Implementing evaluation strategies for continuous real functions

TL;DR

This paper presents a comprehensive implementation of evaluation strategies for continuous real functions, covering various representations and operations, with a focus on exact real computation and efficient algorithms.

Contribution

It introduces multiple representations for continuous functions, including oracle-based and polynomial approximations, and discusses local hybrid approaches for efficient evaluation.

Findings

Multiple function representations enable precise evaluation strategies.

Local hybrid representations improve efficiency for function approximation.

The paper analyzes the complexity of function division in polynomial representations.

Abstract

We give a technical overview of our exact-real implementation of various representations of the space of continuous unary real functions over the unit domain and a family of associated (partial) operations, including integration, range computation, as well as pointwise addition, multiplication, division, sine, cosine, square root and maximisation. We use several representations close to the usual theoretical model, based on an oracle that evaluates the function at a point or over an interval. We also include several representations based on an oracle that computes a converging sequence of rigorous (piecewise or one-piece) polynomial and rational approximations over the whole unit domain. Finally, we describe "local" representations that combine both approaches, i.e. oracle-like representations that return a rigorous symbolic approximation of the function over a requested interval sub-domain with a requested effort. See also our paper "Representations and evaluation strategies for feasibly approximable functions" which compares the efficiency of these representations and algorithms and also formally describes and analyses one of the key algorithms, namely a polynomial-time division of functions in a piecewise-polynomial representation. We do not reproduce this division algorithm here.