1IntroductionToinvokeMaximainaconsole,typemaximante;&#xr000;Thecom - PDF document

1IntroductionToinvokeMaximainaconsole,typemaximante;&#xr000;Thecomputerwilldisplayagreetingofthesort:DistributedundertheGNUPublicLicense.SeethefileCOPYING.DedicatedtothememoryofWilliamSchelter.ThisisadevelopmentversionofMaxima.Thefunctionbug_report()providesbugreportinginformation.(%i1)The(%i1)isa\label".Eachinputoroutputlineislabelledandcanbereferredtobyitsownlabelfortherestofthesession.ilabelsdenoteyourcommandsandolabelsdenotedisplaysofthemachine'sresponse.Neverusevariablenameslike%i1or%o5,asthesewillbeconfusedwiththelinessolabeled.Maximadistinguishesloweranduppercase.Allbuilt-infunctionshavenameswhicharelowercaseonly(sin,cos,save,load,etc).Built-inconstantshavelowercasenames(%e,%pi,inf,etc).IfyoutypeSIN(x)orSin(x),Maximaassumesyoumeansomethingotherthanthebuilt-insinfunction.User-denedfunctionsandvariablescanhavenameswhichareloweroruppercaseorboth.foo(XY),Foo(Xy),FOO(xy)arealldierent.2Specialkeysandsymbols1.ToendaMaximasession,typequit();.2.ToabortacomputationwithoutleavingMaxima,type^C.(Here^standsforthecontrolkey,sothat^CmeansrstpressthekeymarkedcontrolandholditdownwhilepressingtheCkey.)Itisimportantforyoutoknowhowtodothisincase,forexample,youbeginacomputationwhichistakingtoolong.Forexample:(%i1)sum(1/x^2,x,1,100000)$^CMaximaencounteredaLisperror:Interactiveinterruptat#x7FFFF74A43C3.Automaticallycontinuing.ToenabletheLispdebuggerset*debugger-hook*tonil.(%i2)3.InordertotellMaximathatyouhavenishedyourcommand,usethesemicolon(;),followedbyareturn.Notethatthereturnkeyalonedoesnotsignalthatyouaredonewithyourinput.2 However,itisoftenusefultoexpressaresultindecimalnotation.Thismaybeaccom-plishedbyfollowingtheexpressionyouwantexpandedby\,numer":(%i4)%,numer;(%o4)82.01219330881977Notetheusehereof%torefertothepreviousresult.InthisversionofMaxima,numergives16signicantgures,ofwhichthelastisoftenunreliable.However,Maximacanoerarbitrarilyhighprecisionbyusingthebfloatfunction:(%i5)bfloat(%o3);(%o5)8.201219330881976b1ThenumberofsignicantguresdisplayediscontrolledbytheMaximavariablefpprec,whichhasthedefaultvalueof16:(%i6)fpprec;(%o6)16Hereweresetfpprectoyield100digits:(%i7)fpprec:100;(%o7)100(%i8)''%i5;(%o8)8.20121933088197564152489730020812442785204843859314941221\2371240173124187540110412666123849550160561b1Notetheuseoftwosinglequotes('')in(%i8)torepeatcommand(%i5).Maximacanhandleverylargenumberswithoutapproximation:(%i9)100!;(%o9)9332621544394415268169923885626670049071596826438162146859\2963895217599993229915608941463976156518286253697920827223758251\1852109168640000000000000000000000004AlgebraMaxima'simportanceasacomputertooltofacilitateanalyticalcalculationsbecomesmoreevidentwhenweseehoweasilyitdoesalgebraforus.Here'sanexampleinwhichapolynomialisexpanded:(%i1)(x+3*y+x^2*y)^3;23(%o1)(xy+3y+x)(%i2)expand(%);63432335232(%o2)xy+9xy+27xy+27y+3xy+18xy2423+27xy+3xy+9xy+x4 Notethatthedisplayconsistsofa\list",i.e.,someexpressioncontainedbetweentwobrackets[...],whichitselfcontainstwolists.Eachofthelattercontainadistinctsolutiontothesimultaneousequations.TrigonometricidentitiesareeasytomanipulateinMaxima.Thefunctiontrigexpandusesthesum-of-anglesformulastomaketheargumentinsideeachtrigfunctionassimpleaspossible:(%i10)sin(u+v)*cos(u)^3;3(%o10)cos(u)sin(v+u)(%i11)trigexpand(%);3(%o11)cos(u)(cos(u)sin(v)+sin(u)cos(v))Thefunctiontrigreduce,ontheotherhand,convertsanexpressionintoaformwhichisasumofterms,eachofwhichcontainsonlyasinglesinorcos:(%i12)trigreduce(%o10);sin(v+4u)+sin(v-2u)3sin(v+2u)+3sin(v)(%o12)---------------------------+-------------------------88Thefunctionsrealpartandimagpartwillreturntherealandimaginarypartsofacomplexexpression:(%i13)w:3+k*%i;(%o13)%ik+3(%i14)w^2*%e^w;2%ik+3(%o14)(%ik+3)%e(%i15)realpart(%);323(%o15)%e(9-k)cos(k)-6%eksin(k)5CalculusMaximacancomputederivativesandintegrals,expandinTaylorseries,takelimits,andobtainexactsolutionstoordinarydierentialequations.Webeginbydeningthesymbolftobethefollowingfunctionofx:(%i1)f:x^3*%e^(k*x)*sin(w*x);3kx(%o1)x%esin(wx)Wecomputethederivativeoffwithrespecttox:(%i2)diff(f,x);3kx2kx(%o2)kx%esin(wx)+3x%esin(wx)3kx+wx%ecos(wx)6 Maximaalsopermitsderivativestoberepresentedinunevaluatedform(notethequote):(%i9)'diff(y,x);dy(%o9)--dxThequoteoperatorin(%i9)means\donotevaluate".Withoutit,Maximawouldhaveobtained0:(%i10)diff(y,x);(%o10)0Usingthequoteoperatorwecanwritedierentialequations:(%i11)'diff(y,x,2)+'diff(y,x)+y;2dydy(%o11)---+--+y2dxdxMaxima'sode2functioncansolverstandsecondorderODE's:(%i12)ode2(%o11,y,x);-x/2sqrt(3)xsqrt(3)x(%o12)y=%e(%k1sin(---------)+%k2cos(---------))228 [-1a1][][1-aa][][11-1](%o4)-----------------a+1In(%i4),themodierdetoutkeepsthedeterminantoutsidetheinverse.Asacheck,wemultiplymbyitsinverse(notetheuseoftheperiodtorepresentmatrixmultiplication):(%i5)m.%o4;[-1a1][][1-aa][01a][][][11-1](%o5)[101].-----------------[]a+1[110](%i6)expand(%);[a1][-----+-----00][a+1a+1][][a1](%o6)[0-----+-----0][a+1a+1][][a1][00-----+-----][a+1a+1](%i7)factor(%);[100][](%o7)[010][][001]Inordertondtheeigenvaluesandeigenvectorsofm,weusethefunctioneigenvectors:10 echoesit,tomakesure---------------------------------------------------------------------------*/print("f=",f),/*---------------------------------------------------------------------------producesalistwiththetwopartialderivativesoff---------------------------------------------------------------------------*/eqs:[diff(f,x),diff(f,y)],/*---------------------------------------------------------------------------producesalistofunknowns---------------------------------------------------------------------------*/unk:[x,y],/*---------------------------------------------------------------------------solvesthesystem---------------------------------------------------------------------------*/solve(eqs,unk))$Theprogram(whichisactuallyafunctionwithnoargument)iscalledcritpts.EachlineisavalidMaximacommandwhichcouldbeexecutedfromthekeyboard,andwhichisseparatedbythenextcommandbyacomma.Thepartialderivativesarestoredinavariablenamedeqs,andtheunknownsarestoredinunk.Hereisasamplerun:(%i1)batch("critpts.max");batching#p/home/robert/tmp/maxima-clean/maxima/critpts.max(%i2)critpts():=(print("programtofindcriticalpoints"),f:read("enterf(x,y)"),print("f=",f),eqs:[diff(f,x),diff(f,y)],unk:[x,y],solve(eqs,unk))(%i3)critpts();programtofindcriticalpointsenterf(x,y)%e^(x^3+y^2)*(x+y);23y+xf=(y+x)%e(%o3)[[x=.4588955685487001%i+.3589790871086935,y=.4942017368275118%i-.1225787367783657],[x=.3589790871086935-.4588955685487001%i,y=-.4942017368275118%i-.1225787367783657],[x=.4187542327234816%i-.6923124204420268,y=0.455912070111699-.8697262692814121%i],[x=-.4187542327234816%i-.6923124204420268,y=.8697262692814121%i+0.455912070111699]]12 expand(a)Algebraicallyexpandsa.Inparticularmultiplicationisdistributedoverad-dition.exponentialize(a)Transformsalltrigonometricfunctionsinatotheircomplexexpo-nentialequivalents.factor(a)Factorsa.freeof(a,b)Istrueifthevariableaisnotpartoftheexpressionb.grind(a)Displaysavariableorfunctionainacompactformat.WhenusedwithwritefileandaneditoroutsideofMaxima,itoersaschemeforproducingbatchleswhichincludeMaxima-generatedexpressions.ident(a)Returnsanaaidentitymatrix.imagpart(a)Returnstheimaginarypartofa.integrate(a,b)Attemptstondtheindeniteintegralofawithrespecttob.integrate(a,b,c,d)Attemptstondtheindeniteintegralofawithrespecttob.takenfromb=ctob=d.Thelimitsofintegrationcanddmaybetakenisinf(positiveinnity)ofminf(negativeinnity).invert(a)Computestheinverseofthesquarematrixa.kill(a)RemovesthevariableawithallitsassignmentsandpropertiesfromthecurrentMaximaenvironment.limit(a,b,c)Givesthelimitofexpressionaasvariablebapproachesthevaluec.Thelattermaybetakenasinfofminfasinintegrate.lhs(a)Givestheleft-handsideoftheequationa.loadfile(a)Loadsadisklewithlenameafromthecurrentdefaultdirectory.Thedisklemustbeintheproperformat(i.e.createdbyasavecommand).makelist(a,b,c,d)Createsalistofa's(eachofwhichpresumablydependsonb),con-catenatedfromb=ctob=dmap(a,b)Mapsthefunctionaontothesubexpressionsofb.matrix(a1,a2,...,an)Createsamatrixconsistingoftherowsai,whereeachrowaiisalistofmelements,[b1,b2,...,bm].num(a)Givesthenumeratorofa.ode2(a,b,c)Attemptstosolvetherst-orsecond-orderordinarydierentialequationaforbasafunctionofc.part(a,b1,b2,...,bn)Firsttakestheb1thpartofa,thentheb2thpartofthat,andsoon.14