NAMF - Numerical Analysis MATLAB Functions

A comprehensive collection of MATLAB functions for numerical analysis, organized by mathematical method categories.

📋 Table of Contents

• Overview
• Project Structure
• Function Categories
• Installation and Usage
• Examples
• Contributing
• License

🎯 Overview

NAMF is a collection of MATLAB implementations of fundamental numerical analysis algorithms, developed for educational and research purposes. The functions cover three main areas:

• Quadrature Formulas: Numerical methods for computing definite integrals
• Root Finding: Iterative algorithms for finding function zeros
• Linear Systems: Direct and iterative methods for system solving

📁 Project Structure

NAMF/
├── Quadrature_Formulas/           # Numerical integration methods
├── Root_Finding/                  # Root finding algorithms
├── Linear_Systems/                # Linear system solving
│   ├── Factorizations/           # Factorization methods
│   └── Iterative_Methods/        # Iterative methods
├── LICENSE
└── README.md

🔧 Function Categories

📊 Quadrature Formulas

Methods to estimate definite integral values without computing the antiderivative:

Function	Description
trap_comp_integr_dop.m	Double integration with composite trapezoids
NewCot_chiu_semp.m	Simple closed Newton-Cotes
NewCot_chiu_comp.m	Composite closed Newton-Cotes
NewCot_aper_semp.m	Simple open Newton-Cotes
NewCot_aper_comp.m	Composite open Newton-Cotes
NCcc_Num_iter.m	Newton-Cotes iteration number calculation
NewCot_chiu_semp_pol_carat.m	Simple closed Newton-Cotes for characteristic polynomial

🎯 Root Finding Functions

Iterative algorithms to find α ∈ [a,b] such that f(α) = 0:

Function	Description
Bisect.m	Bisection method
rad_newton_vett.m	Vectorial Newton method
rad_pol_newton.m	Newton method for polynomials
Horner.m	Horner's algorithm
Frobenius.m	Frobenius method
rad_minmax.m	Min/max root search
max_iter_trapezi.m	Maximum iterations for trapezoidal rule
newton_mat_trid_sim.m	Newton method for symmetric tridiagonal matrix eigenvalues
val_pol_carat_trid.m	Characteristic polynomial evaluation for tridiagonal matrices

🔢 Linear Systems

Factorizations

Function	Description
Fatt_LU.m	LU factorization with pivoting
Fatt_LU_NOPIV.m	LU factorization without pivoting
Fatt_QR.m	QR factorization

Iterative Methods

Function	Description
Jacobi.m	Jacobi method
GaussSeidel.m	Gauss-Seidel method

System Solving

Function	Description
RSL_SI.m	Lower triangular system solving (forward substitution)
RSL_SA.m	Upper triangular system solving (backward substitution)
RisolSisMatTrid.m	Tridiagonal matrix system solving

🚀 Installation and Usage

1. Clone the repository:

   git clone https://github.com/username/NAMF.git

2. Add to MATLAB path:

   addpath(genpath('path/to/NAMF'))

3. Use the functions:

   % Example: Bisection method
   f = 'x^2 - 2';
   [x, fx, n] = Bisect(f, 0, 2, 1e-6);
   
   % Example: Iterative methods
   A = [4 -1; -1 4]; b = [1; 2]; x0 = [0; 0];
   [x, iter] = Jacobi(A, b, x0, 1e-6, 100);

4. Run tests:

   NAMF_tests  % Run comprehensive test suite

📝 Examples

Numerical Integration

% Double integral with composite trapezoids
f = @(x,y) x.*y + sin(x.*y);
result = trap_comp_integr_dop(0, 1, 0, 1, 10, 10, f);

Root Finding

% Bisection to find √2
f = 'x^2 - 2';
[root, fval, iter] = Bisect(f, 1, 2, 1e-10);
fprintf('√2 ≈ %.10f (iterations: %d)\n', root, iter);

Linear Systems

% LU factorization
A = [4 3; 6 3];
[L, U] = Fatt_LU(A);

🤝 Contributing

Contributions are welcome! To contribute:

1. Fork the project
2. Create a feature branch (git checkout -b feature/AmazingFeature)
3. Commit your changes (git commit -m 'Add AmazingFeature')
4. Push to the branch (git push origin feature/AmazingFeature)
5. Open a Pull Request

📄 License

This project is distributed under the MIT License. See the LICENSE file for more details.

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Note: This project is developed for educational and research purposes in numerical analysis.