I have a text which claims the following (a) is Newton's method for cube roots, where $y$ is an approximation to the cube root of $x$:
$$\large{\frac{x / y^2 + 2y}{3}} \label{eqn:a}$$
It's my understanding the form can be derived from:
$$\large{x_1 = x_0 – \frac{f(x_0)}{f'(x_0)}}$$
where each $x_n$ is a better approximation of the root with each iteration and $x_0$ being an initial approximate guess.
I'm confused when deriving form (a). Can anyone explain how that form is obtained? My derivation shows:
$f(x) = x^3 – a$ and $f'(x) = 3x^2$ where $a$ is the cube, then we find $x$ for $f(x) = 0$
$x_1 = x_0 – ((x_0)^3 – a)/(3(x_0)^2)$
UPDATE:
One of the derivatives was calculated incorrectly. When the numerator and denominator are divided now by $x^2$ instead, the problem is resolved.
Thank you
Best Answer
In the (k+1)-th iteration $$x_{k+1} = x_k- \frac{x_k^3-a}{3x_k^2}=x_k - \frac{1}{3}x_k + \frac{a}{3x_k^2} = \frac{a/x_k^2 + 2x_k}{3}$$