Java Reference
In-Depth Information
arithmetic
.Intervalarithmeticisbasedonexecutingaseriesofcalculations
twice:onceroundingupandonceroundingdown.Thisallowsthedetermina-
tionoftheupperandlowerboundsoftheerror.Usingintervalarithmetic,itis
possible,inmanycases,tocertifythatthecorrectresultisavaluenotlarger
thantheresultobtainedwhileroundingup,andnosmallerthantheresultob-
tainedwhileroundingdown.Thisplacestheexactresultwithinacertain
boundary.
AlthoughIEEE754doesnotspecificallymentionintervalarithmetic,it
doesrequiredirectedroundingmodes.Intervalarithmeticcanbeapowerful
numericaltool,althoughthereareexceptionalcasesinwhichtheseresults
arenotvalid.Notallmathematicalcalculationscanbesubjecttointerval
analysis.Thefundamentalrulesareasfollows:
1.Theoperationmustconsistofmultiplesteps.
2.Atleastoneintermediateresultinthecalculationsmustbesubjectto
rounding.
3.Thevaluezeroshouldnotbeintheerrorrange,thatis,bothresultsmusthave
thesamesign.Thesubsequentpossibilityofdivisionbyzeroorbyaverysmall
numberintroducesotherpotentialproblemsthatarenotevidentininterval
arithmetic.
4.Thecalculationsshouldnotbe,inthemselves,amethodforapproximatingre-
sults.Compoundedapproximationsrenderinvalidintervals.
Treatmentofinfinity
Theconceptofinfinityarisesinrelationtotherangeofasystemofrealnum-
bers.Oneapproach,calleda
projectiveclosure
,describesinfinityasanun-
signedrepresentationforverysmallorverylargenumbers.Whenprojective
infinityisadopted,thesymbol
∞
isusedtorepresentanumberthatiseither
toosmallortoolargetobeencodedinthesystem.
Analternativeapproach,called
affineclosure
,recognizesthedifference
betweenvaluesthatexceedthenumbersystembybeingtoolarge(+
∞
)or
jective and the affine methods for the closure of a number system.
According to the standard, infinity must be interpreted in the affine sense.
That is, any representable finite number
x
shall be located
-
∞
(x) +
∞
