## Problem : Finding roots with Newton-Raphson method

Newton's method is a way to find roots of any continuos function, e.g. to find an $$x$$ where $$f(x)=0$$. The method starts of with some guess, $$x_i$$ and in each iteration the guess is improved unless $$-\epsilon < f(x_i) < \epsilon$$. Improvement is done as: $x_{i+1}=x_i - \frac{f(x_i)}{f'(x_i)}$

When $$f(x)=x^n-A$$ what we find x as root $$f(x)=0$$ is $$x^n=A$$, ie. nth root of $$A$$. Since $$f'(x)=n\dot x^{n-1}$$, the iteration becomes $x_{i+1}=x_i - \frac{x_i^n-A}{n\cdot x^{n-1}}$

The initial guess, $$x_0$$ is usually taken as 1.

## solution

    (define (Newton a n xi e)
(cond
((< (abs (- (expt xi n) a)) e) xi)
(else (Newton a n (- xi (/ (- (expt xi n) a) (* n (expt xi (- n 1))))) e))))

(Newton 2 2 1 0.01)
(Newton 27 3 1 0.01)