Linear Equation Solver: Gauss-Jordan and LU Decomposition

This repository contains Python implementations for solving systems of linear equations using two numerical methods:

1. Gauss-Jordan Elimination
2. LU Decomposition

These methods can be applied to solve any square system of linear equations. The code is written in a modular way for clarity and easy integration.

Features

• Solve a system of linear equations using the Gauss-Jordan Elimination method.
• Solve the same system using LU Decomposition (with forward and backward substitution).
• Handles errors for systems with no unique solutions.

Requirements

The following Python libraries are required:

• numpy

Install the dependencies using pip:

pip install numpy

Usage

Clone the repository and run the solver.py script:

git clone <repository-url>
cd <repository-directory>
python solver.py

Example Input

The script solves a system of equations, such as:

2x + y - z = 8
-3x - y + 2z = -11
-2x + y + 2z = -3

Represented in matrix form:

A = [[2,  1, -1],
     [-3, -1,  2],
     [-2,  1,  2]]

b = [8, -11, -3]

Output

The script provides solutions using both methods:

Solution using Gauss-Jordan:
[ 2.  3. -1.]

Solution using LU Decomposition:
[ 2.  3. -1.]

Functions

Gauss-Jordan Method

• Function: gauss_jordan(a, b)
• Description: Solves a system of equations by transforming the coefficient matrix into reduced row echelon form.

LU Decomposition Method

• Functions:
  • lu_decomposition(a)
  • forward_substitution(L, b)
  • backward_substitution(U, y)
  • solve_lu(a, b)
• Description: Decomposes the matrix into lower (L) and upper (U) triangular matrices, solves using substitution.

Error Handling

• Raises errors for singular matrices or systems with no unique solutions.

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Feel free to modify and expand the code for your specific use cases!