Computer arithmetic and validity: Theory, implementation, and applications

451 Pages·2013·3.86 MB·English

by  Kulisch U.

De Gruyter Studies in Mathematics 33
Editors
CarstenCarstensen,Berlin,Germany
NicolaFusco,Napoli,Italy
FritzGesztesy,Columbia,USA
NielsJacob,Swansea, UnitedKingdom
Karl-HermannNeeb,Erlangen,Germany

Ulrich Kulisch
Computer Arithmetic and
Validity
Theory, Implementation, and Applications
2nd Edition
De Gruyter
MathematicalSubject Classification 2010: Primary: 06, 65, 68, 94; Secondary: 65G, 65Y,
68M,94C.
Additionalkeywords:order,semimorphism,roundinginvariantstructures,floating-pointarith-
metic,intervalarithmetic,exactdotproduct,reliablecomputing,computingwithguarantees,
enclosuremethods.
ISBN978-3-11-030173-1
e-ISBN978-3-11-030179-3
Set-ISBN978-3-11-030180-9
ISSN0179-0986
LibraryofCongressCataloging-in-PublicationData
ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress.
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliotheklists this publicationin theDeutscheNationalbibliografie;
detailedbibliographicdataareavailableintheinternetathttp://dnb.dnb.de.
© 2013WalterdeGruyterGmbH,Berlin/Boston
Typesetting:PTP-BerlinProtago-TEX-ProductionGmbH,www.ptp-berlin.de
Printingandbinding:Hubert&Co.GmbH&Co.KG,Göttingen
Printedonacid-freepaper
PrintedinGermany
www.degruyter.com
LassetunsamAlten,
soesgutist,halten
unddannaufdemaltenGrund
NeuesschaffenStundumStund.
HouseinscriptionintheBlackForest.

Foreword to the second edition
Whenthefirsteditionofthisbookwaspublishedintervalarithmeticorcomputingwith
guarantieswasasubjectforspecialists.Butonlyafewmonthsafterpublicationofthe
book the IEEE Computer Society founded a new standard committee IEEE P1788
forintervalarithmeticinAugust2008.Itsays:Theaimofthestandardistoimprove
theavailabilityofreliablecomputinginmodernhardwareandsoftwareenvironments
bydefiningthebasicbuildingblocksneededforperformingintervalarithmetic.This
eventhasmovedthecontentsofthisbookrightintothecenter ofinterestandofsci-
entificcomputing.Workonthestandardbeganimmediatelyandisstillinprogress.A
majornumberofscientistsfromallovertheworldareparticipatinginitsdevelopment.
Theworkisnowapproachingitsend.Sothereisgreathopethatfutureprocessorswill
betteranduniquelysupportintervalarithmeticandcomputingwithguarantees.
Followingthecontentsandtheintentionsofthebook,theauthorhaspreparedand
presentedseveralmotionstothestandardcommitteeIEEEP1788.Theyhaveallbeen
accepted. These motions include the basic arithmetic operations for floating-point
intervals, the comparison and lattice operations, an exact dot product and complete
arithmetic, roundingcontrol for floating-pointoperations and from complete format
to floating-point and interval formats, denotations for the fundamental spaces of in-
terest, and for the central operations of Kaucher arithmetic. This is an extension of
conventionalinterval arithmetic byso-called improperintervals with great potential
forapplicationsincomputergeometry.
In the older and classic books of interval arithmetic by R.E. Moore [457, 458],
G.Alefeld andJ.Herzberger [28,29],or E. Hansen[219, 222]real intervals are de-
notedbyuppercaselettersandtheboundsbylowercaseletterswherethelowerbound
carriesanindex1andtheupperboundanindex2,forinstance,AD Œa ,a (cid:2).Thisno-
1 2
tationiswellestablishedinintervalarithmeticanditisusedthroughoutthisbook.We
mentionthis here because a coupleof years ago a group of scientists recommended
denotinga real interval by a bold face lower case letter, the lower boundby under-
liningandtheupperboundbyoverliningtheletter, forinstance,a=Œa,a(cid:2).Although
thedraftofthestandardP1788currentlyusesthisnotation,wedonotfollowthisrec-
ommendationin this bookand prefer the older notation.There are goodreasons for
doingthis:Itisdifficulttousethenewnotationinhandwrittentexts.Furthermore,the
contents of the book are not, unlike the standard P1788, restricted to real intervals.
It also considers intervals over the complex numbers and intervals over vectors and
matrices of real andcomplexnumbers.Here the older notationallows to stay closer
toconventionalnotationsofmathematics.The indexnotationalsoismoresuitedfor
viii Forewordtothesecondedition
provingseveral theorems in the book.Further, an improper interval is just a pair of
realnumberswherethefirstcomponentmaybegreaterthanthesecond.Here,under-
andoverliningthecomponentsmaybeaconstantsourceofconfusion.Afterall,num-
beringthecomponentsneedsfewer keystrokesthanunder-andoverlining.
Thelaborofpreparingandwritingtheneweditionofthebookhasbeenenormous.
Minor changes had to be made throughoutthe entire book. The developmentof the
standard IEEE P1788 brought different notations and the time and communications
withmembersofthestandardcommitteeP1788broughtnewinsights.Thuslargepor-
tionsofseveralchaptershadtobenewlywritten.
I wish to express mydeep appreciation to thosemany colleagues who have com-
mentedonmycontributionstothedevelopmentofthestandardIEEEP1788.Although
I didnotfollow hissuggestionscompletely,I owe particular thankstoStefan Siegel
for his help rewriting Chapter 7 on hardware support for interval arithmetic. Klaus
BrauneagainwasagreathelpwheneverproblemswiththeLATEXsystemappeared.I
alsoagainthankBoEinarssonforproofreadinggreatportionsofthebook.Mythanks
alsogotothepublisherforaskingmeintimetoprepareasecondeditionofthebook.
Sotheworkcouldbedonewithoutgettingundertoohighpressure.
Karlsruhe,November2012 UlrichW.Kulisch
Preface
In general, a computer isunderstoodto be an enginethat deals with data. These are
stored by binary digits or bits. The data can represent all kinds of objects such as
wholenumbers,letters ofanalphabet,thebooksofalibrary,theaccountsofabank,
the inhabitants of a town, the text of a law, or the features of a disease. If the com-
puter isfunctioningcorrectly, thenits processingisfree of error. Acomputer thatis
programmedproperlyisthereforegenerallyconsideredtobeaninfallibletool.
It isquiteironical thatthisviewisoften notjustifiedwhenthedigitalcomputeris
used for its original purpose – numerical or scientific computation.This booktreats
thearithmeticprocessingofrealnumbersincomputers.Itdescribeshowthecomputa-
tionalerrorcanbereducedandavoidedbyspecialdesignofthecomputer’sarithmetic.
Arealnumberlike(cid:3),forinstance,consistsofaninfiniteb-adicexpansion,aninfi-
nitestringofdigits.Onacomputer,however,onlyfinitepartsoftheseexpansionscan
berepresented. Thusarithmetic for real numberscan onlybe approximated.Fornu-
merical orscientific computing,floating-pointnumbersandfloating-pointarithmetic
havebeenusedsincetheearlydaysofelectroniccomputers.Afloating-pointnumber
e
takestheformm(cid:2)2 .Heremisthemantissaorsignificandande istheexponent.
Computers designedand builtby Konrad Zuse, the Z3 (1941) andthe Z4 (1945),
wereamongthefirstcomputersthatusedthebinarynumbersystemandfloating-point
for number representation. In the Z4, a floating-point number was represented by a
32bitwordandarithmeticwasusedinamannerverysimilartothedataformatsingle
precisionontoday’scomputers.Overtheyearscomputertechnologywaspermanently
improved.Thispermittedanincreaseofthewordsize.
In 1985 a standard for floating-point arithmetic was internationally adopted. It is
knownastheIEEE754floating-pointarithmeticstandard.Itspecifiestwodataformats
for single precision of 32 bits and double precision of 64 bits. The format double
precisionuses53bitstorepresentthemantissaand11bitsfortheexponent.Arevision
ofthestandardIEEE 754,publishedin2008,addedanotherwordsizeof128bits.
The IEEE floating-point arithmetic standard is undoubtedlythorough, consistent,
andwelldefined.Ithasbeenwidelyacceptedandhasbeenusedinalmosteveryproces-
sordevelopedsince1985.Thishasgreatlyimprovedtheportabilityoffloating-point
programs.Untiltodaythemostusedfloating-pointformatisdoubleprecision.Itcorre-
spondstoabout16decimaldigits.Floating-pointarithmetichasbeenusedsuccessfully
in the past. Every computer user is trained and familiar with all details of floating-
pointarithmeticincludingallitsexceptionslikeunderflow,overflow,(cid:3)1,C1,NaN
(not a number), (cid:3)0,C0, and so on.Seventy years of extensive use of floating-point