I would like to know how to calculate first approximation in N-R division algorithm. I want to find the inverse of R. Here is the formula:
$x_{i+1} = x_{i}(2-R*x_{i})$
I'm trying to implement it in my program, so I need some algorithm to calculate $x_0$. Here is the example: https://www.youtube.com/watch?v=fBIQXHgj7j4
Thanks, for every kind of help.
We want to solve the equation $$f(x):={1\over x}-R=0$$ for given $R>0$ by means of Newton's algorithm. The resulting recursion formula is $$x_{n+1}=x_n(2- R\,x_n)\qquad(n\geq0)\ .$$ The problem is a proper choice of $x_0$.
The graph of $f$ intersects the $x$-axis at the desired root $\xi:={1\over R}$. From the convexity of the graph we can immediately deduce that $0<x_0\leq\xi$ results in (quadratic) convergence $x_n\to\xi$. Choosing $x_0>\xi$ gives an $x_1$ with $0<x_1<\xi\>$ (which then will lead to $x_n\to\xi$) if $$2-{x_0\over\xi}>0\ ,$$ or if $x_0<2\xi$.
It follows that any $x_0$ with $0<x_0<{2\over R}$ will lead to convergence of the $x_n$ to ${1\over R}$.
