I'm attempting to use Newton Raphson method to calculate the square root of fixed point numbers.
The mathematics I understand - and, using this question I easily managed the normal;
$x_{n+1} = \frac{1}{2}(x_n+\frac{a}{x_n})$ to generate $\sqrt{a}$
And then, because I will be using this algorithm for computing, decided to try for the more complex reciprocal algorithm that uses only multiplication:
$x_{n+1} = x_n(1.5 - 0.5 a x_n^2)$ to generate $\frac{1}{\sqrt{a}}$
Which, to check, I also derived normally from the Newton Raphson equation shown in the question linked above.
However, whilst the first equation converges as expected, the second, does not, although I cannot find anywhere the rules for this convergence. For example:
$a = 100$, and $ x_0 = 16$
$x_1 = 16(1.5 - 0.5\times 100 \times 16^2) = -204776$ $x_2 = -204776(1.5 - 0.5\times 100 \times (-204776)^2) = -2.62\times10^9$
As I'm sure you'd agree - this is not converging to 10 - clearly I'm doing something wrong and yet I followed the normal Newton Raphson procedure in deriving it, and it works for the simpler formula. What are the conditions for this one?
Thanks very much!