Numerical Methods Analysis Suite

Version en Espanol

Academic numerical analysis project built in MATLAB. This repository implements, compares, and visualizes five families of numerical methods applied to approximation, root finding, interpolation, integration, and ordinary differential equation problems.

The workflow is reproducible by design: running main.m executes all problem modules, generates high-resolution plots, exports .csv tables, and updates a consolidated report in results/summary_report.txt.

Table of Contents

• Objective
• Implemented methods
• Project structure
• How to run
• Key results
• Generated artifacts
• Technical design
• Authors

Objective

Apply classic numerical methods to modeled problems related to system behavior, demand, and performance, comparing accuracy, convergence, and experimental error through tabular and visual outputs.

The project covers:

• Local approximation with Taylor series.
• Root finding with closed and open methods.
• Polynomial interpolation for intermediate estimates.
• Numerical integration over traffic-related functions.
• Numerical solution of a logistic ODE.

Implemented Methods

Category	Methods	Main files	Output
Approximation	Taylor series	methods/taylor_approximation.m	Maximum error, mean error, and error near the expansion point
Root finding	Bisection and Newton-Raphson	methods/bisection_method.m, methods/newton_raphson_method.m	Approximate root, iterations, and final residual
Interpolation	Lagrange and Newton	methods/lagrange_interpolation.m, methods/newton_interpolation.m	Interpolated values, coefficients, and divided differences
Integration	Trapezoidal rule and Simpson's rule	methods/trapezoidal_rule.m, methods/simpson_rule.m	Approximate integral, step size, and absolute error
ODEs	Euler and Runge-Kutta 4	methods/euler_method.m, methods/runge_kutta_4.m	Final approximation and maximum absolute error

Project Structure

.
|-- main.m
|-- methods/
|   |-- bisection_method.m
|   |-- euler_method.m
|   |-- lagrange_interpolation.m
|   |-- newton_interpolation.m
|   |-- newton_raphson_method.m
|   |-- runge_kutta_4.m
|   |-- simpson_rule.m
|   |-- taylor_approximation.m
|   `-- trapezoidal_rule.m
|-- problems/
|   |-- problem_taylor.m
|   |-- problem_roots.m
|   |-- problem_interpolation.m
|   |-- problem_integration.m
|   `-- problem_ode.m
|-- results/
|   |-- figures/
|   |-- tables/
|   `-- summary_report.txt
|-- utils/
|   |-- ensure_results_directories.m
|   |-- save_figure_file.m
|   |-- save_table_file.m
|   `-- write_summary_report.m
`-- Entrega 1_ Planteamiento de Problemas y Contexto_ Metodos Numericos.pdf

How to Run

Requirements

• MATLAB with support for table, writetable, and exportgraphics.
• Recommended: MATLAB R2020a or newer.

Local execution

Clone the repository:

git clone https://github.com/Estebangmz666/numerical_method_v1.git
cd numerical_method_v1

Open MATLAB at the project root and run:

main

The main script is responsible for:

• Clearing the execution environment.
• Adding methods, problems, and utils to the MATLAB path.
• Creating result directories automatically when needed.
• Running the five problem modules.
• Exporting plots, tables, and the final summary report.

Key Results

Problem	Main finding
Taylor	The 6th-order polynomial reduces the error near the expansion point down to 0.019097.
Root finding	Bisection and Newton-Raphson converge to 60.909843; Newton-Raphson reaches it in 5 iterations.
Interpolation	Lagrange and Newton produce the same values at x = 11 and x = 13, confirming the same interpolating polynomial.
Integration	Simpson's rule with 24 subintervals reaches an absolute error of about 0.00018729.
ODE	Runge-Kutta 4 with h = 0.25 achieves a maximum error of about 0.00029596, far below Euler.

Generated Artifacts

All outputs are stored under results/:

• results/summary_report.txt: consolidated report with conclusions and main tables.
• results/figures/*.png: exported high-resolution visualizations.
• results/tables/*.csv: numeric metrics for each problem.

Available tables:

• problem_1_taylor_metrics.csv
• problem_2_root_metrics.csv
• problem_3_interpolation_estimates.csv
• problem_3_lagrange_coefficients.csv
• problem_3_newton_divided_differences.csv
• problem_4_integration_metrics.csv
• problem_4_reference_integral.csv
• problem_5_ode_metrics.csv

Technical Design

The repository separates responsibilities to keep the code maintainable and scalable:

• methods/ contains reusable implementations of the numerical methods.
• problems/ defines the applied cases, parameters, visualizations, and exported tables.
• utils/ centralizes directory creation, figure export, table export, and summary report generation.
• main.m acts as the single entry point.

The main functions return results and metadata that support convergence, error, and iteration analysis without coupling the numerical logic to presentation concerns.

Authors

Project developed for a Numerical Analysis course.

• Esteban Gomez Leon
• Juan Antonio Betancourt Parra

Universidad del Quindio, Systems and Computer Engineering Program.