MCS320IntroductiontoSymbolicComputationSpring2007MapleLecture18.Symbol - PDF document

MCS320IntroductiontoSymbolicComputationSpring2007MapleLecture18.SymbolicandAutomaticDi®erentiationThedistinctionbetweenaformulaandafunctionisveryimportantinthislecture.Formulasaredi®erenti-atedsymbolically,whileautomaticdi®erentiation[1]producesderivativefunctions.Numericaldi®erentiationisnotconsideredhere.Thislecturecorrespondsto[2,Chapter9].18.1SymbolicDi®erentiationSymbolicdi®erentiationisthecalculationofthederivativeofaformula,verymuchaccordingtorulesyoulearnedincalculus.TheMaplecommandforsymbolicdi®erentiationisdi®,andits\inert"versionisDi®.�[expsin:=exp(sin(x));The\inert"versionofdi®,theDi®,simplyechoesthecommand,whiledi®executes:�[Diff(expsin,x)=diff(expsin,x);Wecancomputethesecond,third,fourth,...derivativeapplyingdi®repeatedly,butdoingthis25timesiscumbersome.Therefore,Maplehasthedollaroperator:�[x$10;#shortcuttobuildasequence�[diff(expsin,x$10);#herewehavethe10-thderivativeInmanyapplications,whatwewishtoderiveisnotde¯nedexplicitly,butimplicitlybyanequation.Torecallwhatimplicitdi®erentiationis,we¯rstdoitthelongwayontheequationofacirclex2+y2=1.Theequation\circle"relatesytox.We¯rsttellMapleweviewyasafunctionofx:�[alias(y=y(x)):�[circle:=x^2+y^2=1;Togetdi®(y,x),wedi®erentiatethede¯ningequation.Remember:thedi®erentiationofanequationisanequation!(Alsodonotforgetthatdiffusesaremembertable.)�[equ:=diff(circle,x);Nowweneedtosolveforthederivativeofywithrespecttox:�[solve(equ,diff(y,x));Theshortwaygoeslikethis:�[implicitdiff(x^2+z^2=1,z,x);#implicitdifferentiation�[implicitdiff(x^2+z^2=1,z,x$3);#3-rdderivative18.2AutomaticDi®erentiationAutomaticdi®erentiationisthecalculationofthederivativeofafunction;theresultisagainafunction.TheMaplecommandforautomaticdi®erentiationisD.Weneedautomaticdi®erentiationfortworeasons.First:noteveryfunctioncanberepresentedbyaniceformula.Second,evenifthereisaformula,wemayhavetodealwithhugeexpressionswellwhichrenderstheresultofsymbolicdi®erentiationverydi±culttouse.Firstweillustratethedi®erencebetweendi®andD,takingtheexampleexpsinfromabove:�[funexpsin:=unapply(expsin,x);�[derfunexpsin:=D(funexpsin);�[derfunexpsin(1.2);Forafunctionfinseveralvariables,D[2$3](f)givesthefunctionwhichreturnsthethirdderivativewithrespecttothesecondvariable.Itisinstructivetolookatthefollowingcommands:JanVerschelde,March2,2007UIC,DeptofMath,Stat&CSLecture18,page1 MCS320IntroductiontoSymbolicComputationSpring2007�[diff(cos(t),t);#differentiationoftheformulacos(t)�[diff(cos,t);#wrong!�[D(cos(t));#alsowrong�[D(cos);#differentiationofthecosinefunctionThefollowingillustratestheavoidanceofexpressionswellbyautomaticdi®erentiation:�[jf:=z�-z^2+1/4;#aquadraticmap�[f10:=jf@@10;#iteratethemaptentimesThef10isafunctionwhichappliesthemapÄjftentimes.Wecanseetherecipeforthisfunctionsymbolicallybyevaluatingatsomesymbolz:�[sf10:=f10(z);#symbolicformulaExpandingsf10reallyleadstoexpressionswell,sowedonotdothishere,butevendi®erentiationonlyonce,leadstoalargerexpression:�[dsf10:=diff(sf10,z);#symbolicdifferentiation�[fsdf10:=unapply(dsf10,z);#turnformulaintofunction�[fsdf10(0.3);#evaluatethederivativeIfweareonlyinterestedinthevalueofthederivative,thenwebetterapplyautomaticdi®erentiation:�[Df10:=D(f10);#automaticdifferentiation�[Df10(0.3);#evaluatethederivativeVerifytheresultsobtainedbyfsdf10(0.3)andDf10(0.3)andcomparethesizesoftheprocedures:�[eval(fsdf10);eval(Df10);#comparesizes18.3Assignments1.GivetheMaplecommandstogetthevalueofthe7-thderivativeofex10+2atx=1.Alsogivethevalueyouobtained.2.GivetheMaplecommand(s)tocompute@8f@5x@3yforf(x;y)=e2x+cos(y).3.Forf:=(x;y)!cos(xy),computethederivativefunctionwhichreturns@5f@2x@3y.4.Considerthecurvede¯nedbyf(x;y)=3+2x+y+2x2+2xy+3y2=0.Locallyonthecurvewecanviewyasafunctionofx,i.e.:y=y(x).Computeformulasforthe¯rstandsecondderivativeofywithrespecttox.5.Computethederivativeofthefunctionf(x)=min(x2+1;2x+3).ReferencesA.Griewank.EvaluatingDerivatives:PrinciplesandTechniquesofAlgorithmicDi®erentiation.SIAM,2000.[2]A.Heck.IntroductiontoMaple.Springer-Verlag,thirdedition,2003.JanVerschelde,March2,2007UIC,DeptofMath,Stat&CSLecture18,page2