Newton (Iterative) Vs. Babylonian (Direct) For Roots Is Newton's iterative method for finding a square root more efficient then the Babylonian method? Considering most roots are irrational, which method would get me within, say 16 decimal places, the fastest? I need a high level of precision so I was curious which would be more efficient.

The Newton-Raphson process for the case of the square root is in effect the same algorithm as the Babylonian Method, aka Heron's Method.