Fortran Numerical Methods Toolkit

A small, modular Fortran project demonstrating five classic numerical methods. Built and tested with gfortran (GNU Fortran 13, but any Fortran 2008+ compiler will work).

Files

File	Purpose
numerical_methods.f90	Module containing all five methods
main.f90	Driver program that runs each method on a worked example
README.md	This file

How to compile and run

gfortran -o numerical_toolkit numerical_methods.f90 main.f90
./numerical_toolkit

(The module file must be compiled before main.f90, since main.f90 depends on it — listing them in that order in the command above handles this automatically.)

Expected output

==========================================
   FORTRAN NUMERICAL METHODS TOOLKIT
==========================================

--- Root finding: f(x) = x^3 - x - 2 ---
 Bisection:      root =   1.521380  (19 iterations)
 Newton-Raphson: root =   1.521380  (3 iterations)

--- Integration: f(x) = 4/(1+x^2) on [0,1]  (exact = pi) ---
 Trapezoidal rule: integral =   3.141576
 Simpson's rule:   integral =   3.141593
 True value (pi):              3.141593

--- Gauss elimination: solving A*x = b ---
 Solution:
   x(1) =   2.000000
   x(2) =   3.000000
   x(3) =  -1.000000

==========================================

What each method does

• Bisection method — repeatedly halves an interval [a, b] where the function changes sign, narrowing in on a root. Slow but always converges if the starting interval is valid.
• Newton-Raphson method — uses the function's derivative to jump toward the root much faster than bisection, starting from a single guess. Converges quickly but needs a decent starting point and a derivative.
• Trapezoidal rule — approximates the area under a curve by summing trapezoids between sample points.
• Simpson's 1/3 rule — fits parabolas through pairs of intervals instead of straight lines, which is why it lands closer to the true value of pi (3.141593) than the trapezoidal rule does above.
• Gauss elimination — solves a system of linear equations by reducing the matrix to upper-triangular form (forward elimination), then solving from the last equation upward (back substitution).

Notes / possible extensions

• The root-finding and integration routines currently work on fixed demo functions (f_root and f_integrand in numerical_methods.f90). To try a different function, edit those two functions directly.
• gauss_elimination does not use partial pivoting, so it assumes no zero pivots turn up during elimination — fine for the demo system here, but worth mentioning as a limitation if this goes in a report.
• Natural next steps if you want to extend the project: add partial pivoting, let the user type in their own function or matrix at runtime, or add another method (e.g. Runge-Kutta for ODEs, or Lagrange interpolation).