TDMA - Tridiagonal Matrix Algorithm (Thomas Algorithm)

Overview

This repository contains a MATLAB implementation of the Tridiagonal Matrix Algorithm (TDMA), also known as the Thomas Algorithm, for solving systems of linear equations of the form AX = B where A is a tridiagonal matrix.

TDMA is a computationally efficient algorithm that reduces the complexity of solving such systems, making it especially useful in engineering and scientific computations where tridiagonal matrices arise naturally, such as in:

• Heat conduction problems
• Fluid flow simulations
• Finite Difference Method (FDM)
• Finite Volume Method (FVM)

Files

• TDMA.m: The main MATLAB function implementing the Thomas Algorithm. It accepts a tridiagonal matrix A and a constants vector B, and returns the solution vector X.

How It Works

Methodology

The algorithm consists of two main steps:

1. Forward Elimination: Eliminates the sub-diagonal entries and modifies the diagonal and constant vectors.
2. Backward Substitution: Solves for the unknowns starting from the last equation and moving upwards.

Function Signature

X = TDMA(A, B)

• A: Input tridiagonal coefficient matrix (square matrix).
• B: Input column vector of constants.
• X: Output solution vector.

Example Usage

% Example tridiagonal matrix A and vector B
A = [4, 1, 0;
     1, 4, 1;
     0, 1, 4];

B = [5; 6; 5];

% Solve using TDMA
X = TDMA(A, B);

% Display result
disp(X);

Prerequisites

• MATLAB (preferably R2018b or later)
• Basic understanding of numerical methods and matrix operations.

Getting Started

Installation

1. Clone the repository:

   git clone https://github.com/iammohith/TDMA-Tridiagonal-Matrix-Algorithm-in-MATLAB.git

2. Navigate to the project directory:

   cd TDMA-Tridiagonal-Matrix-Algorithm-in-MATLAB

3. Open the TDMA.m script in MATLAB:

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

Acknowledgements

I would like to acknowledge the following sources for their contributions to this project:

• MATLAB Documentation for providing comprehensive resources on matrix operations and numerical methods.
• Numerical Methods for Engineers by Steven C. Chapra and Raymond P. Canale for foundational concepts and problem-solving techniques in numerical methods, including algorithms for solving tridiagonal systems.

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
   This textbook provides in-depth coverage of numerical algorithms, including the Thomas Algorithm (TDMA), used in solving engineering problems.

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.