Fast computation of the Nth root

Abstract

AbstractA new class of iterative methods for computing a differentiable function is proposed, which is based on Páde approximation to Taylor's series of the function. It leads to a faster algorithm than Newton's method for x1N and a different interpretation of Newton's method. This algorithm uses 3rd degree approximation of continued fraction expansion (CFE) to Taylor's series for x1N, with adaptive expansion point for every iteration. Its major computational cost is O[(2 log2 N)(log4 M)] multiplications asymptotically, M is the number of precision bits desired, as opposed to existing bound of O[(2 log2 N)(log2 M)] for Newton's method. A new periodic CFE for special case x12 is obtained which is simple and free from error propagation. New algorithm is not applied to the square root case for N is too small to be practical; instead, the 3rd degree approximation is used as the starting value for Newton's method which has high precision and low computational cost