Conclusion¶

This project used three numerical methods — Bisection, Newton–Raphson, and Secant — to find the root of nonlinear equations that cannot be solved exactly.
Each method was tested and compared to study how fast and accurately it converges to the correct solution.

6.1 Summary of Findings¶

All three methods found the same root for ( f(x) = x^2 - 4 ), which is x = 2.
The Bisection Method was the most reliable since it always converges when the function changes sign in the interval.
The Newton–Raphson Method was the fastest, reaching the correct answer in the fewest steps.
The Secant Method gave accurate results without needing the derivative and was faster than Bisection.

These results agree with what was expected from the course material and slides — Newton–Raphson has the highest speed, while Bisection guarantees convergence.

6.2 Reflection¶

Through this project, we learned how: - Iterative methods gradually move closer to the root.
- Each method has different trade-offs between speed, accuracy, and stability.
- The choice of initial guess and tolerance affects how well the method performs.

By using GNU Octave, we were able to apply the mathematical theory directly through coding.
This helped connect what was learned in class with practical problem-solving using software tools.

6.3 Future Work¶

To improve this project in the future, we could: - Add graphical visualizations to show how the root changes with each iteration.
- Apply the methods to different functions or systems of equations.
- Compare these methods with other techniques like Fixed-Point Iteration or Hybrid Methods to study more complex cases.

6.4 Final Thoughts¶

This project shows how numerical methods are powerful tools in mathematics and engineering.
Even when a problem has no exact solution, these techniques allow us to find accurate approximations.
Among the three methods, Newton–Raphson proved to be the fastest, but Bisection remains the most dependable when accuracy and guaranteed convergence are needed.