Toward accurate polynomial evaluation in rounded arithmetic (short report)

Abstract

Given a multivariate real (or complex) polynomial $p$ and a domain $cal D$,
we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately 
for all $x in {cal D}$ using rounded  real (or complex) arithmetic. 
Here ``accurately'' means with relative error less than 1, i.e., with some correct 
leading digits. The answer depends on the model of rounded arithmetic:
We assume that for any arithmetic operator  $op(a,b)$, for example $a+b$ or 
$a cdot b$,  its computed value is $op(a,b) cdot (1 + delta)$, 
where $| delta |$ is bounded by some constant $epsilon$ where $0 < epsilon ll 1$, 
but $delta$ is otherwise arbitrary. This model is the traditional one used to analyze 
the accuracy of floating point algorithms.

Our ultimate goal is to establish a decision procedure that, for any $p$ and $cal D$, 
either exhibits an accurate algorithm or proves that none exists. In contrast to the 
case where numbers are stored and manipulated as finite bit strings (e.g., as floating 
point numbers or rational  numbers)  we show that some polynomials $p$ are impossible to 
evaluate accurately.  The existence of an accurate algorithm will depend not just
on $p$ and $cal D$, but on which arithmetic operators and constants are available 
to the algorithm  and whether branching is permitted in the algorithm. 

Toward this goal, we present necessary conditions on $p$ for it to be 
accurately evaluable on open real or complex domains ${cal D}$.
We also give sufficient conditions, and describe progress toward
a complete decision procedure. We do present a complete 
decision procedure for homogeneous polynomials $p$ with integer coefficients,
${cal D} = C^n$, using only arithmetic operations
$+$, $-$ and $cdot$.