Univariate amenable functions

TL;DR

This paper explores the properties of univariate real analytic functions related to amenability and compatibility, providing tests and tables to determine these properties, which are crucial for ensuring the stability of composed algorithms.

Contribution

It extends the theory of amenable functions to univariate real analytic functions, offering practical tests and classifications for elementary functions.

Findings

Identifies which elementary functions are amenable or compatible.

Provides simple tests for amenability and compatibility.

Includes tables classifying functions based on these properties.

Abstract

The concepts of amenable and compatible functions have been introduced in a recent work, in order to state precise mathematical theorems that guarantee that a backward stable algorithm is also forward stable, and that the composition of two stable algorithms results in an stable algorithm. In this work, we elaborate in this theory for univariate real analytic functions, providing simple tests for both concepts and producing tables for a number of elementary functions which are or fail to be amenable.