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1 Appliedand Computational NISTIR5916 Mathematics Division ComputingandAppliedMathematicsLaboratory ServiceforSpecialFunctions AProposedSoftwareTest DanielW.Lozier October1996 NationalInstituteofStandardsandTechnology U.S.DEPARTMENTOFCOMMERCE TechnologyAdministration Gaithersburg,MD20899

2 ThispaperwillappearinnalforminTheQualityofNumericalSoftware: AssessmentandEnhancement,RonaldF.Boisvert,editor,Chapman&Hall, PREPRINT London,1997.ItwaspresentedorallyattheIFIP/TC2/WG2.5WorkingConference7,heldatSt.Catherine'sCollege,Oxford,England,July7{13,1996.

3 ThisisaproposaltodevelopasoftwaretestserviceattheNationalInstituteof StandardsandTechnologyforuseintestingtheaccuracy,ornumericalprecision, ofmathematicalsoftwareforspecialfunctions.theservicewouldusetheworld ABSTRACT willbeofpracticalutilitytoanyonewhousesspecialfunctionsinphysicsorother runonanetworkofworkstationsattheinstitute.itishopedthatsuchaservice WideWebtoreceivetestrequestsandreturntestresults.Thetestswouldbe applications,andthatitwillstimulatetheinterestofappliedmathematicians whoareinterestedinthecomputationofspecialfunctionsaswellascomputer scientistswhoareinterestedininnovativeusesoftheinternet.theauthor solicitscommentsonanyaspectoftheproposedservice.

4 1Mathematicalsoftwareisdeeplyembeddedinthecomputingenvironment.Since thisenvironmentisevolvingrapidly,itsimpactonmathematicalsoftwareneeds toberevisitedregularly. Introduction algorithms,particularlyincomputationallinearalgebra.theearlierintroduc- highbandwidthcommunicationlinks,oneresultofwhichisimprovedaccess tionofvectorcomputershadasimilareect.recentadvancesincommunica- tionsandnetworkinghaveledtotheglobalinterconnectionofcomputersby Progressinparallelcomputinghasstimulatedmuchreworkingofnumerical Web.Forexample,electroniccatalogsandrepositoriessuchasxnetlib[8]are toinformationaboutmathematicalsoftwareviatheinternetandworldwide nowconsultedroutinelyforhelpinlocatingandobtainingmathematicalsoftwarepackagesforspecictasks. [4]and[10].However,mathematicalfunctionsseemparticularlyappropriatefor putationofmathematicalfunctions.somereferencescanbecited,forexample demonstratinganewandpotentiallyvaluableuseoftheweb:mathematical softwaretesting.theproblemoftestingisintrinsicallysimplerformathematicalfunctionsthanforotherkindsofnumericalcomputation.theinputand outputeuclideanspaceshavelowdimension.incontrast,numericallinearalgebradealswitheuclideanspacesofhighdimension,andmostothernumerical Vectorandparalleldevelopmentshavehadonlyamodestimpactonthecom- formathematicalfunctionscanbedevisedthatapply,intheory,toallpossible computationsdealwithfunctionspacesofinnitedimension.testprocedures vantages,comparedtocurrenttestingpractice,inusingthewebtotestmath- ematicalsoftware?ouransweristhattestscanbetailoredtosuitaparticular inputs. need,andtheycanbeperformedondemand. Aquestionthatneedsansweringatthispointis:Whatwouldbethead- editorsandsoftwaremaintainersalsoplayarole.thesepeoplehavedirectresponsibilityforthecorrectnessofthesoftware.forcommercialsoftware,high licensefeesarejustiedlargelybythehighcostsassociatedwithsoftwaremain- Toseewhythisisuseful,wedividecurrentpracticeintotwocategories. Suppliertestingisperformedbythesoftwarewriterorprojectteam.Referees, timesindependenttestsareconductedandpublishedinjournalarticlesand institutionalreportsasaguideforprospectiveusers. itleadstoanincreasedcondenceinthecorrectnessofthesoftware.some- groups.usersoftenperformthiskindoftestingfortheirownpurposesbecause tenanceandtesting.independenttestingisperformedbyotherindividualsand testsarenevercomplete,theirresultsmaynotapplydirectlytothenumerical havingbeenperformedatsometimeinthepastinacomputingenvironment thatmaybeverydierentfromtheprevailingone.evenmoreunsettling,since computationofcurrentinterest. Publishedtestsofeitherkindhavea`frozen-in-time'qualityaboutthem, 1

5 Herewearefollowingtheconventionalterminologyofcallingthetranscendental ticalbecausethetestprogramisunavailable.notableexceptionsarethetest andcomplexarguments,andofcody[6]forspecialfunctionsofrealargument. programsofcodyandwaite[7]andcody[5]forelementaryfunctionsofreal Justthesimplerepetitionofamathematicalfunctiontestisoftenimprac- functionsmetincalculuscourseselementaryfunctionsandthehigherfunctions thatappearinadvancedapplicationsspecialfunctions.thegeneralunavailabilityoftestprogramsisundoubtedlyrelatedtotheconsiderableeortthatis ofrandomnessincluded. someimplementationaldetailsaregiven.theemphasisisonspecialfunctions, programsapplyonlytoabuiltinsetoftestarguments,oftenwithanelement distribution.anotherproblemislackofgenerality.forexample,mosttest requiredtoraisethemtoanacceptablyhighstandardforpublicationorpublic becausethisiswheretheneedisgreatest,buttheservicewillapplyequallywell tailorteststospecicrequirements.thereforeitshouldbeofinterestinboth toelementaryfunctions.theservicewillprovideatoolthatcanbeusedto Inthispaperasoftwaretestserviceforspecialfunctionsisproposedand supplierandindependenttesting. 2Thepurposeoftheproposedsoftwaretestingserviceforspecialfunctionsisto assesstheaccuracy,ornumericalprecision,ofcomputedfunctionvaluesthrough theuseofacomparisonmethod.testvalueswillbecomparedagainstreference Proposal andtechnology.thetestswillbeconductedattheinstituteusingsoftware developedforthepurpose.thetestresultswillbereturnedtotherequesterin valuescomputedinhigherprecisionbyhighlyaccuratealgorithms.testrequestswillbesubmittedtoawebserveratthenationalinstituteofstandards theformofanappropriatedocumentonthewebserver. ReferenceSoftwareThiswillconsistofhighlyaccurateandreliable,butnot Thekeycomponentsoftheservicewillbe referencevaluesofspecialfunctionsoververyextensiveargumentdomains. necessarilyecient,numericalproceduresforgeneratinghigh-precision rithmsbecauseitwillbeembeddedinacomputingenvironmentthatmit- igatesthecomputerarithmeticliabilities(underow,overow,andlimited Thereferencesoftwarewillbeanexcellentrepositoryforadvancedalgo- ComparisonSoftwareThiswillservethepurposeoforchestratingthegenerationofreferencevaluesanddeterminingtheprecisionoftestvalues.The precision)ofconventionalcomputingenvironments. comparisonsoftwarewillutilizeparallelmethodsviathesimpledeviceof domainpartitioning.anappropriatemeasureofprecisionwillbedened intermsofintervalmathematics. 2

6 CommunicationInterfaceThiswillbeanappropriatelydesignedWebdoc- alternativeapproach,numericalvericationofidentities,isadvocatedandused Thecomparisonmethodwaschosenbecauseofitsconceptualsimplicity.An umentwithassociatedsubdocumentsforacceptingtestrequestsandre- turningtestresultsviatheinternet. byw.j.codyandhiscoworkers.itavoidstheneedforhigherprecisionbutit algorithmusedintheimplementationofthefunction.also,caremustbetaken conclusionsthatcouldariseiftheidentitywerenotentirelyindependentofthe requirescarefulattentioninthechoiceofidentitytoguaranteeagainstincorrect toseparatetheerrorinthefunctionevaluationfromtheerrorintheevaluation oftheidentity.thesecomplicationswillbeavoidedinthetestservicebytaking andcommunication. 3fulladvantageofthetremendouspowerofcurrentcapabilitiesforcomputation forspecialfunctions.notonlymustitbehighlyaccurate,adeniteboundon theerrorineachcomputedreferencevalueisessential.otherwise,noonecan Thereferencesoftwareisattheheartoftheproposedsoftwaretestingsystem ReferenceSoftware becertainoftheresultsofatest.forthisreason,thereferencesoftwareshould bewrittenusingintervaltechniques.anintroductiontointervalcomputations errorbounds. isgiveninthebookbyalefeldandherzberger[2].however,asidefromthe numericalanalyststodevelopintervalalgorithmsthatgeneratetherequired specicmathematicalfunctions.anopportunityandaneedexistsherefor elementaryfunctions,verylittlehasbeenpublishedonintervalalgorithmsfor software.thusitisappropriatetowritethereferencesoftwareinmultiple precision.thefortranpackageofbailey[3]isavailableandapplicableforthis puterarithmeticsystems,bailey'spackagerelievestheneedtobecarefulabout purpose.becauseofitsvastexponentrangeincomparisontoconventionalcom- Theservicemustbeabletotestdouble-precisionaswellassingle-precision underowandoverow.theoccurrenceoftheseconditionscancompletelyinvalidateanotherwisepristinecomputation.thealgorithmswilltakefullyinto considerationstabilityandroundoquestionsbecausethesetoo,ifignored,can destroyacomputation. precision-limitedbecauseitemployspolynomialorrationalapproximationsthat areconstructedwithrespecttoaxedtargetprecision.flexibilityismoreimportantthaneciencyforreferencesoftware.ideally,referencealgorithmswill Highlyecientsoftware,atleastforfunctionsofonevariable,isusually acceptanarbitrarytolerancespecicationsothatthesameprogramscanbe executedinincreasedprecisionwithoutamajoreorttogenerateapproximationcoecientsforthehigherprecision.thismeansthatmethodswillbe 3

7 constructedfromtaylorexpansions,asymptoticexpansions,dierentialordifferenceequations,integralrepresentations,andotheranalyticalproperties,just asisdoneinmuchexistingsoftwareforfunctionsoftwoormorevariables. Thecomparisonsoftwarehasamathematicalcomponentandacomputersciencecomponent.Themathematicalcomponentisconcernedwithmeasuring ComparisonSoftware 4theerrorintestvalues.Thiscouldbedonesimplywithpointwiseabsoluteor relativeerrorbutanintervalformulationismoreappropriate.thecomputer ues.thisisanaturalapplicationforparallelprocessingwithalooselycoupled sciencecomponentisneededtocollectandprocessthetestandreferenceval- networkofcomputerworkstations. describe,atleastwhenallvariablesarereal.letusconsiderafunction Onlythemathematicalcomponentwillbeconsideredhere.Itiseasyto whererdenotesthesetofrealnumbers.letfbethesetofrealnumbers thatarerepresentableexactlyintheformatofaparticularcomputerarithmetic system,excludinganynonnumericalsymbolicrepresentationssuchas1,0 y=f(x); x2rm; y2r; (1) andnan(not-a-number).thusanapproximatingfunction isdenedbythesoftwaretobetested.ourproblemistomeasuretheerror committedwhen~yistakenasanapproximationtoy. Thepointwiseabsoluteerror,denedforx2Fm,isjustjy?~yj.Because ~y=~f(x); x2fm; ~y2f (2) absoluteerrorisnaturallyassociatedwithxed-pointcomputation,andnot tionf.insteadofrelativeerror=j(~y?y)=yj,weprefertouserelativeprecision oating-point,relativeerrorismoreappropriateexceptnearzerosofthefunc- thisdenitionwasintroducedin[13].sincerp(y;~y)=+o(2),relative precisionandrelativeerrorarenearlythesamewhen~yisagoodapproximation rp(y;~y)=jln(~y=y)jify~y>0; undenedotherwise; (3) (0;1)and(?1;0),respectively. toy.butrelativeprecisionhastheadvantagefordetailederroranalysesthatit isametriconr+andr?,wherethesesymbolsdenotetheopenrealintervals Y=[y`;yu]wherey`;yuaretwoconsecutiveelementsofF.Acriterionthat approximatefunctionvalue~y=~f(x)tosatisfyeither~y=y`or~y=yu.this isappliedsometimesintheconstructionofcomputersoftwareistorequirethe Givenx2Fm,theexactfunctionvaluey=f(x)determinestheinterval 4

8 willbecalledthecriterionoffullprecision.itcanbeexpressedinadierent way.firstwedenethemachineepsilon wheret+denotesthesuccessoroftinf.thentheapproximatingfunction~f satisesthecriterionoffullprecisionif,andonlyif,rp(y;~y)forallx2fm =max t2frp(t;t+): (4) suchthatf(x)and~f(x)havethesamesign.itiscustomarytoemployfull precision,ornearlyfullprecision,insoftwareforelementaryfunctions. plementationsofoating-pointarithmeticaswellasbysomesoftwareforele- mentaryfunctions,particularlywhensuppliedwithfortrancompilers. Thestrongestpossiblecriterioniscorrectrounding.Requiredby[9]for standardoating-pointarithmeticoperations,itismetbymostup-to-dateim- denoteitspredecessorandsuccessor(wheretheorderingisdenedcomponentwise).ifweregardxasarepresentativeofthemultivariateinterval Thecriterionoffullprecisionisquiterigorous.Ifx2Fm,letx?andx+ thentheuncertaintyinxisreectedintherangeoffasitsargumentsvary unnecessarytorequirefullprecisionin~f(x).infact,thecomputationof~f(x) throughoutx.ifapartialderivativeoffislarge,itcanbearguedthatitis X=[x`;xu]=12[x+x?;x+x+]; (5) tofullprecisionisunwarrantedifitrequiresaninordinateamountofexecution setofallclosedintervalsubsetsofr.alefeldandherzberger[2]showthat,if time.thispenaltyislikelytobeespeciallysevereforspecialfunctions. A=[a`;au]andB=[b`;bu]aretwointervals,thenthefunction Amoreappropriatecriterionofprecisioncanbedened.LetI(R)bethe isametric.also,sinceq([a;a];[b;b])=ja?bj,themetricqgeneralizestheusual metricinr.arithmeticoperationsa+b;a?b;abanda=baredenedin I(R)byoperatingontheendpointsoftheintervals.Theyarecontinuousinthe q(a;b)=maxfja`?b`j;jau?bujg; A;B2I(R) (6) topologyoffi(r);qg.similarly,itispossibletodenecontinuousintervalextensionsofcontinuousrealfunctions.forexample,forthelogarithmicfunction, theintervalextensionln(a)=[lna`;lnau]isdenedandcontinuousoni(r+). Next,wedeneintervalrelativeprecision Thisiseasytocompute,sinceitcanbeshownthat rp(y;~y)=8<:q(lny;ln~y)ify;~y2i(r+); undened rp(?y;?~y)ify;~y2i(r?); otherwise: (7) rp(y;~y)=maxfrp(y`;~y`);rp(yu;~yu)g: 5 (8)

9 IntervalrelativeprecisionisametriconI(R+)andI(R?),anditgeneralizes pointwiserelativeprecisionsincerp([y;y];[~y;~y])=rp(y;~y). multivariateintervalx=[x`;xu],andassumethefunctionfiscontinuouson X.LetYbetherangeoffonX: Nowconsiderthetestargumentx2Fm,againasarepresentativeofthe Finally,letthetestfunctionvalue~y=~f(x)2Frepresenttheinterval Y=[y`;yu]=f(X)=ff(x)jx`xxug: ~Y=[~y`;~yu]=12[~y+~y?;~y+~y+]: (10) (9) Thenwewillsaythattheapproximatingfunction~fsatisestheintervalcriterionofprecisionif forallxsuchthattherelativeprecisionsaredened.therightsideofthis inequalityprovidesastandardofcomparison.ittakesintoaccountthebehavior offasitsargumentsvarythroughouttheneighborhoodrepresentedbyx.it rp(y;~y)maxf;rp(y`;yu)g (11) establishestheallowablerangeofrelativeerrorsoverthisneighborhood.the leftsidemeasuresthedistancebetweentheallowablerangeoffandtheinterval representedbythetestfunctionvalue.iftheintervalcriterion(11)issatised, satised.inallcaseswhen(11)issatised,asimpleinterpretationintermsof thenthesetintersectiony\~yisnonempty.if~yyory~y,then(11)is pointwiserelativeerrorcanbegiven.thiswillbediscussedinafuturepaper. errorscausedbytruncatinginniteprocesses.thisproblemwillneedtobe estimatedsubstantially.also,itisnecessarytoconstructstrictboundsforall doesnotnecessarilyproducetherange;tothecontrary,therangemaybeover- ofrealfunctions.evaluationofexplicitexpressionsusingintervalarithmetic Afundamentalprobleminintervalmathematicsishowtocomputetherange facedinthedesignandconstructionofreferencesoftwareforthesoftwaretest service. 5function.Foreachtest,thetestrequesterprovidesanargumentsettogether Forthesoftwaretestsystem,anargumentsetisasubsetofthedomainofa withcorrespondingfunctionvaluestothecommunicationinterface.thenthe CommunicationInterface testrequester'sfunctionvalues,andnallythecommunicationinterfacereturns argumentset,thecomparisonsoftwarecomparesthereferencevaluesagainstthe atableorplotoftheintervalrelativeprecisiontothetestrequester. referencesoftwarecomputesthefunctiontohigherprecisionatallpointsinthe sionprocessesbetweenarbitrarybasesareconsideredindetailinmatula[11] processesofdecimal-to-binaryandbinary-to-decimalconversion.baseconver- Acarefuldevelopmentofthesoftwaretestservicerequiresattentiontothe 6

10 and[12].ap-digit,base-signicancespaceisthesetspofallp-digitnormalizedoating-pointnumbersinthebase,excludingzeroandwithoutregardto size.letspandsqbetwosignicancespaces.theroundingconversionmappingrqfromspintosqisthemappingthatisdenedbyconvertingx2sp intoits-aryexpansiontosucientlyhighprecision,thenroundingittoq base-digits.thetruncationconversionmappingtqisdenedsimilarly.the thenbacktosp.matulaprovedtwotheorems: compositionisanin-and-outconversionmappingwhichmapsspintosq,and Theorem1(BaseConversionTheorem)Ifi6=jforanypositiveintegersi;j,thenthebaseconversionmappingsRq:Sp!SqandTq:Sp!Sq are:1.one-to-oneontotheirrangesifandonlyifq?1p?1; Theorem2(In-and-OutConversionTheorem)Ifi6=jforanypositiveintegersi;j,then 2.ontoifandonlyifp?1q?1. compositionofbaseconversionmappingsispossible.aninterestingkindof 2.RpTq:Sp!Spistheidentityifandonlyifq?12p?1. 1.RpRq:Sp!Spistheidentityifandonlyifq?1>p,and whenthebasesandareintegralpowersofacommonbase.underthe conditionsoftheorem2,rqandtqareone-to-oneontotheirrangesandtheir inversemappingscoincidewithrp:sq!sp. Theconditioni6=jforanypositiveintegersi;jexcludesthetrivialcase arithmeticasdenedin[9].then=2,p=24,=10,andqistobe determinedaccordingtosomecriterion.theroundingconversionmapping Rq10:S24 Asanexample,considerdecimaloutputfromsingle-precisioncomputer 1.one-to-oneontoitsrangeifandonlyifq9; 2.ontoifandonlyifq6; 2!Sq10is exceed6digitsifthecompletesetofq-digitdecimalnumbersistobecovered. similarlyfortq10.thusdecimaloutputprecisionqneednotexceed9digitsif eachinternalnumberistohaveauniquedecimalrepresentation,anditcannot Also,eitherofR24 computerarithmeticf.letsq10bethedecimalsignicancespacewithminimumqsuchthatthenecessaryandsucientconditioninpart1oftheorem2 issatised.finally,letsp0 NowletSpdenotethesignicancespaceassociatedwiththetestrequester's 2Rq10orR24 2Tq10istheidentitymappingifandonlyifq9. 0denotethesignicancespaceassociatedwiththe 7

11 referencesoftware.weassume0andarepositiveintegralpowersoftwo,and weassumep0issuchthatspsp0 representanargumentsetinbinaryordecimal.ifwechoosebinary,thebase conversionmappingfromsptosp0 ThentheIn-and-OutConversionTheoremallowsustochoosewhetherto 0. thischoiceleadstoprogrammingcomplicationsthatarenotentirelytrivial. Also,conversiontodecimalisnecessaryforhumaninterpretation.Thereforeit conversionmappingsarecorrectlyimplementedinthecomputingenvironmentof seemsthatthedecimalchoiceshouldbeconsidered.assumingthatrounding 0istrivial.Itmustbenoted,however,that thetestrequesterandalsointhecomputingenvironmentusedbythesoftware executionspeed)whetherargumentsetsarerepresentedinbinaryordecimal. testservice,itisimmaterial(exceptpossiblyforpracticalconcernsinvolving values,thatarepassedthroughthecommunicationinterfacefromfmtothe Thesameremarkistrueconcerningothertestdata,suchascomputedfunction referencesoftwareorviceversa. inatinginoneoftwoways: DecimalOriginationThefunctiony=f(x)istobetestedtoobtainageneralimpressionofitsaccuracyoverpartsofitsdomain.Hereknowledge Tosummarize,atestrequesterwillwanttoconsiderargumentsetsasorig- BinaryOriginationThefunctiony=f(x)istobetestedatasetofexactlymachine-representablearguments.Forexample,iffisusedinan Heredecimalrepresentationisstillpermissible,providedtheconditionsof fattheexactargumentsthatariseinaparticularprogramexecution. ofexactbinaryrepresentationsisnotimportant,soitisnaturaltospecify theargumentsetindecimal. applicationprogram,itmightbeusefultohavethecapabilityoftesting argumentsexplicitly.lets:[0;1]m!rmbeamonotonicfunction,where monotonicityisdenedcomponentwise.togeneratejtestarguments,the Fordecimalorigination,useofatestgeneratoravoidstheneedtosupply Theorem2aremet. s(t)=x0(1?t)+xjtandthelogarithmicgenerators(t)=x1?t isused.examplesofunivariatetestgeneratorsaretheequidistantgenerator formula generatorsproducejtestargumentsinthex-interval[x0;xj]withequidistantor xj=s(j=(j+1)); j=1;2;:::;j 0xtJ.These (12) logarithmicspacing.anelementofrandomnessisintroducedbyusingapseudorandomnumbergeneratortoproduceat-sequencet1<t2<:::<tjinstead whenexecutedeitherbythetestrequesterorthesoftwaretestservice. thesoftwaretestservice,andassumingallroundingbaseconversionmappings areimplementedcorrectly,testgeneratorswillproduceidenticalargumentsets ofthet-sequencedenedbytj=j=(j+1).withsucientlycarefulcodingof 8

12 tospecifyargumentsetsonapropersubmanifoldofthefunctiondomain.let Weconcludethissectionbyintroducingageneralapproachthatcanbeused Df=D(f) 1D(f) 2:::D(f) Thechiefreasonforsuchaprocedureisthatsomeofthevariablesmaybe denotethedomainofafunctionf.ifthedimensionmexceeds1,itmay bedesirabletoholdoneormorevariablesxedforthedurationofthetest. mrm: (13) xedintheapplicationthatgaverisetothetest.astraightforwardapproach m-dimensionaldomain,wherek<m,anditalsocanbeusedtochangethe canbegeneralizedtopermittestingonak-dimensionalsubmanifoldinthe thisunnecessaryspecicationofxedvariables.ithastheadvantagethatit variablesremainingconstantthroughout.however,anotherapproachavoids wouldbetolisttheargumentsetwiththecomponentscorrespondingtoxed coordinatesystem. wehave theonesthatwillvary;theremainingm?kareheldconstant.wesupposethat kisgivensuchthat1km.letbeapermutation,orrearrangement,of (1;2;:::;m),andlet=?1.Denotingthereorderedvariables1;2;:::;m, Firstwereorderthevariablessothattherstkofthemintheneworderare andthetestisappliedtothefunction y=g()=f(x); r=xr; xr=r 2Rk; (r=1;2;:::;m); x2rm; y2r; (14) Dg=D(g) compareeq.(1).thetestrequesterprovidestheintegerk,thepermutation,thexedargumentsk+1;k+2;:::;m,andtheargumentsetinthedomain 1D(g) 2:::D(g) k. (15) Letusconsiderasanexampletheincompletegammafunction (a;z)=z Denea=x1+ix2,z=x3+ix4.Then Z0e?tta?1dt (<a>0): (16) Supposeatestiswantedinwhichx1isheldconstant.Thenk=3,therequired permutationsare=(2;3;4;1),=(4;1;2;3),andthefunctiongisdenedby f(x1;x2;x3;x4)=(x1+ix2;x3+ix4): (17) Alternatively,supposeatestiswantedinwhichthevariablesarerestrictedto realvalues.then==(1;3;2;4)andg(1;2)=(1+i3;2+i4). g(1;2;3)=(4+i1;2+i3): (18) 9

13 Theproposedsoftwaretestserviceisundergoingactivedevelopmentatthe 6NationalInstituteofStandardsandTechnology.Theinitialemphasisisonthe constructionofthecommunicationinterfaceandassociatedwebdocuments. ConcludingRemarks Thissubstantialprogrammingtaskisbeingaccomplishedwiththeassistanceof M.A.McClainoftheAppliedandComputationalMathematicsDivision. page'forthetestservice.itwillpresentamenuoffunctionsfromwhichthe willfollowtheclassicationthatisusedin[1].specialfunctionsaresubjectto testrequesterwillchoosebyclickingthemouse.initiallyatleast,themenu alternativedenitionsarisingfromvaryingnormalizationcriteria,modication TheenvisionedcommunicationinterfacewillbeaccessedasaWeb`home posedanidenticationprobleminexistingsoftwareforevaluatingandtesting byscalingfunctions,andotherpracticalortheoreticalconsiderations.thishas specialfunctionsbecauseoftheseverelyrestrictedcharactersetusedincomputing.animportantfeatureofwebdocumentsisthattheysupportthefullrange ofmathematicalnotation.thisfeatureisbeingusedtoavoidanyambiguityin theidenticationoffunctionsinoursoftwaretestservice.italsofacilitatesthe possibilityofoeringawiderangeofalternativefunctiondenitionsfortesting. notberelevanttoallusageoftheservice.thereferencesoftwareisessential toallusage,anditisverydemandingtoprovide.itwillrequirealong-term ondemandaswellastoevaluatesoftware.thusthecomparisonsoftwarewill researchanddevelopmenteort.however,symboliccomputingenvironments Thesoftwaretestservicewillbeabletosupplynumericalfunctionvalues existthatsupportnumericalcomputingtoarbitraryprecision.somehaveextensivesupportforspecialfunctions,includingcomputingnumericalvaluesto stricterrorbounds,theyprobablyrepresentthebestcurrentlyavailablesource erencevalues.althoughtheydonotmeetourrequirementsfortheprovisionof highprecision.initiallyatleast,theseenvironmentswillbeusedtosupplyref- usefulinimprovingtheproposedsoftwaretestservice.theauthorwouldbe mostgratefulforreceivingallsuchcomments. paper,readersmayhaveopinions,recommendationsorcriticismsthatwouldbe ofreferencesoftware. Finally,inviewofthenewapproachtotestingthatisintroducedinthis References [1]M.AbramowitzandI.A.Stegun,editors.HandbookofMathematicalFunc- [2]G.AlefeldandJ.Herzberger.IntroductiontoIntervalComputations.AcademicPress,1983. PrintingOce,Washington,DC,1964. tionswithformulas,graphsandmathematicaltables,volume55ofna- tionalbureauofstandardsappliedmathematicsseries.usgovernment 10

14 [3]D.H.Bailey.Algorithm719:Multiprecisiontranslationandexecutionof [4]R.F.BoisvertandB.V.Saunders.PortablevectorizedsoftwareforBessel functionevaluation.acmtrans.math.software,18:456{469,1992.for FORTRANprograms.ACMTrans.Math.Software,19:288{319,1993. [5]W.J.Cody.Algorithm714.CELEFUNT:Aportabletestpackagefor corrigendumseesamejournalv.19(1993),p.131. [6]W.J.Cody.Algorithm715.SPECFUN:AportableFortranpackageof complexelementaryfunctions.acmtrans.math.software,19:1{21,1993. [7]W.J.CodyandW.Waite.SoftwareManualfortheElementaryFunctions. specialfunctionroutinesandtestdrivers.acmtrans.math.software, 19:22{32,1993. [8]J.Dongarra,T.Rowan,andR.Wade.Softwaredistributionusingxnetlib. PrenticeHall,1980. [9]IEEE.IEEEstandardforbinaryoating-pointarithmetic.ANSI/IEEE Std ,TheInstituteofElectricalandElectronicsEngineers,New ACMTrans.Math.Software,21:79{88,1995. [10]D.W.LozierandF.W.J.Olver.AiryandBesselfunctionsbyparallel SixthSIAMConferenceonParallelProcessingforScienticComputing, integrationofodes.inr.f.sincovecetal.,editors,proceedingsofthe York,1985. [11]D.W.Matula.Baseconversionmappings.InProceedingsofthe1967 SpringJointComputerConference,volume30,pages311{318,1967. Philadelphia,1993. volume2,pages531{538.societyforindustrialandappliedmathematics, [12]D.W.Matula.In-and-outconversions.Comm.ACM,11:47{50,1968. [13]F.W.J.Olver.Anewapproachtoerrorarithmetic.SIAMJ.Numer. Anal.,15:368{393,