Oscar Lanzi III wrote: > > There is none, but there is a simple way to approximate it. To otain > sqrt(a), let x be any positive number, and compute > > x' = (a+x/a)/2 > > and keep repeating, using x' in plce os x each time. Soon you'll get a > highly accurate approximation. > > An efficient way to implement this is to set x = p/q where p and q are > integers and the fraction is in lowest terms. Then: > > x' = (a+x/a)/2 = (p^2 + a* q^2)/(2pq) > > which will again be in lowest terms. You can interpret the numerator > and denominator as new values of p and q for reuse. With this > implementation, you don't have to do any division (You keep calculating > p and q separately; you don't need to compute p/q itself) ) until > you're ready to end the process with your final approximation. > > --OL
If one requirement is not to use division, then just use Newton's method for the reciprocal of the square root, and multiply by the number itself at the end. Useful on computers that do not have divide instructions (showing my age!); or, if a divide is horribly slower than a multiply and speed is important. Watch the radius of convergence!
Another technique, if it's still of interest to the person who started this thread.