Fast reciprocal square root algorithm

[Svein]

I quoted Fröberg since it was on the required list for my Numerical Algorithms exam back in 1964. After I referred to it, I started to get very unsure of the validity of the algorithm - and, sure enough, it converges badly for a starting value of 1 if a is greater than 1. I have checked and found that the best way is to modify the start value according to this algorithm;

Start with x0=2
if a>1
   repeat
      x0=x0/2
      t=(3-x0*x0*a)
   until (t<2) and (t>0)

Then use the Fröberg algorithm with the resulting x0.