A Unified Root Algorithm in Bignum Arithmetic

Abstract

In bignum arithmetic, the cube root and higherorder root operations are conventionally treated as elementary functions, meaning that for them, correct rounding is not required, usually even without an error bound. In this paper we provide a unified algorithm RootRem to find the (arbitrary-ordered) root of bignums. This algorithm first finds the reciprocal root by using Newton iteration, and then obtains the root through a reciprocal operation. It has guaranteed correct rounding in all cases. RootRem(2), which is to find the square root, is currently found to be less efficient than the existing algorithm SqrtRem proposed by Brent and Zimmermann. However, RootRem(p) with $p\geq 3$ fill an important theoretical gap and are major improvements over the existing algorithms.