Newton Iteration and Integer Division

Integer Division in Residue Number Systems

This contribution to the ongoing discussion of division algorithms for residue number systems (RNS) is based on Newton iteration for computing the reciprocal. An extended RNS with twice the number of moduli provides the range required for multiplication and scaling. Separation of the algorithm description from its RNS implementation achieves a high level of modularity, and makes the complexity ...

Multiprecision division: Expanded Version

This paper presents a study of multiprecision division on processors containing word-by-word multipliers. It compares several algorithms by first optimizing each for the software environment, and then comparing their performances on simple machine models. While the study was originally motivated by floating-point division in the small-word environment, the results are extended to multiprecision...

Note on fast division algorithm for polynomials using Newton iteration

The classical division algorithm for polynomials requires O(n) operations for inputs of size n. Using reversal technique and Newton iteration, it can be improved to O(M(n)), where M is a multiplication time. But the method requires that the degree of the modulo, x, should be the power of 2. If l is not a power of 2 and f(0) = 1, Gathen and Gerhard suggest to compute the inverse, f, modulo x r⌉,...

On Fast Division Algorithm for Polynomials Using Newton Iteration

A full Nesterov-Todd step interior-point method for circular cone optimization

In this paper, we present a full Newton step feasible interior-pointmethod for circular cone optimization by using Euclidean Jordanalgebra. The search direction is based on the Nesterov-Todd scalingscheme, and only full-Newton step is used at each iteration.Furthermore, we derive the iteration bound that coincides with thecurrently best known iteration bound for small-update methods.