{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article1288","name":"Toward accurate polynomial evaluation in rounded arithmetic (short report)","abstract":"Given a multivariate real (or complex) polynomial $p$ and a domain $cal D$,\r\nwe would like to decide whether an algorithm exists to evaluate $p(x)$ accurately \r\nfor all $x in {cal D}$ using rounded real (or complex) arithmetic. \r\nHere ``accurately'' means with relative error less than 1, i.e., with some correct \r\nleading digits. The answer depends on the model of rounded arithmetic:\r\nWe assume that for any arithmetic operator $op(a,b)$, for example $a+b$ or \r\n$a cdot b$, its computed value is $op(a,b) cdot (1 + delta)$, \r\nwhere $| delta |$ is bounded by some constant $epsilon$ where $0 < epsilon ll 1$, \r\nbut $delta$ is otherwise arbitrary. This model is the traditional one used to analyze \r\nthe accuracy of floating point algorithms.\r\n\r\nOur ultimate goal is to establish a decision procedure that, for any $p$ and $cal D$, \r\neither exhibits an accurate algorithm or proves that none exists. In contrast to the \r\ncase where numbers are stored and manipulated as finite bit strings (e.g., as floating \r\npoint numbers or rational numbers) we show that some polynomials $p$ are impossible to \r\nevaluate accurately. The existence of an accurate algorithm will depend not just\r\non $p$ and $cal D$, but on which arithmetic operators and constants are available \r\nto the algorithm and whether branching is permitted in the algorithm. \r\n\r\nToward this goal, we present necessary conditions on $p$ for it to be \r\naccurately evaluable on open real or complex domains ${cal D}$.\r\nWe also give sufficient conditions, and describe progress toward\r\na complete decision procedure. We do present a complete \r\ndecision procedure for homogeneous polynomials $p$ with integer coefficients,\r\n${cal D} = C^n$, using only arithmetic operations\r\n$+$, $-$ and $cdot$.","keywords":["Accurate polynomial evaluation","models or rounded arithmetic"],"author":[{"@type":"Person","name":"Demmel, James","givenName":"James","familyName":"Demmel"},{"@type":"Person","name":"Dumitriu, Ioana","givenName":"Ioana","familyName":"Dumitriu"},{"@type":"Person","name":"Holtz, Olga","givenName":"Olga","familyName":"Holtz"}],"position":8,"pageStart":1,"pageEnd":15,"dateCreated":"2006-01-31","datePublished":"2006-01-31","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Demmel, James","givenName":"James","familyName":"Demmel"},{"@type":"Person","name":"Dumitriu, Ioana","givenName":"Ioana","familyName":"Dumitriu"},{"@type":"Person","name":"Holtz, Olga","givenName":"Olga","familyName":"Holtz"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.05391.8","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume583","volumeNumber":5391,"name":"Dagstuhl Seminar Proceedings, Volume 5391","dateCreated":"2006-01-31","datePublished":"2006-01-31","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article1288","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume583"}}}