Rounding Error Analysis of Linear Recurrences Using Generating Series

TL;DR

This paper introduces a new analytical toolbox leveraging generating series and analytic combinatorics to analyze rounding errors in linear recurrences, with applications to Bernoulli number computation and interval arithmetic.

Contribution

It presents a novel approach combining generating series and analytic methods for error analysis of linear recurrences, including new algorithms and worst-case bounds.

Findings

New worst-case analysis for Bernoulli number computation

A novel algorithm for evaluating differentially finite functions in interval arithmetic

Demonstrated effectiveness of the approach through multiple applications

Abstract

We develop a toolbox for the error analysis of linear recurrences with constant or polynomial coefficients, based on generating series, Cauchy's method of majorants, and simple results from analytic combinatorics. We illustrate the power of the approach by several nontrivial application examples. Among these examples are a new worst-case analysis of an algorithm for computing Bernoulli numbers, and a new algorithm for evaluating differentially finite functions in interval arithmetic while avoiding interval blow-up.