Understanding Maple

233 Pages·2017·1.597 MB·english

by  Ian Thompson

UNDERSTANDING MAPLE
IAN THOMPSON
UniversityofLiverpool
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
4843/24,2ndFloor,AnsariRoad,Daryaganj,Delhi-110002,India
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CambridgeUniversityPressispartoftheUniversityofCambridge.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781316628140
10.1017/9781316809761
©IanThompson2017
Firstpublished2017
PrintedintheUnitedKingdombyClays,StIvesplc,November2016
ISBN978-1-316-62814-0Paperback
Additionalresourcesforthispublicationatwww.cambridge.org/maple
Contents
Acknowledgements pageviii
1 Introduction 1
1.1 WhyThisBook? 2
1.2 TheMapleInterface 3
1.3 HowtoReadThisBook 4
2 GettingStarted 6
2.1 ConfiguringtheInterface 6
2.2 TheHelpSystem 9
2.3 StatementsandExecution 10
2.4 Spaces,LineBreaksandComments 16
2.5 ExecutionGroups 18
2.6 Sections 19
2.7 DisplayedResultsandReturnValues 19
2.8 ObtainingApproximateResults 21
2.9 ElementaryFunctions 24
2.10 ComplexNumbers 27
2.11 Variables 31
2.12 Names 34
2.13 AutomaticSimplificationandEvaluation 37
2.14 Concatenation 42
2.15 RelationalOperators 44
2.16 Sequences 48
2.17 SetsandLists 49
2.18 Indices 53
2.19 Element-wiseOperations 56
2.20 Theseq,addandmulCommands 58
2.21 Types 62
2.22 Packages 65
3 AlgebraandCalculus 71
3.1 ManipulatingExpressions 71
3.2 ExtractingPartsofanExpression 76
3.3 Substitutions 79
3.4 Functions 81
3.5 Limits 87
3.6 SummingSeries 89
3.7 Differentiation 92
3.8 Integration 94
3.9 SeriesExpansions 97
3.10 Assumptions 99
4 SolvingEquations 103
4.1 SolvingSingleEquations 103
4.2 SolvingMultipleEquations 107
4.3 SolvingApproximately 109
4.4 DifferentialEquations 113
5 LinearAlgebra 117
5.1 CreatingMatricesandVectors 117
5.2 AccessingVectorandMatrixEntries 120
5.3 DisplayingMatricesandVectors 122
5.4 Addition,MultiplicationandScalarProducts 123
5.5 VectorProductsandNorms 125
5.6 OtherMatrixOperations 127
5.7 SolvingLinearSystems 129
5.8 CopyingMatricesandVectorsandTestingfor
Equality 129
6 Graphics 133
6.1 CreatingBasicPlots 133
6.2 CustomisingaPlot 135
6.3 ParametricandPolarPlots 137
6.4 Three-DimensionalPlots 138
6.5 CombiningPlots 140
6.6 PlotsfromData 141
6.7 Animations 144
7 Programming 147
7.1 ConditionalStatements 147
7.2 DoLoops 150
7.3 Nestingandprintlevel 158
7.4 TheprintandprintfCommands 159
7.5 Arrays 162
7.6 Tables 168
8 Procedures 174
8.1 ABasicProcedure 174
8.2 TheStructureofaProcedure 176
8.3 LocalandGlobalVariables 178
8.4 ArgumentsandParameters 183
8.5 CheckingArgumentValidity 189
8.6 DataReturnedbyProcedures 191
8.7 ReturningUnevaluated 192
8.8 OutputDisplayedfromWithinProcedures 195
8.9 RememberTablesandRecursion 195
8.10 ViewingaProcedureDefinition 197
9 ExamplePrograms 200
9.1 Pascal’sTriangle 200
9.2 TheCollatzProblem 203
9.3 ANewton–RaphsonIteration 205
9.4 SortingData 208
9.5 QuadratureFormulae 211
9.6 Necklaces 215
AppendixA OtherWaystoRunMaple 219
AppendixB TerminatingCharacters 223
IndexofMapleNotation 225
1
Introduction
Hewouldkeepontryingtodothisorthatwithagrimpersistencethat
waspainfultowatch...
JohnWyndham,‘TheDayoftheTriffids’1
Mapleisacomputerprogramcapableofperformingawidevarietyof
mathematicaloperations.Itoriginatedintheearly1980sasacomputer
algebra system, but today this description doesn’t really do it justice.
Maplehasfacilitiesforalgebra,calculus,linearalgebra,graphics(two-
and three-dimensional plots, and animations), numerical calculations
toarbitraryprecision,andmanyotherthingsbesides.Itiswidelyused
in universities across the world, and is particularly useful for tasks
that are tedious and error-prone when performed by humans, such as
manipulating complicated series expansions and solving large sets of
simultaneousequations.Usedcorrectly,Maplecansavetimeandquickly
solveproblemsthatwouldotherwisebeintractable.Usedincorrectly,it
canleadtofrustration,andthedestructionofexpensiveITequipment.
Atthetimeofwriting,thecurrentversionisMaple2016.Versions
before Maple 2015 were numbered starting from 1; the last of these
wasMaple18.NewfeaturesintroducedineachversionfromMaple4.0
onwardscanbeviewedusingthehelpsystem(seeSection2.2).Forthe
mostpart,recentchangeshavebeenrelativelyminor,atleastasfaras
thematerialinthisbookisconcerned.Consequently,alloftheexamples
workwithbothMaple2015andMaple2016.Infact,mostwillwork
in older versions as well, though naturally the number of exceptions
increasesthefurtherbackonegoes.Twosubstantialnewfeaturesare
the dataplot command, discussed in Section 6.6, and the new rules
concerningterminatingcharacters,describedinAppendixB(seealso
Section2.3).BothofthesewereintroducedinMaple2015.
1 PenguinBooks,1954.ReprintedbypermissionofPollingerLimited
(www.pollingerltd.com)onbehalfoftheEstateofJohnWyndham.
1
2 Introduction
1.1 WhyThisBook?
Thisbookisintendedforstudents,teachersandresearcherswhowill
ultimatelywishtouseMapleforadvancedapplications.Here,‘advanced’
meanssomethingmorecomplexthanevaluatingasingleintegral,but
not necessarily designing and running a simulation of the latest jet
engine.Thebookissuitableforundergraduatesandpostgraduatestaking
a course in which Maple is used, and for researchers who intend to
use Maple for part of their work. It can also serve as a consolidation
guideforuserswhoalreadyhavesomeknowledgeofMaple,butfind
themselvesunabletodecipherandeliminatecertainerrormessages,or
who currently rely on apparently magic recipes for solving problems,
basedoncommandsoroperationswhosemeaningisnotclear.There
isnorelianceonmagicrecipeshere.Everyfeatureweuseisproperly
explained,withreferencestotheonlinedocumentationwhereappropriate.
Thebookisnotacomprehensivereferenceguide(alreadyavailablevia
the help system; see Section 2.2), nor is it a beginner’s guide in the
normalsense.Itmostcertainlyisnota‘guidefordummies’.Westartfrom
thebeginning,assumingnopriorknowledgeofthesubjectwhatsoever,
butwhereadvancedtopicsarecentraltounderstandingMapletheyare
tackledhead-on,evenasearlyasChapter2.Inparticular,theevaluation
rules are a regular feature throughout. These determine the order in
whichinputisprocessedbyMaple.Inmostcircumstancestheyarefairly
simple(Section8.4discussessomemorecomplexsituations),butthey
areabsolutelycrucial.
Usingthisbook,itispossibletotacklemanycomplexproblemswithout
anyadditionalMapledocumentation.Readerscanquicklyprogressto
usingpackagesandcommandsthathavenotbeendiscussed,becausethe
principlesintroducedapplyacrossthewholesystem.Wherethebook
aloneisnotsufficient,thosewhohavereaditwillfindthemselvesable
tofullyunderstandMaple’shelppages,whichcanberathertechnical,
andtheMapleProgrammingGuide(seeSection2.2),whichissquarely
aimedatveryadvancedusers.
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1.2 TheMapleInterface 3
1.2 TheMapleInterface
UsersmayinteractwithMapleinseveraldifferentways.Onemayproceed
bytypingcommands,byusinginterfacedrivenmethodsbasedonpalettes
andcontextmenus,orbyacombinationofthetwo.Theadvantagesand
disadvantagesofaparticularmethodarenotalwaysobvious,thoughthey
maybehugelysignificant.
Interfacedrivenmethodscanbeverytemptingtonewusers,because
theymayappeartoeliminatetheneedtostudyamanual.Forexample,
withthefactorysettingforinput(2-DMath;seeSection2.1)itdoesn’ttake
muchefforttoworkouthowtosumseriesandevaluateintegrals,using
theCalculusandExpressionpalettesatthesidesofthescreen(iftheseare
notvisible,theycanberevealedbychoosing Palettes (cid:2) Expand Docks
fromthe View menu).However,mostuserswilleventuallywanttotry
somethingalittlemorecomplicated,inwhichcasethingsarenotquiteso
straightforward.Forallitspower,Mapleisonlyacomputerprogram,and
assuchitcanonlyunderstandmathematicalinputthatisstructuredinthe
correctway.Justasitispossibletotypeincorrectcommands,itisalso
possibletousethepalettesincorrectly,andconstructsomethingthata
humanmathematicianmightunderstand,butMapledoesnot.Onewayor
another,technicalissueswillsometimesarise,andasolidunderstanding
ofMapleisneededtodealwiththeseeffectively.Inviewofthis,interface
drivenmethodsdon’tsavemuchtime,unlessthesoftwareistobeused
exclusivelyforsolvingelementaryproblems.Therewillbenofurther
discussionofsuchmethodshere(searchfortheUserManualinthehelp
systemformoreinformationonthesubject).
This book takes a command driven, or programmatic, approach to
Maple, with the focus on the language rather than the interface. This
hastwoprincipaladvantages.First,itscalesupveryeasily:thesimple
building blocks that make up the Maple language can be assembled
to solve complex problems in an efficient way. This is where the real
powerofMaplelies.Second,thereistransparency:aMapleworksheet
constructed using a sequence of mouse clicks and menu selections is
opaqueinthatauseropeningitcannotseeimmediately(ifatall)howit
wascreated,orhowitcouldbemodifiedandadaptedtohis/herneeds.
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4 Introduction
On the other hand, a worksheet composed from typed commands is
100% transparent. It may be that some users who master the Maple
languagelaterdecidethatmoreinterfacedrivenmethodsaresuitablefor
someoralloftheirwork.However,suchuserswillcontinuetofindthis
bookvaluable,becauseunderstandingtheMaplelanguagemakesthe
behaviourofitsinterfacefarmoretractable.
1.3 HowtoReadThisBook
Thetechnicalmaterialinthisbookisintendedtobereadinorderand
initsentirety.Greatefforthasbeenexpendedtokeepthecontentshort,
while still covering all of the key points. Time has also been spent
minimisingthenumberofsituationsinwhichconceptsareusedbefore
theyareproperlyintroduced.Inafewplaces,thesestructuralaberrations
turn out to be unavoidable (or the lesser of two evils). Where this is
the case, the simplest possible examples have been used to illustrate
theissueathand,andareferencetoalatersection,inwhichtheoutof
placeconceptisdealtwithindetail,isalwaysgiven.Attheveryleast,
everyreadershouldstudyChapter2.Mostofthisisverybasic,butmany
fundamentalaspectsofMaplearedescribedhere,andwithoutknowledge
of these its behaviour can seem mysterious at best, and infuriating at
worst.Chapters3–6dependheavilyonChapter2,butlesssouponeach
other.ThemajorityofuserswillneedthematerialinChapters3and4,
whichintroduceMaple’ssymboliccomputationfacilities.Chapters7–9
reallyneedtobereadinsequence.Withouttheideastheycontain,solving
someproblemswillnecessitatetedious,repetitivework,suchasentering
largenumbersofverysimilarcommands,whichisnotanefficientway
touseMaple.
Throughoutthebook,itemsthatappearinmenusordialogueboxes
are shown in a rounded box with a grey background, such as Help .
Keystrokes such as return have a sharp-cornered box with a white
background. Icons for toolbar buttons are shown as they appear in
Maple2016;theMaple2015versionisalsoshownordescribedifitis
significantlydifferent.Smallblocksoftextmarkedwiththesymbol(cid:3)are
tips.Theseareuseful(oftenveryuseful)butnotvitalpiecesofinformation
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1.3 HowtoReadThisBook 5
oradvice.Insomecasesitispossibletodeducethemfromotherpartsof
thebook.Inputandoutputisshowninthesamestyleinwhichitappearsin
Mapleitself(providedtheconfigurationprocessexplainedinSection2.1
isfollowed).Inparticular,Maplecommandsandstatementsareshownin
atypewriter typeface.Inafewplaces,Maplestatements,orparts
thereof,havebeenomittedinordertoillustratealargerormoregeneral
structure.Textinitalicsisusedtogiveanindicationastowhatismissing.
Outputisomittedifshowingitrequiresaninordinateamountofspace.
ThisconventionisusedextensivelyinChapter6,whereplotsdrawnby
Maplearenotshown.
Togetthemostoutoftheexamplesthroughoutthebook,itisnecessary
to execute them in Maple, and to experiment by modifying them. To
save time for readers, the relevant files have been made available for
download.2Typingtheexamplesmanuallyisfine,butitmaybenecessary
toinsertadditionalrestartcommands(seeSection2.11)betweensome
of them to prevent unintended interactions, especially if the ordering
ischanged.Theonlineworksheetsalreadycontainsufficientrestart
commandstoensurethateverythingworksexactlyasshown.Copying
andpastingfromanelectronicversionofthebookmayleadtounexpected
results,andisthereforenotrecommended.
[email protected].
2 pcwww.liv.ac.uk/∼itho17/understanding_maple
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