Editing Newton's method (section)

==Code==
The following is an example of a possible implementation of Newton's method in the [[Python (programming language)|Python]] (version 3.x) programming language for finding a root of a function <code>f</code> which has derivative <code>f_prime</code>.

The initial guess will be {{math|1={{var|x}}{{sub|0}} = 1}} and the function will be {{math|1={{var|f}}({{var|x}}) = {{var|x}}{{sup|2}} − 2}} so that {{math|1={{var|{{prime|f}}}}({{var|x}}) = 2{{var|x}}}}.

Each new iteration of Newton's method will be denoted by <code>x1</code>. We will check during the computation whether the denominator (<code>yprime</code>) becomes too small (smaller than <code>epsilon</code>), which would be the case if {{math|{{var|{{prime|f}}}}({{var|x}}{{sub|{{var|n}}}}) ≈ 0}}, since otherwise a large amount of error could be introduced.

<syntaxhighlight lang="python3" line="1">
def f(x):             
	return x**2 - 2   # f(x) = x^2 - 2

def f_prime(x):
	return 2*x        # f'(x) = 2x

def newtons_method(x0, f, f_prime, tolerance, epsilon, max_iterations):
    """Newton's method

    Args:
      x0:              The initial guess
      f:               The function whose root we are trying to find
      f_prime:         The derivative of the function
      tolerance:       Stop when iterations change by less than this
      epsilon:         Do not divide by a number smaller than this
      max_iterations:  The maximum number of iterations to compute
    """
    for _ in range(max_iterations):
        y = f(x0)
        yprime = f_prime(x0)

        if abs(yprime) < epsilon:       # Give up if the denominator is too small
            break

        x1 = x0 - y / yprime            # Do Newton's computation

        if abs(x1 - x0) <= tolerance:   # Stop when the result is within the desired tolerance
            return x1                   # x1 is a solution within tolerance and maximum number of iterations

        x0 = x1                         # Update x0 to start the process again

    return None                         # Newton's method did not converge
</syntaxhighlight>