, and applies elementary row operations to obtain a row echelon form. The final solution is computed via back substitution. A graphical interface is provided to facilitate user interaction.

Introduction

This project implements the method programmatically using Python, with the following objectives:

• Accept arbitrary linear systems as input (string format)
• Convert symbolic equations into matrix representation
• Apply Gaussian Elimination to obtain a Row Echelon Form (REF)
• Compute solutions using Back Substitution
• Provide a simple graphical user interface (GUI)

Problem Representation

A linear system is represented as:

$$ A\mathbf{x} = \mathbf{B} $$

Where:

• $A \in \mathbb{R}^{n \times n}$ : coefficient matrix
• $\mathbf{x} \in \mathbb{R}^n$ : vector of variables
• $\mathbf{B} \in \mathbb{R}^n$ : constant vector

The system is internally transformed into an augmented matrix:

$$ [A \mid \mathbf{B}] $$

Gaussian Elimination

The algorithm transforms the augmented matrix into Row Echelon Form (REF) using elementary row operations:

• Row Swapping (pivoting)
• Row Scaling (normalizing pivots)
• Row Elimination (zeroing values below pivots)

This process is implemented in the primary routine: row_echelon_form(A, B)

Back Substitution

Once the matrix is in row echelon form, back substitution is applied to compute the solution vector.

Implemented in: back_substitution(M)

Steps:

1. Traverse rows in reverse order.
2. Eliminate coefficients above pivots.
3. Extract the solution from the last column.

Input Parsing

User-provided equations are parsed using symbolic computation: string_to_augmented_matrix(equations)

Responsibilities:

• Extract variable names dynamically.
• Convert expressions using symbolic algebra.
• Build the coefficient matrix $A$ and vector $\mathbf{B}$.

Example Input:

3*x + 6*y + 6*w + 8*z = 1
5*x + 3*y + 6*w = -10
4*y - 5*w + 8*z = 8
4*w + 8*z = 9

Graphical Interface

A simple GUI is implemented using PySimpleGUI:

• Multiline input for equations
• Submit / Cancel controls
• Popup display for results or errors

Features:

• Input validation
• Automatic enabling/disabling of controls
• Error handling for singular or invalid systems

Results

The system successfully computes solutions for valid, non-singular linear systems.

Output is formatted as:

x = -1.5414
y = -0.5223
w = -0.1210
z = 1.1855

Invalid or singular systems trigger appropriate error messages.

Limitations

• Only supports systems with a unique solution.
• Numerical instability may occur for ill-conditioned matrices (no full pivoting).