Newton's method

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Newton's method approximates the zeroes of differentiable functions.

Given a function $f$ and an initial guess $x$, repeatedly apply the following equation to get an improved guess: $x = x - \frac{f(x)}{f'(x)}$. When $f(x) \approx 0$, we are done.

Simple examples

To find the square root of $a$, we can solve the equation $x^2 = a$ (i.e., find the zero of $f(x) = x^2 - a$). In code:

def square_root(a):
    """Return the square root of A with a tolerance of 1e-15."""
    f = lambda x: x**2 - a
    df = lambda x: 2*x
    x = 1
    while abs(f(x)) > 1e-15: # loops until f(x) is within 1e-15 of 0
        x = x - f(x)/df(x)
    return x

To find the cube root of $a$, use f = lambda x: x**3 - a and df = lambda x: 3*(x**2) instead.

Implemented with iterative improvement

def newton_solve(f, df, guess, tolerance):
    def close(x):
        return abs(f(x)) < tolerance
    def newton_update(x):
        return x - f(x) / df(x)
    return iter_solve(guess, close, newton_update, 1000000000)