The complexity of multiple-precision arithmetic

TL;DR

This paper investigates the computational complexity of multiple-precision arithmetic, providing bounds on operation counts and analyzing the efficiency of methods for solving nonlinear equations with variable precision.

Contribution

It establishes upper and lower bounds on multiple-precision operation counts and explores their implications for solving nonlinear equations efficiently.

Findings

Bounds on single-precision operation counts for multiple-precision arithmetic

Analysis of efficiency differences in methods for nonlinear equations

Implications for variable-length multiple-precision computation

Abstract

In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision required increases as the computation proceeds. We give upper and lower bounds on the number of single-precision operations required to perform various multiple-precision operations, and deduce some interesting consequences concerning the relative efficiencies of methods for solving nonlinear equations using variable-length multiple-precision arithmetic. A postscript describes more recent developments.