Newton's Method (Fortran)

Root Finding

Newton's method (also known as the Newton-Raphson method) is an iterative root-finding algorithm. This Fortran implementation demonstrates the method for finding roots of a function.

Newton's Method

Given a function $f (x)$ , Newton's method finds a root by iterating:

x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}

The method converges quadratically when the initial guess is sufficiently close to the root.

Implementation

This implementation finds the root of $f (x) = x^{2} - 4$ , which has roots at $x = \pm 2$ .

!https://en.wikipedia.org/wiki/Newton%27s_method
program newton_method
    implicit none
    integer, parameter:: dp=kind(0.d0) ! double precision
    real(dp) :: f_val, f_prime_val, epsilon, tol, x0, x1
    integer :: max_iter, i

    max_iter = 1000
    tol = 0.001
    x0 = 4.0_dp
    epsilon = 0.00001_dp

    do i = 1,max_iter
        f_val = f(x0)
        f_prime_val = f_prime(x0)
        if (abs(f_prime_val) < epsilon) then
            exit
        end if
        x1 = x0 - f_val/f_prime_val
        if (abs(x1-x0) <= tol) then
            print *, x1
            exit
        end if
        x0 = x1
    end do

contains
    function f(x)
        real(dp) :: f
        real(dp) :: x
        f = x**2 - 4
    end function f

    function f_prime(x)
        real(dp) :: f_prime
        real(dp) :: x
        f_prime = 2*x
    end function f_prime

end program newton_method

Key Features

Uses double precision for numerical accuracy
Checks for division by zero (when $f^{'} (x) = 0$ )
Convergence check based on tolerance
Maximum iteration limit to prevent infinite loops
Uses contains to define internal functions

Output

For the initial guess $x_{0} = 4$ , the method converges to $x = 2$ :

   2.0000000000000000

Trace it by hand

ProblemNewton's method for √2

Apply Newton's method to f(x) = x² − 2 = 0 starting from x₀ = 1.5. Compute x₁ and x₂ explicitly. How close are you to √2 ≈ 1.41421356 at each step?