LU Decomposition

Overview

This repository contains a MATLAB implementation of LU Decomposition, a method used for solving systems of linear equations, computing matrix inverses, and finding determinants. The algorithm decomposes a square matrix A into a lower triangular matrix L and an upper triangular matrix U, facilitating efficient solutions to linear systems.

Files

• LU_Decomposition.m: The main MATLAB function implementing the LU Decomposition algorithm. It takes a square coefficient matrix A and a constants vector B, and returns the solution vector X, along with matrices L and U.

How It Works

Methodology

1. Decomposition: The algorithm splits matrix A into L and U such that: [ A = LU ]

2. Forward Substitution: Solves the equation (LC = B) for intermediate vector C.

3. Backward Substitution: Solves the equation (UX = C) for the final solution vector X.

Function Signature

[X, L, U] = LU_Decomposition(A, B)

• A: Input square matrix (coefficient matrix).
• B: Input column vector (constants).
• X: Output solution column vector.
• L: Lower triangular matrix.
• U: Upper triangular matrix.

Example Usage

% Example coefficient matrix A and constants vector B
A = [4, 2, 1;
     2, 3, 1;
     1, 1, 2];

B = [1; 2; 3];

% Solve using LU Decomposition
[X, L, U] = LU_Decomposition(A, B);

% Display results
disp('Solution X:');
disp(X);
disp('Lower triangular matrix L:');
disp(L);
disp('Upper triangular matrix U:');
disp(U);

Prerequisites

• MATLAB (preferably R2018b or later)
• Basic understanding of linear algebra concepts.

Getting Started

1. Clone the repository:

   git clone https://github.com/iammohith/LU_Decomposition-in-MATLAB.git

2. Navigate to the project directory:

   cd LU_Decomposition-in-MATLAB

3. Open the LU_Decomposition.m script in MATLAB:

   • Launch MATLAB and navigate to the project folder.
   • Open LU_Decomposition.m to run the algorithm or modify it as needed.

Acknowledgements

• MATLAB Documentation for providing comprehensive resources on matrix operations and numerical methods.
• Various online resources and textbooks on numerical linear algebra for foundational concepts.

References

1. MATLAB Documentation
   https://www.mathworks.com/help/matlab/

2. Numerical Methods for Engineers, 8th Edition
   Steven C. Chapra, Raymond P. Canale
   ISBN: 978-0073397924

Contributing

Contributions are welcome! Feel free to fork this repository, make improvements, and submit a pull request.

License

This project is open source and available under the MIT License.