Functional Calculation 0 Numerical Apéritif ABSTRACT

NevilleHolmes

SchoolofComputing

UniversityofTasmaniaLaunceston7250Australia

Email:Neville.Holmes@utas.edu.au

ABSTRACT

Thisarticleisthefirstinaseriesexpoundingthejoysoffunctionalcalculation.Functionalcalculationdoeswithoperationsappliedtofunctionsandnumberswhatnumericalcalculationdoeswithfunctionsappliedtonumbers.Thefunc-tionalnotationusedasthevehicleinthisseriesisprovidedbyafreelyavailablecalculationtoolcalledJ.

ThisarticlegivesadetailedillustrationofafewnumericalusesofthenotationsupportedbytheJinterpreter.Thisexplanationispreliminarytofollowingar-ticles,whichwillreviewsomeofthattool’ssupportfornumericalcalculation.TheillustrationtakesasimplefunctionandshowshowJ,oranyothertoolforfunctionalcalculation,canbeusedtoinvestigatespecialvaluesofthefunction.

CALCULATION

Calculationisanhonourableandsatisfyingoccupationwithalonghistoryandarichtradition[1].

Relativelyrecentdevelopmentsinelectroniccomputationhaveledtotheadop-tionofthreeclassesofaidstocalculation,namelyelectronicorprogrammedcalculatorsbaseddirectlyontraditionalcalculation,spreadsheetprograms,andcomplexmathematicalsystems.Electroniccalculatorsareunreliableandverypoorlydesigned[2].Spreadsheetprogramsareverypowerfulandusefulforthekindofcalculationthatsuitsthem,butclumsyandlimitedotherwise.Thecom-plexmathematicalsystemsareextremelypowerful,butinanelephantineway.

Thekindoffunctionalcalculationdescribedinfollowingarticlesislikeanenor-mousbutsympatheticextensionoftraditionalcalculation.Theextensionispar-ticularlyeffectiveindealingwiththekindsoftablestowhichspreadsheetpro-gramsaresuitedbutallowingconvenientlyfortablesofmanydimensions.Itisalsousefulforcollectionsoftablesofdifferentkinds.Theextensionisofsurpris-ingexpressivepowersothatproblemswhichwouldotherwiserequirethemightofacomplexmathematicalsystemcanbetackledsimply.

Thisarticlewillgothroughsomesimplenumericalcalculationsstepbystep,ex-plainingthemindetail.Readerswillgetmostbenefitoutofthedescriptioniftheygettheirowninterpreterandthemselvescarryoutthecalculations(andanyotherssuggestedbytheexamples)astheyarebeingdescribed.

ThenotationusedhereiscalledsimplyJ,andprograms(formanyofthemorepopularcomputingplatforms)toevaluateexpressionsinthenotationcanbedownloadedfromInternetlocation

ftp://archive.uwaterloo.ca/languages/jandhelpwithitcanbegotfromtheUsenetgroup

news:comp.lang.apl

BecausetheprogramswhichevaluateJexpressionsastheyarekeyedinseemtointerpretthenotation,theyarecalledinterpreters.TheJinterpretersthatcanbegotforfreefromtheInternetareoldversions,butinformationaboutup-to-dateversions,andmuchmuchmore,canbefoundat

http://www.jsoftware.com/Theprimitivefunctions,thosefunctionswithsymbolsfornames,usedinthisarticlearefew,andverymanymoresuchareprovidedbytheinterpreter,andarelistedandbrieflyexplainedbytheinterpreter’shelpfacilities.

Thepurposeofthisarticleistoshowhowsimplefunctionscanbeusedsimplytogetusefulresults,thekindofresultsthatshowtheinterpreterinthelightofautilitycalculator.Toallowthistobebetterappreciated,exampleswhichinvolvetheuseoflistsofnumbershavebeenchosen.

Thecalculationsusedhereasanexamplearebasedonanexplorationofafunc-tionnamedfinthedescription.Anyreadercarryingouttheexamplecalcula-tionscandefinefbykeyingin

f=:ˆ-o.

butthisdefinitionwillnotbefurtherexplainedherebecausetheexpressionusedtodefinefisafunctionalexpressionand,assuch,willbeexplainedinlaterarti-clesofthisseries.

FINDINGZEROES

Giventhefunctionfandaneedtoexploreitsbehaviour,thefirststepistonameausefullistoftestvalues,andthentoapplythefunctiontothosevalues.Keyingintheexpression

nn=:i.11

usestheprimitivefunctioni.(pronouncedintegers)toproducethefirstelevennonnnegativeintegers,towhichthecopula=:(pronouncedgetsorbecomes)givesthenamenn.

Thecopulaisonlyusedforconferringnames,anditisnotafunctionassuch.Noresultisshownbecausetheinterpreterisjustnaming.Howeveranynamed

2

resultcanbelookedatjustbykeyinginitsname,inthiscase

nn

0123456710

showsnntobealistoftenintegers.

Somegeneralremarksareappropriateatthispoint.TheJinterpretertakesintheexpressionsitevaluatesindented,buttheresultsareshownunindented.Forthefirstexpressionshownabove,thespacesbetweenthennandthe=:,andbetweenthe=:andthei.,arenotrequiredbytheinterpreter,buthavebeenputtheretomaketheexpressionmorereadable.Morethanonespacecouldhavebeenusedwithsameresult,andspacescouldhavebeenputbetweenthei.andthe11withoutaffectingtheresult.However,puttingaspacewithinthe=:orthei.wouldhaveaffectedtheresultjustasputtingaspacewithinthe11wouldhave,becausethenotationusessuffixeddotsandcolonstoextendtheASCIIcharacterset,sothatineffect=and=.and=:arethreedifferentcharacters.

Thefunctionfcannowbeevaluatedfortheargumentnn,anditsresultdis-played,by

fnn

1_0.4233111.1058710.660842.0318132.705384.5791074.2955.838074.8121995

whichshowsaquitedistinctivebehaviourindeed.

However,theinconvenientlysmallfontforcedontheprintingaboveoftheresultsoffnnmakesitdifficultheretoappreciatefullythenatureofthatbehaviour,soitisusefultodefinewithscantcomment

show=:\".&(0.1&\":)

whichnamesafunctionthatcanbeusedasin

showfnn

1_0.41.110.742132.7384.61074.62955.88074.821995toforceeachvaluetobedisplayedwithatmostonedecimalplace.Thisisper-hapsnotsousefulhere,butwillbesolater.

Oneofthevaluesintheresultlistisnegative.Noticethatthenotationdisam-biguatesthehyphenbyrepresentingthepropertyofnegativitybythesign_sothatthehyphencanbekepttostandforthenegationandsubtractionfunctions.Toinvestigatehowfbehavesfornegativearguments,thetestvaluescanhavetensubtractedfromthembeforeapplyingfbykeyingin

showfnn-10

31.428.325.12218.915.712.69.46.43.51

wherethevaluesdisplayedhavebeenroundedtoonedecimalplacebyshow.Tomakecompletelyplainwhattheargumentvalueisforeachresultvalue,thelaminatefunction,:canbeusedin

a,:showfa=:nn-10_10_9_8_7_6_5_4_3_2_1031.428.325.12218.915.712.69.56.43.51

wherethetestvaluesaretemporarilynamedasothattheycanbestackedby,:

3

ontopoftheresults.Caremustbetakeninputtingtogetherexpressionsliketheonekeyedinabove.Forexample,thecopula=:givesthenametoitslefttoitspredicate,whichiseverythingtoitsright,justasinEnglishsentencesthepredicateofaverbiseverythingtoitsright.Providingthereisaspacebetweenthefandthea,thentherearetwonames,andthecopulaonlyusesthenamenexttoit.Ifthere’snospace,thenameisfaandthereisnofunctionfanymore.Similarly,thepredicateofthelaminatefunctioniseverythingtoitsright,whichistheresultofapplyingftothevaluesofa.

Thesametechniquecanbeusedtostackthevaluesofnnontheresultoffnnby

nn,:showfnn01234567101_0.41.110.742132.7384.61074.62955.88074.821995whichshowsquitedifferentbehaviourfromthatfornegativeargumentvalues.

Themostinterestingfeaturehereisthatthefunctionhastwozeroes,onebetweenargumentsof0and1,andonebetween1and2.Thesezeroescannowbeex-ploredoneatatime,thoughtheexplorationofthesecondzeroishereleftasanexerciseforthereader.

THEFIRSTZERO

Thesimplestzerotoexploreistheonebetween0and1,andby

a,:showfa=:nn%10

00.10.20.30.40.50.60.70.80.9110.80.60.40.20.1_0.1_0.2_0.3_0.4_0.4

theelevenvaluesbetweenandincluding0and1aredisplayed.BecausetherestrictionsoftheASCIIcharactersetrequirethe%symboltobeusedforthedividefunction,ratherthanthemoreusualsymbol,theexpressionnn%10asksfornntobedividedbyten.Thus,theexpressionnn%10givesalistofvaluesbetween0and1incrementingbytenths,andfromtheresultofapplyingftonn%10itcaneasilybeseenthatthezeroliesbetweenargumentvaluesof0.50.6,abouthalfwaybetween.

Thelimitationsofthefunctionshowareevidenthere,butitcaneasilyberede-finedas

show=:\".&(0.2&\":)

whichthenreturnsaseconddecimalplacetosightin

a,:showfa=:nn%10

00.10.20.30.40.50.60.70.80.9110.790.590.410.240.08_0.06_0.19_0.29_0.37_0.42sothatthezerocanbepredictedtobeslightlycloserto0.6thanto0.5.Notonlyarefractionalvalueslike0.5and0.6alwaysdisplayedwithaleadingzero,buttheymustalsobekeyedinwithone.Hencethenextstepinfindingthezero,goingdowntohundredths,is

a,:showfa=:0.5+nn%100

0.50.510.520.530.540.550.560.570.580.590.60.080.060.050.030.020.01_0.01_0.02_0.04_0.05_0.06

wheretheleadingzeroofthe0.5oftheexpressionbeingkeyedinisrequired,notoptional.Thenewdisplayoftheresultsshowsthatthezeroisbetweenargumentvalues0.55and0.56.

Theresultvaluesarebecomingrathertooclosetozero,andwillnowneedtobescaledup,soin

a,:show1000*fa=:0.55+nn%1000

0.550.5510.5520.5530.5540.5550.5560.5570.5580.5590.565.383.972.561.16_0.24_1._3.04_4.44_5.83_7.23_8.62

theresultsaremultipliedbyathousandbeforebeingdisplayed,the*symbolbeingusedtostandforthemultiplicationfunctionbecausethesymbolisnotprovidedintheASCIIcharacterset!Intheabsenceofparentheses,therightargu-mentofafunctionistheentireexpressiontotherightofthefunctionsymbol,sothatthe*in1000*takestheresultoftheapplicationoffasitsrightargument.

Bynow,theargumentvaluesaregettingabittoolongtoprintoutcomfortably,andthewaytheycorrespondshouldbegettingtobeclear,sotheywon’tbeshownstackedanylonger.Theargumentoffnowbeingfrom0.55to0.56,andthezerobeinglocatedbetween0.553and0.554,thenextnarrowingdowncanbedoneby

show10000*f0.553+nn%10000

11.610.28.797.395.994.583.181.780.38_1.02_2.42whichnowplacesthezerobetweenarguments0.5538and0.5539.

TheJinterpretermakesiteasyenoughtobringdownthepreviouslineandsim-plyinsertthethreeextradigitsneededateachstageofthecalculation.Butbynowthestringsofzerodigitsarebecomingabithardtokeepvisualtrackof,so,usinge-notationforthelargescalingintegerandthesmallintervalsize,thenextstepbecomes

show1e5*f0.5538+nn%1e5

3.792.390.99_0.42_1.82_3.22_4.62_6.02_7.42_8.83_10.23

Thisprocessofnarrowingdownhasnowbecomesomewhatrepetitive.Ofcourse,thewholeprocesscouldbemadeautomatic,butbydoingitstepbystepthesuccessionofnumbersgivesabetterfeelforhowthefunctionfisbehaving.Nevertheless,therepeatedpartoftheexpressionusedjustabovecouldrelativelyeasilybeencapsulated(byanexperiencedJuser)asin

g=:]*f@(+nn&%)

5

sothattheinvestigationcannowproceedasfollows.

show0.55382g1e6

9.868.467.065.6.262.851.450.05_1.35_2.75_4.15

show0.553827g1e7

0.51_0._2.29_3.69_5.09_6.49_7.9_9.3_10.7_12.1_13.5

show0.5538270g1e8

5.143.732.330.93_0.47_1.87_3.27_4.68_6.08_7.48_8.88

show0.55382703g1e9

9.317.916.515.113.712.310.9_0.5_1.9_3.3_4.7

Forsometimenowtheresultshavebeenshowingastrangeuniformity.Itmightbesuspectedeitherthatthearithmeticisbeginningtogiveout,orthatthevari-ationbetweensuccessivevalueshasbecomeuniform.Thiscanbecheckedbytakingtheresultvaluesinpairsandsubtractingthem.Thisisdone(withoutex-planationhere)by

show2-/\\0.55382703g1e9

1.41.41.41.41.41.41.41.41.41.4

whichimmediatelyshowsthepostulateduniformity.Indeed,keyingintheex-pressionwithouttheshowgivestenvalues1.40169soitisveryuniform.Thiscouldallowafairlyaccurateinterpolationbyhandofjustwherethezerois,but,forthepurposesofexploration,theestimationprocessiscontinuedhereasbe-fore,exceptthatnow

show=:\".&(0.1&\":)

becauseonlyonedecimalplaceisnowneededtobeshown.

show0.553827036g1e10

97.66.24.83.420.6_0.8_2.2_3.6_5

show0.5538270366g1e11

6.24.83.420.6_0.8_2.2_3.6_5_6.4_7.8

show0.553827036g1e12

6.34.93.52.10.7_0.7_2.1_3.5_4.9_6.3_7.7

show0.55382703g1e13

7.25.84.431.60.2_1.2_2.6_4_5.4_6.8

show0.553827035g1e14

1.90.5_0.9_2.3_3.7_5.1_6.5_7.9_9.3_10.7_12.1

show0.5538270351g1e15

4.93.320.7_0.7_2_3.3_4.7_6.2_7.5_8.9

show0.55382703513g1e16

6.74.46.72.2000_2.2_6.7_2.2_6.7

Thevalueshavebecomeerratic,somuchsothatdoubtmuchattachtoanyat-tempttoaddfurtherdigitstothevaluefoundforthefirstzero.However,check-ingwith

f0.55382703510.553827035130.5538270351354.88498e_156.66134e_160

suggeststhatatleastthefirstfifteendecimalplacesshouldbereliable,providedthecalculationbuiltintotheinterpreterisreliabletosuchprecision.

6

Itisimportanttorealisethatthebarrierthathasbeenencounteredhereisoneofprecision,notrange,since

4*4e324e_321.6e331.6e_31

showsthatamuchgreaterrangecanbesatisfactorilyused.

FINDINGTHELOWESTVALUE

Whilethevalueofthesecondzerocanbeaseasilyfoundasthefirst,betweenthetwozeroesallthevaluesoffarenegative,itmightbepresumed.Thiscanbeexploredby

show0.5g5

0.4_0.9_1.8_2.3_2.1_1.20.73.67.813.721.6

wheretheintervalbetweenargumentsis0.2andstartingfrom0.5.Thissug-gestsaminimumvalueforfofabout_2.3%5whentheargumentisabout1.1,probablyjustabitlargerthan1.1.

Beforegoingfurther,itisconvenienttoredefinefasitsnegation,thus

f=:-&(ˆ-o.)

sothatthevaluesbeinglookedatwillbefreeofthenegativesign,thoughnowwhatwaslowestvaluewillhavebecomethehighest.Consequently,

show1.1g100

45.245.345.445.445.545.545.445.445.345.145

suggeststhat,whilethenowhighestvalueoffis0.455,theargumentcorre-spondingtothisvalueisroundabout1.14.Butthisisnowestimation,andonedecimalplaceforshowistoofew.Thenextstepsincalculationaretherefore

show=:\".&(0.2&\":)show1.14g1000

454.65454.654.67454.68454.68454.68454.68454.67...

whichverifiesthehighestvaluetobeabout0.45468butit’smainlyaguessthattheargumentforthatvalueisabout1.144.Evenwhenweissue

show=:\".&(0.3&\":)

togetthreedecimalplaces,whichthenallows

show_45468+10*1.144g10000

0.1510.1720.1910.2060.2180.2260.2320.2340.2340.230.223

wherethe_45468+10*reducesthesizeofnumbersdisplayedandgetsanextradecimalplaceatthesametime.Theprocedurecanbecontinued,thus

show_4546823+100*1.1447g100000

0.4470.4550.460.4610.460.4550.4470.4360.4220.4050.384

whichaddsanextradigit(abituncertainly—isit2,or3?)tothevalueoftheargument,andtwoextradigitstothevalueofthefunction.Theproceduregoeson,heresplittingthedifferencebetweenthe2andthe3.

7

show_454682346+1000*1.144725g1000000

0.1020.1160.1260.1340.1380.1390.1370.1320.1240.1130.098

whichconfirms2,anditssuccessor9,andagainaddsacoupleofdigits,13,maybeeven138,tothevalueofthefunction.Now

show_454682346138+100000*1.144729g1e7

0.1320.3950.6260.8250.9941.1311.2361.3111.3531.31.344

showsthatthelastdigitofthepreviousfunctionvalueshouldhavebeen9ratherthan8,butstillgivesanothertwofunctionvaluedigitsforthesingleargumentdigit.Inthenextstep

show_45468234613935+1e6*1.1447298g1e8

0.2730.5470.7660.9841.1171.2971.3441.3911.431.4771.477

leadsimmediatelyto

show_45468234613937+1e7*1.144729g1e9105_4511_411_4

whereeverythingnowblowsupatapointwherethefunctionvalueisknowntoabout16digits,buttheargumentvaluetoonly9.

Thismismatchbetweentheaccuracyoftheresultandthatofitsargumentisagoodillustrationoftheinherentunreliabilityofsimplecomputerarithmetic.Moreextremeexamplesofmismatchcouldeasilybeconcocted.

CONCLUSION

Theproceduresdescribedinthisarticleshowtheway,orthestyle,inwhichcal-culationscanbecarriedoutinteractivelyusingatoollikeJ.Ifthecalculationtoolismastered,mostcalculationscanbecarriedoutinthesamestyle.Thebarrierisattainingthatmastery.

Indeedtherepeatedsearchingforananswerbyakindoftrialanderrorisnotcharacteristicoffunctionalcalculation,sincemasterywillenablefunctionstobedefinedwhichcandothesearchingautomatically.Suchvirtuositywouldnotbeappropriatelydemonstratedinanap´eritif.Theinteractivestyleofcalculationdependsontwofactorsillustratedinthisar-ticle.Thefirstissimplytheabilitytokeyinanexpressionandhaveitevaluatedonthefly.Thesecondistheabilitytogiveanametoanexpressionthenusethatnamefromthenonasthoughitwerethatexpression.

Thisarticleisonlyanap´eritiftothemainmealofgettingtogripswitharamifiedfunctionalcalculationtool.

References

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8

2ndedition(asNumberWordsandNumberSymbolsbyMITPressin1969).

2.Thimbleby,H.W.(1995)ANewCalculatorandWhyitisNecessary,The418–433.9

˜/tex/fc/screeds/mt0/na.tex(0/3)

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