Root Finding Using Newton-Raphson Method in Fortran

This repository contains a Fortran implementation of the Newton-Raphson Method for finding the root of a given equation. The Newton-Raphson Method is a numerical technique used to find the root of a function by iteratively improving the approximation of the root.

Newton-Raphson Method Overview

The Newton-Raphson Method works by using the derivative of the function to approximate the root. It starts with an initial guess and iteratively refines this guess using the formula:

[ x_{n+1} = x_n - \frac{f(x_n)}{f’(x_n)} ]

where:

( x_{n+1} ) is the new approximation of the root,
( x_n ) is the current approximation,
( f(x_n) ) is the value of the function at ( x_n ),
( f’(x_n) ) is the value of the derivative of the function at ( x_n ).

Steps of the Newton-Raphson Method

Initial Guess: Start with an initial guess ( x_0 ).
Iterative Update: Compute the next approximation using ( x_{n+1} = x_n - \frac{f(x_n)}{f’(x_n)} ).
Convergence Check: Repeat the update until the difference between successive approximations is within a specified tolerance or the maximum number of iterations is reached.

Fortran Implementation

File Structure

newton_raphson.f90: The main Fortran program implementing the Newton-Raphson Method.
README.md: This file, providing an overview and instructions.

Program Description

The Fortran program newton_raphson.f90 defines a function f(x) and its derivative f_prime(x) whose root is to be found, and implements the Newton-Raphson Method to approximate the root.

Function and Derivative Definitions

The function f(x) and its derivative f_prime(x) are defined within the program. You can modify these functions according to the specific equation you are trying to solve.

Input and Output

Input: The program takes the initial guess, the tolerance for convergence, and the maximum number of iterations.
Output: The program outputs the approximated root and the value of the function at the root.

Example Usage

Modify the Function: Update the f(x) and f_prime(x) functions in newton_raphson.f90 to represent the equation for which you want to find the root.
Compile the Program: Use a Fortran compiler to compile the program:
```
 gfortran -o newton_raphson newton_raphson.f90
```
Run the Program: Execute the compiled program:
```
 ./newton_raphson
```
Provide Input: Enter the initial guess, tolerance, and maximum iterations when prompted.

Sample Code

Below is a sample code snippet from newton_raphson.f90:

program newton_raphson
  implicit none
  real(8) :: x, tol, fx, fpx
  integer :: max_iter, iter

  ! Function prototypes
  real(8) :: f, f_prime
  real(8) :: x_init

  ! Define the function f(x)
  contains
  function f(x)
    real(8) :: f
    real(8), intent(in) :: x
    ! Example function: f(x) = x^2 - 4
    f = x**2 - 4.0
  end function f

  ! Define the derivative f'(x)
  function f_prime(x)
    real(8) :: f_prime
    real(8), intent(in) :: x
    ! Derivative of the example function: f'(x) = 2x
    f_prime = 2.0 * x
  end function f_prime

  ! Read input values
  print *, 'Enter the initial guess:'
  read *, x_init
  print *, 'Enter the tolerance:'
  read *, tol
  print *, 'Enter the maximum number of iterations:'
  read *, max_iter

  ! Newton-Raphson Method
  x = x_init
  iter = 0
  do while (iter < max_iter)
    fx = f(x)
    fpx = f_prime(x)
    if (abs(fpx) < tol) then
      print *, 'Derivative is too small. Exiting.'
      stop
    endif
    x = x - fx / fpx
    if (abs(fx) < tol) exit
    iter = iter + 1
  end do

  ! Output the result
  print *, 'Approximated root:', x
  print *, 'Function value at root:', f(x)
  print *, 'Number of iterations:', iter

end program newton_raphson

Watch the full tutorial video:

Link of the downloaded script file: Examples Directory