Chapter 2 Differential Algebraic Techniquesbt.pa.msu.edu/pub/papers/AIEP108book/chap2.pdf · Chapter 2 Differential Algebraic Techniques ... Campbell, and Petzold 1989 ... mination

ADVANCESIN IMAGING AND ELECTRON PHYSICS,VOL. 108

Chapter 2

Differ ential Algebraic Techniques

In this chapter, we discussa techniquethat is at the coreof further discussionsin thefollowing chapters.Differentialalgebraictechniquesfind theirorigin in theattemptto solve analytic problemswith algebraic means. Oneof the initiatorsof thefield wasLiouville (Ritt 1948)in connectionwith theproblemof integrationof functionsanddifferentialequationsin finite terms.It was thensignificantlyenhancedbyRitt (1932),whoprovidedacompletealgebraictheoryof thesolutionof differentialequationsthatarepolynomialsof thefunctionsandtheirderivativesandthathavemeromorphiccoefficients.Furtherdevelopmentin thefield is duetoKolchin (1973)and,with aneyeon thealgorithmicaspect,to Risch(1969,1970,1979).

Currently, themethodsform thebasisof many algorithmsin modernformulamanipulators,in which the treatmentof differential equationsand quadratureproblemscallsfor thesolutionof analyticproblemswith algebraicmeans.Otherimportantcurrentwork relying on differentialalgebraicmethodsis thepracticalstudyof differentialequationsunderalgebraicconstraints,so-calleddifferentialalgebraicequations((AscherandPetzold1998); (GriepentrogandMarz 1986);(Brenan,Campbell,andPetzold1989;Matsuda1980)).

For our purposes,we will concentrateon theuseof differentialalgebraictech-niques((Berz 1986),(1986,1987b,1989))for the solutionof differentialequa-tionsandpartialdifferentialequations,in particularwediscusstheefficientdeter-minationof Taylor expansionsof the flow of differentialequationsin termsofinitial conditions.Themethodsdevelopedherehave takentheperturbative treat-ment of flows

�� of dynamicalsystemsfrom the customarythird(Brown 1979a;Wollnik, Brezina,and Berz 1987; Matsuoand Matsuda1976;Dragt,Healy, Neri, andRyne1985)or fifth order(Berz,Hofmann,andWollnik1987)all the way to arbitraryorder in a unified andstraightforwardway. Sinceits introduction,the methodhasbeenwidely utilized in a large numberof newmapcodes((Makino andBerz1996a;Berz,Hoffstatter, Wan,Shamseddine,andMakino 1996);(Berz1992b),(1992b,1995a);(Michelotti 1990;Davis, Douglas,Pusch,andLee-Whiting1993;van Zeijts andNeri 1993;van Zeijts 1993;Yan1993;YanandYan1990;Iselin1996)).

81 Copyright c�

1999by Martin BerzAll rightsof reproductionin any form reserved.

82 DIFFERENTIAL ALGEBRAIC TECHNIQUES

2.1 FUNCTION SPACES AND THEIR ALGEBRAS

2.1.1 Floating Point Numbers and Intervals

Thebasicideabehindthis methodis to bring thetreatmentof functions andtheoperationson themto the computerin a similar way as the treatmentof num-bers. In a strict sense,neitherfunctions(e.g.,thosethatare �� , infinitely oftendifferentiable)nor numbers(e.g.,the real � ) canbetreatedon a computersinceneithercanin generalberepresentedby a finite amountof information;afterall,a realnumber“really” (punintended)is anequivalenceclassof boundedCauchysequencesof rationalnumbers.

However, from theearlydaysof computerswe areusedto dealingwith num-bers by extracting information deemedrelevant, which in practiceusuallymeansthe approximationby floating point numbers with finitely many digits.In a formal sense,this is possiblesincefor every oneof the operationson realnumbers,suchasadditionandmultiplication,we cancraft an adjoint operationon thefloatingpoint numberssuchthatthefollowing diagramcommutes:�� ! "�� "�#�%$�&

' ((((((((()((((((((() *� ' � ��! � "� * " �

(2.1)

Of course,in realitythediagramscommuteonly “approximately,” whichtypicallymakestheerrorsgrow over time.

Theapproximatecharacterof theseargumentscanberemovedby representinga real numbernot by onefloating point numberbut ratherby interval floatingpoint numbersproviding a rigorousupperand lower bound((Hansen; Moore1979),(1988);(Alefeld andHerzberger1983;KaucherandMiranker1984;Kear-fott andKreinovich 1996)).

By roundingoperationsdown for lower boundsandup for upperbounds,rig-orousboundscanbe determinedfor sumsandproducts,andadjoint operationscanbemadesuchthatdiagram(2.1)commutesexactly. In practice,while alwaysmaintainingrigor, themethodsometimesbecomessomewhatpessimisticbecauseover time theintervalsoftenhave a tendency to grow. This so-calleddependencyproblemcanbe alleviatedsignificantly in several ways,anda ratherautomatedapproachis discussedin (Makino andBerz1996b).

2.1.2 Representations of Functions

Historically, the treatmentof functions in numericshasbeendonebasedon thetreatmentof numbersand,asaresult,virtually all classicalnumericalalgorithms

FUNCTIONSPACESAND THEIR ALGEBRAS 83

arebasedonthemereevaluationof functionsatspecificpoints.As aconsequence,numericalmethodsfor differentiation,whichoffer onewayto attemptto computeTaylor representationsof functions,arevery cumbersomeandproneto inaccura-ciesbecauseof cancellationof digits, andthey arenot usefulin practicefor ourpurposes.

Thesuccessof thenew methodsis basedontheobservationthatit is possibletoextractmoreinformationaboutafunctionthanits merevalues.Indeed,attemptingto extendthe commutingdiagramin Eq. (2.1) to functions,onecandemandtheoperation� to betheextractionof theTaylor coefficientsof a prespecifiedorder+ of the function. In mathematicalterms, � is an equivalencerelation,and theapplicationof � correspondsto thetransitionfromthefunctionto theequivalenceclasscomprisingall thosefunctionswith identicalTaylorexpansionto order + .

SinceTaylor coefficientsof order + for sumsandproductsof functionsaswellasscalarproductswith realscanbe computedfrom thoseof the summandsandfactors,it is clearthatthediagramcanbemadeto commute;indeed,exceptfor theunderlyinginaccuracy of thefloatingpointarithmetic,it will commuteexactly. Inmathematicalterms,this meansthat the setof equivalenceclassesof functionscanbeendowedwith well-definedoperations,leadingto theso-calledtruncatedpower seriesalgebra (TPSA) ((Berz1986),(1986,1987b)).

This factwasutilized in thefirst paperon thesubject(Berz1987b),which ledto a methodto extractmapsto any desiredorderfrom a computeralgorithmthatintegratesorbits numerically. Similar to the needfor algorithmswithin floatingpoint arithmetic,thedevelopmentof algorithms for functions followed,includ-ing methodsto performcompositionof functions,to invert them,to solve non-linear systemsexplicitly, andto introducethe treatmentof commonelementaryfunctions;many of thesealgorithms((Berz 1999), (1999,1991b))will be dis-cussedlater.

However, it becameapparentquickly ((Berz 1988b),(1988b,1989))that thisrepresentsonly a halfway point, andone shouldproceedbeyond mere arith-metic operationson functionspacesof additionandmultiplicationandconsidertheir analytic operations of differ entiation and integration. This resultedinthe recognitionof the underlyingdiffer ential algebraic structur e andits prac-tical exploitation,basedon thecommutingdiagramsfor addition,multiplication,anddifferentiationandtheir inverses:, �.- ��/��. $0�213 � � ((((() ((((() 4 ��5,76 - �/��. � $985:1

, �.- ��/��2 $0�21; ��< ((((() ((((() = �/>,@?<A- �/��2 � $CB>D1

84 DIFFERENTIAL ALGEBRAIC TECHNIQUES, ��/��. $E � EGFIH ((((() ((((() EKJ � E FLHJE , � EGFIH , �/��. � EKJ $0� E FIHJ $ (2.2)

Thetheoryof this approachwill bedevelopedin detail in this chapter.Of course,thequestionof whatconstitutes“information deemedrelevant” for

functionsdoesnot necessarilyhave a uniqueanswer. Formulamanipulators,forexample,attackthe problemfrom a differentperspective by attemptingto alge-braically expressfunctions in termsof certainelementaryfunctions linked byalgebraicoperationsandcomposition.In practice,the Achilles’ heelof this ap-proachis thecomplexity thatsuchrepresentationscantakeafteronly afew opera-tions.Comparedto themereTaylor expansion,however, they have theadvantageof rigorouslyrepresentingthefunctionunderconsideration.ModernextensionsoftheTaylormethod(MakinoandBerz1996b)havetheability to overcomethisdif-ficulty by providing fully rigorousboundsfor remainderterms;but thesemethodsgo beyondthescopeof this book.

2.1.3 Algebras and Differential Algebras

Before proceeding,it is worthwhile to put into perspective a variety of differ-ent conceptsthat wereintroducedto the field in connectionwith the previouslydiscusseddevelopments.We do this in parallel to establishingthe scopeof thefurther developmentsin which differentialalgebraictechniqueswill be appliedextensively.

The first and simplest structurethat was introduced((Berz 1986), (1986,1987b))is TPSA. This is the structurethat resultswhentheequivalenceclassesof functionsare endowed with arithmeticsuch that the diagramsin Eq. (2.1)commutefor the basic operationsof addition, multiplication, and scalarmul-tiplication. Addition and scalarmultiplication lead to a vector space, and themultiplication operationturnsit into a commutative algebra. In many respects,togetherwith the polynomialalgebras,this structureis an archetypalnontrivialalgebra,and in fact it can be embeddedinto many larger andmore interestingalgebras.

TPSA canbe equippedwith an order, andthenit containsdifferentials,i.e.,infinitely small numbers.This fact triggeredthe studyof suchnonarchimedeanstructuresin moredetail andled to the introductionof a foundationof analysis((Berz1996),(1990a,1994,rint)) on a largerandfor suchpurposesmuchmoreusefulstructure,theLevi–Civita field discussedin Section2.3.5.TheLevi–Civitafield is the smallestnonarchimedeanextensionof the real numbersthat is alge-braicallyandCauchycomplete,andmany of thebasictheoremsof calculuscanbeprovedin asimilarwayasin �NM Furthermore,conceptssuchasDeltafunctionsandthe ideaof derivativesasdifferentialquotientscanbeformulatedrigorously

FUNCTIONSPACESAND THEIR ALGEBRAS 85

andintegratedseamlesslyinto thetheory. Onthepracticalend,basedonthelatterconcept,therearealsoseveralimprovementsregardingmethodsof computationaldifferentiation((ShamseddineandBerz1996),(1997)).

Thefinal conceptthatis connectedto ourmethodsandworth studyis thetech-niqueof automatic differ entiation (Berz,Bischof,Griewank,Corliss,andEds.1996;Griewank,Corliss,andEds.1991;Berzrint). Thepurposeof thisdisciplineis the automatedtransformationof existing codein sucha way that derivativesof functionalrelationshipsbetweenvariablesarecalculatedalongwith the orig-inal code.Besidesthe significantlyincreasedcomputationalaccuracy comparedto numericaldifferentiation,a striking advantageof this approachis thefact thatin theso-calledreversemodeit is actuallypossiblein principle to calculategra-dientsin O variablesin afixed amount of effort ; independentof O , in theoptimalcasethe entiregradientcanbe obtainedwith a costequallingonly a few timesthecostof theevaluationof theoriginal functions,in starkcontrastto numericaldifferentiationwhich requires� O 3QP � timestheoriginal cost.

In practice,multivariate automaticdifferentiation is almost exclusively re-strictedto first order, andassuchis not directly useful for our purposes.Onereasonfor this situationis the fact thatconventionalnumericalalgorithmsavoidhigher derivatives as much as possiblebecauseof the well-known difficultieswhentrying to obtainthemvia numericaldifferentiation,which for a long timerepresentedtheonly availableapproach.On theotherhand,thepreviously men-tionedsavingsthatarepossiblefor linearderivativesaremuchharderto obtaininthesameway for higherorders.

Altogether, the challengein automaticdifferentiationis more reminiscentofsparsematrix techniquesfor managementandmanipulationof Jacobiansthanof a power seriestechnique.It is perhapsalsoworth mentioningthatbecauseoftheneedfor coderestructuringin orderto obtainperformance,thereis a certainreluctancein the communitytoward the useof the word “automatic.” Mostly inorderto avoid theimpressionof makingfalsepretence,thetechniquehasrecentlybeenreferredto ascomputational differ entiation.

Only veryrecentlyareothergroupsin computationaldifferentiationpickingupat leastonsecondorder(Abate,Bischof,Roh,andCarle1997),but sofar theonlysoftwarefor derivativesbeyondordertwo listed in the automaticdifferentiationtool compendium(Bischof and Dilley ) is in fact the packageDAFOR ((Berz1991a),(1987a,1990c))consistingof the FORTRAN precompilerDAPRE andthe arbitraryorderDA packagethat is also usedas the power seriesengineinthe codeCOSY INFINITY (Berz 1997a;et al. ; Makino andBerz 1999a;Berz,Hoffstatter, Wan,Shamseddine,andMakino1996;MakinoandBerz1996a).

As alludedto previously, the power of TPSA canbe enhancedby the intro-ductionof derivations

Eand their inverses,correspondingto the differentiation

and integration on the spaceof functions.The resultingstructure,a differ en-tial algebra (DA), allows thedirecttreatmentof many questionsconnectedwithdifferentiationand integrationof functions,including the solutionof the ODEsR �SL< RUT � �, � �SV� T � describingthemotionandPDEsdescribingthefields.


2.2 TAYLOR DIFFERENTIAL ALGEBRAS

2.2.1 The Minimal Differential Algebra

We begin our studyof differentialalgebraicstructureswith a particularlysimplestructurewhich is of historicalsignificanceandconstitutesthesimplestnontrivialdifferentialalgebra(Berz 1989).In fact, the realsundertheir regular arithmeticwith a derivation

Ethat vanishesidentically trivially form a differentialalgebra,

but it isnotaveryinterestingone.To obtainthefirst nontrivial differentialalgebra,wehaveto moveto �XWUM Let usconsiderthesetof all orderedpairs � Y[Z\�2Y H � , whereY[Z and Y H arereal numbers.We defineaddition,scalarmultiplicationandvectormultiplicationasfollows:� Y Z �2Y H � 3 ��] Z �.] H �^�_�`Y Z 3 ] Z �2Y H 3 ] H � (2.3)T ; �`YaZb�cY H �^�_� T ; YaZU� T ; Y H � (2.4)�`Y[Z\�2Y H � ; ��]aZ\�.] H �^�_�`YaZ ; ]dZb�cY[Z ; ] H 3 Y H ; ]aZA� M (2.5)

Theorderedpairswith thearithmeticarecalled HfegH M Thefirst two operationsarethefamiliar vector spacestructureof �XWhM Themultiplication,on theotherhand,looks similar to that in the complex numbers;except here,as one seeseasily,�ì�� P � ; �ì�� P � doesnot equal � � P �2i\� but rather �ì��2ib� . Therefore,theelementRkjalnm��ì�� P � (2.6)

plays a quite different role than the imaginaryunit o in the complex numbers.Themultiplicationof vectorsis seento have � P �cib� astheunity elementandto becommutative,i.e., � Y[Z\�2Y H � ; ��]aZ\�.] H �^�_��]dZb�2] H � ; � Y[ZU�cY H � M It is alsoassociative, i.e.,�`YaZU�cY H � ;�p ��]dZb�2] H � ; �rqdZb��q H ��st� p � Y[ZU�cY H � ; �`]aZu�.] H �/s ; � qdZb��q H � M It is alsodistributivewith respectto addition,i.e., � Y[Z\�2Y H � ;dp �`]aZU�2] H � 3 �rqaZb�cq H �/st�v� Y[Zb�2Y H � ; ��]dZb�2] H � 3�`YaZU�cY H � ; � qdZb��q H � , andsothetwo operations

3and ; form a (commutative) ring .

Together, thethreeoperationsform analgebra. Like thecomplex numbers,theydo form anextensionof the real numbers; because��]��cib� 3 �rqb�2ib�w�9��] 3 qb�2i\�and ��]��2i\� ; � qb�cib�w�C�`] ; qb�cib�/� thepairs �`]��2i\� behave like realnumbers,andthusasin � � therealscanbeembedded.

However HfegH is not a field. Indeed,it canbeshown that � Y Z �2Y H �� HxeyH hasamultiplicati ve inversein HfegH if andonly if Y Z{z�:i . If Y Z{z�Qi then�`Y Z �cY H � FLH �}| PYaZ � � Y HY WZ�~ M (2.7)

We alsonotethatif Y[Z is positive,then �`YaZb�cY H �� H e H hasa root� �`YaZU�2Y H �� |�� Y[Zb� Y H� � Y Z ~ � (2.8)

TAYLOR DIFFERENTIAL ALGEBRAS 87

assimplearithmeticshows.The structure HfegH plays a quite important role in the theory of real alge-

bras. All two-dimensionalalgebras,which are the smallestnontrivial algebras,areisomorphicto only threedifferentkindsof algebras:the complexnumbers,theso-calleddual numbers in which �ì�� P � ; �ì�� P ��v� P �cib� , andthealgebra HfegH M

In fact,this ratheruniquealgebrahassomeinterestingproperties.Oneof themis that H e H canbe equippedwith an order that is compatiblewith its algebraicoperations.It is this requirementof compatibilityof the orderwith additionandmultiplication that makes � , the field of real numbers,an orderedfield. On theotherhand,while many orderscanbeintroducedonthefield of complex numbers,nonecanbemadecompatiblewith thefield’sarithmetic;wesaythatthecomplexnumberscannotbeordered.

Giventwo elements�`YaZU�2Y H � and ��]dZb�2] H � in H e H , we define�`Y Z �2Y H ��] Z �2] H � if Y Z ��] Z or � Y Z �Q] Z and Y H ��] H ��`Y[Z\�2Y H ��]dZb�2] H � if �`]aZ\�.] H ��`Y[Z\�2Y H ��`Y[Z\�2Y H �^�_��]dZb�2] H � if YaZt��]dZ and Y H �Q] H M (2.9)

It followsimmediatelyfrom thisorderdefinitionthatfor any two elements�`Y Z �cY H �and ��] Z �2] H � in HxeyH � oneandonly oneof � Y Z �2Y H �X�C��] Z �2] H �/�#� Y Z �2Y H �#��] Z �2] H �and �`Y Z �2Y H ��`] Z �.] H � holds.We saythat theorderis total; alternatively, HfegH istotally ordered.Theorderis compatiblewith additionandmultiplication; for all�`YaZb�cY H �/�X��]aZ\�.] H �[�t�rqaZ\�cq H �� H e H , we have �`YaZb�cY H ��]dZb�2] H �X�V��`Y[Z\�2Y H � 3� qdZ\�cq H �N��]dZb�2] H � 3 �rqaZb�cq H �[� and �`YaZb�cY H �N��`]aZ\�.] H � ; and � qdZU��q H �{��ì��cib��i��I��`Y[Z\�2Y H � ; �rqdZU�cq H �{��`]aZ\�.] H � ; �rqdZU�cq H � M We alsoseethat the orderon therealsembeddedin H e H is compatiblewith theorderthere.

ThenumberR

definedpreviouslyhastheinterestingpropertythatit is positivebut smallerthanany positiverealnumber;indeed,we have�ì��2i\�� i�� P ��]��2i\�^�D] M (2.10)

WesaythatR

is infinitely small. Alternatively,R

is alsocalledaninfinitesimal ora differ ential. In fact,thenumber

Ris sosmallthatits squarevanishes,asshown

previously. Sincefor any � Y[Zb�2Y H �� H e H we have�`Y[Z\�2Y H �^�_�`YaZb�cib� 3 �ì��cY H �^�QYaZ 3 R ; Y H � (2.11)

wecall thefirst componentof �`Y Z �cY H � therealpartandthesecondcomponentthedifferentialpart.

ThenumberR

hastwo moreinterestingproperties.It hasneithera multiplica-tive inversenor an + th root in HfegH for any +7� P

. For any �`Y Z �2Y H � in HfegH , wehave �`YaZU�cY H � ; �ì�� P ��_�ì��2YaZA� z�_� P �cib� M (2.12)


Ontheotherhand,oneeasilyshowsby inductionon + that� Y Z �2Y H ��g�v�`Y��Z �2+ ; Y � FLHZ Y H � for all � Y Z �2Y H �� HfegH andall +�� P M (2.13)

Therefore,ifR

hasan + th root �`Y Z �cY H �� HxeyH , then�`Y �Z �.+ ; Y � FLHZ Y H �^�v� i�� P �/� (2.14)

from whichwehavesimultaneouslyY[Z#�:i and + ; Y � FIHZ Y H � P —acontradiction,for +�� P . Therefore,

Rhasno + th root in H e H .

Wenext introduceamapE

from H e H into itself,whichwill proveto beaderiva-tion andwill turn thealgebraH e H into adifferentialalgebra.

DefineE�� H e H�� H e H byE �`YaZU�2Y H ��v�ì��2Y H � M (2.15)

NotethatE p �`Y Z �cY H � 3 �`] Z �.] H ��st� E �`Y Z 3 ] Z �2Y H 3 ] H �^�v� i��2Y H 3 ] H ��_�ì��2Y H � 3 � i��2] H �^� E �`Y Z �cY H � 3 E ��] Z �2] H � (2.16)

andE p �`Y[Zb�2Y H � ; ��]aZ\�.] H �/st� E �`YaZ ; ]dZb�cY[Z ; ] H 3 Y H ; ]dZ��^�_�ì��2YaZ ; ] H 3 Y H ; ]aZ��v� i��2Y H � ; ��]aZ\�.] H � 3 �`YaZb�cY H � ; �ì��.] H �� p E � Y[Zh�cY H ��s ; �`]aZ\�.] H � 3 �`YaZU�cY H � ;hp E �`]aZU�2] H ��s M (2.17)

This holds for all �`Y[Z\�2Y H �[��]dZb�2] H �� H e H . Therefore,E

is a derivation, andhence� H e H � E � is adifferentialalgebra.

Thereis anotherway of introducinga derivationon HfegH � which mapsinto therealnumbers,by setting

EG�! c¡ � Y Z �2Y H ��_Y H M AlsoEG�! 2¡

satisfiesthepreviousaddi-tion andmultiplicationrules;but becauseit doesnot map HfegH into itself, it doesnot introducea differentialalgebraicstructureon HfegH M Thederivation

Eis appar-

ently connectedtoEG�! c¡

byE � R ; EG�! c¡ , andhencehasa resemblenceto a vector

field (Eq.1.7).For the purposesof beamphysics,the mostimportantaspectof H e H is that it

canbe usedfor the automatedcomputationof derivatives.This is basedon thefollowing observation.Let us assumethat we have given the valuesandderiva-tivesof two functions

,and - at the origin; we put theseinto the real anddif-

ferentialcomponentsof two vectorsin H e H , which have the form � , � ib�/� ,£¢ � ib�.�and �x-L�ì\�/�.- ¢ �ìb�2� M Let usassumeweareinterestedin thederivativeof theproduct, ; - , which is givenby

,£¢ � ib� ; -L�ì\� 3 , �ì\� ; - ¢ � ib� M Apparently, thisvalueappears


in thesecondcomponentof theproduct � , �ìb�[� ,£¢ �ì\�.� ; ��-£� ib�[��- ¢ � ib�.� , whereasthefirst componentof theproducthappensto be

, � ib� ; -L�ì\� . Therefore,if two vec-torscontainthevaluesandderivativeof two functions,their productcontainsthevaluesandderivativesof theproductfunction.Definingtheoperation¤�¥ from thespaceof differentiablefunctionsto HfegH via¤ , ¥ �_� , � ib�/� , ¢ � ib�.�[� (2.18)

we thushave ¤ , 3 - ¥ � ¤ , ¥ 3 ¤ - ¥ (2.19)¤ , ; - ¥ � ¤ , ¥ ; ¤ - ¥ � (2.20)

andhencethe operationclosesthe two uppercommutingdiagramsin Eq. (2.2).Similarly, theadditionin H e H is compatiblewith thesumrule for derivatives.

In this light, the derivationsof rulesfor multiplicative inverses(Eq. 2.7) androots(Eq.2.8)appearasa fully algebraicderivationof thequotientandroot rulesof calculus,without any explicit useof limits of differentialquotients;this is asmallexampleof genericdifferentialalgebraicreasoning.

This observationaboutderivativescannow beusedto computederivativesofmany kindsof functionsalgebraicallybymerelyapplyingarithmeticruleson HxeyH ,beginningwith thevalueandderivativeof theidentity function.We illustratethiswith anexampleusingthefunction, ��S£�� PS 3 � P <�S�� M (2.21)

Thederivativeof thefunctionis, ¢ �`S�� P <�S W � � P��S 3 � P <�S£�.� W M (2.22)

Supposewe areinterestedin thevalueof thefunctionandits derivativeat S¦�:§ .We obtain , � §b�^� §P i � , ¢ � §b�^� � ��U¨ M (2.23)

Now take thedefinitionof thefunction,

in Eq. (2.21)andevaluateit at valueandderivativeof theidentity function �`§�� P �^�D§ 3 R . We obtain, �.� §�� P �.�^� P�`§�� P � 3QP <��`§�� P � � P�`§�� P � 3 � P <�§�� P <h©b�� P� P i\<�§��2ªu<�©b� � | §P i � � ª©�« P iUi© ~ � | §P i � � ��U¨ ~ M (2.24)


As canbe seen,after the evaluationof the function the real part of the resultis thevalueof thefunctionat S��7§ , whereasthedifferentialpart is thevalueofthederivativeof thefunctionat S%�:§ .Thisresultis not a coincidence.UsingEq.(2.18),wereadilyobtainthatfor all - with -L�ì\� z�Qi in HfegH ,¤ P <�- ¥ � ¤ P ¥ < ¤ - ¥ � P < ¤ - ¥rM (2.25)

Thus, ¤ , �`S�� ¥ �¬ PS 3QP <�S�® � P¤ S 3DP <�S ¥� P¤ S ¥ 3 ¤ P <�S ¥ � P¤ S ¥ 3QP < ¤ S ¥� , � ¤ S ¥ � M (2.26)

Sincefor a real SG� we have ¤ S ¥ �¯��SG� P �k�CS 3 R � and ¤ , �`S�� ¥ �}� , ��S��[� ,£¢ ��S£�.� ,apparently � , � §b�/� , ¢ � §b�.�°� , �.� § 3 R �.� M (2.27)

The methodcan be generalizedto allow for the treatmentof commonintrin-sic functions - like ±.²´³ ��µ[¶K·I� by setting ±2²!³ � Y[Zb�2Y H �Q�¸� ±.²´³ �`YaZ��[�2Y H£¹[º ± � Y[ZA�2�/�µ[¶K·V� Y[Zb�2Y H ��7�`µ[¶K·G� Y[Z��/�2Y H µ[¶K·V� Y[Z��.� , or moregenerally,-U2� ¤ , ¥ �^� ¤ -U2� , � ¥ or-U��.� Y Z �2Y H �.�^�_�x-Uc�`Y Z �/�cY H - ¢ � Y Z �2� (2.28)

By virtue of Eqs.(2.5) and(2.28),we seethat any function,

representableby finitely many additions,subtractions,multiplications,divisions,andintrinsicfunctionson H e H satisfiestheimportantrelationship¤ , ��S�� ¥ � , � ¤ S ¥ � M (2.29)

Therefore,for all ]N� �_» HfegH , we canwrite� , ��]h�/� , ¢ ��]h�.�� , �`] 3 R �[� (2.30)

from whichweinfer that, ��]h� and

,£¢ ��]h� areequalto therealanddifferentialpartsof, ��] 3 R � , respectively.NotethatEq. (2.30)canberewrittenas, �`] 3 R �^� , ��]h� 3 R ; , ¢ �`]�� M (2.31)

Thisresembles, �`S 3%¼ S��½ , ��S£� 3%¼ S ; , ¢ ��S£� , in whichcasetheapproximation

becomesincreasinglybetterfor smaller¼ S . Here,we chooseaninfinitely small

TAYLOR DIFFERENTIAL ALGEBRAS 91¼ S , andthe error turnsout to be zero.In Section2.3.5we provide a morede-tailedanalysisof this interestingphenomenonandat thesametime obtainsomeinterestingnew calculus.

2.2.2 The Differential Algebra � e{¾In this section,we introducea differentialalgebrathatallows thecomputationofderivativesup to order + of functionsin O variables.Similar to before,it is basedontakingthespace� � � � ¾ �/� thecollectionof + timescontinuouslydifferentiablefunctionson � ¾ . As a vectorspaceover � , � � � � ¾ � is infinite dimensional; forexample,thesetof all monomialsS H is linearly independent.

Onthespace� � � � ¾ �[� wenow introduceanequivalencerelation.For,

andin� � � � ¾ � , we say, � � - if andonly if

, �ì\��-L�ì\� andall thepartialderivativesof,

and - agreeat 0 up to order + . Thenewly introducedrelation � � apparentlysatisfies , � � , for all

, � � � � � ¾ �/�, � � -N�I�¿-{� � , for all, �G-_� � � � � ¾ � and, � � - and -{� ��À �I� , � �NÀ for all, �G-�� À � � � � � ¾ � M (2.32)

Thus, � � is anequivalencerelation. We now groupall thoseelementsthatarerelatedto

,togetherin oneset,theequivalenceclass ¤ , ¥ of thefunction

,. The

resultingequivalenceclassesareoftenreferredto asDA vectorsor DA numbers.Intuitively, eachof theseclassesis then specifiedby a particularcollection ofpartial derivativesin all O variablesup to order + . We call the collectionof alltheseclasses� eN¾ M

Now wetry to carryover thearithmeticon � � � � ¾ � into � e{¾ . We observethatif we know thevaluesandderivativesof two functions

,and - , we caninfer the

correspondingvaluesandderivativesof, 3 - (by mereaddition)and

, ; - (byvirtue of theproductrule).Therefore,we canintroducearithmeticon theclassesin � e{¾ via ¤ , ¥ 3 ¤ - ¥ � ¤ , 3 - ¥ (2.33)T ; ¤ , ¥ � ¤ T ; , ¥ (2.34)¤ , ¥ ; ¤ - ¥ � ¤ , ; - ¥rM (2.35)

Undertheseoperations,� e{¾ becomesanalgebra. For eachÁ � p P � MaMdM � O s , de-fine themap

EKÂfrom � e{¾ to � e{¾ for

,viaEKÂ ¤ , ¥ �¬ÄÃ Â ; E ,E S Â ® � (2.36)


where Ã Â �`S H � MaMdM �2S ¾ �^�QS Â (2.37)

projectsoutthe Á th componentof theidentityfunction.Notethat + thorderderiva-tivesof Ã Â ; � E , < E S Â � involve the productof the valueof Ã Â at the origin with��+ 37P � st orderderivativesof

,at the origin. Even thoughwe do not know the

valuesof theselatterderivativesat theorigin,wedoknow theresultsof theirmul-tiplicationswith thevalueof Ã Â at theorigin, which is i . It is easyto show that,for all Á � P � MdMaM � O andfor all ¤ , ¥ � ¤ - ¥ � � e{¾ , wehave:E Â � ¤ , ¥ 3 ¤ - ¥ �^� E Â ¤ , ¥ 3 E Â ¤ - ¥ (2.38)E Â � ¤ , ¥ ; ¤ - ¥ �^� ¤ , ¥ ; � E Â ¤ - ¥ � 3 � E Â ¤ , ¥ � ; ¤ - ¥rM (2.39)

Therefore,E Â

is a derivation for all Á , andhence � � e ¾ � E H � MaMdM � E ¾ � is a differ -ential algebra. Indeed,� eN¾ is a generalizationof H e H � which is obtainedasthespecialcase+�� P and O � P M

2.2.3 Generators, Bases, and Order

Similar to the caseof H e H � the differential algebra � e{¾ also containsthe realnumbers;in fact, by mappingthe real number ] to the class ¤ ] ¥ of the constantfunctionthatassumesthevalue ] everywhere,we seethatthearithmeticon � e{¾correspondsto thearithmeticon �NM

Next, we wantto assessthedimensionof � eN¾ . We definethespecialnumbersR Âasfollows: R Â � ¤ S Â ¥nM (2.40)

We observe that,

lies in the sameclassasits Taylor polynomial � � of order +aroundtheorigin; they have thesamefunctionvaluesandderivativesup to order+ , andnothingelsematters.Therefore,¤ , ¥ � ¤ � � ¥bM (2.41)

DenotingtheTaylorcoefficientsof theTaylorpolynomial � � of,

as ÅcÆ.Ç�ÈÄÉÄÉÄÉ È ÆnÊ , wehave � � ��S H � MaMaM �.S ¾ �^� ËÆ Ç.Ì�ÍÄÍÄÍ Ì Æ Ê�Î � Å Æ Ç ÈÄÉÄÉÄÉ È Æ Ê ; S Æ ÇH ;b;a;d;h; S Æ Ê¾ � (2.42)

with Å Æ Ç ÈÄÉÄÉÄÉ È Æ Ê � PÏ H�Ð ;U;a;d;h; Ï ¾ Ð ; E Æ Ç�Ì�ÍÄÍÄÍ Ì Æ Ê ,E S Æ�ÇH ;U;a;a;U; E S ÆÑÊ¾ (2.43)


andthus ¤ , ¥ � ¤ � � ¥ �ÓÒÔ ËÆ Ç.Ì�ÍÄÍÄÍ Ì Æ Ê�Î � ÅcÆ�Ç�ÈÄÉÄÉÄÉ È ÆnÊ ; S Æ.ÇH ;U;d;a;h; S ÆÑÊ¾�ÕÖ� ËÆ Ç.Ì�ÍÄÍÄÍ Ì Æ Ê�Î � ÅcÆ.Ç�ÈÄÉÄÉÄÉ È ÆnÊ ; R Æ ÇH ;b;a;d;h; R Æ Ê¾ � (2.44)

where,in thelaststep,usehasbeenmadeof ¤ � 3 � ¥ � ¤ � ¥ 3 ¤ � ¥ and ¤ � ; � ¥ � ¤ � ¥ ; ¤ � ¥rMTherefore,the set p P � R Â � Á � P � � � MdMaM � O s generates� e ¾ ; that is, any elementof � e ¾ can be obtainedfrom

Pand the

R Â’s via addition and multiplication.

Therefore,asan algebra,� e{¾ has � O 3_P � generators. Furthermore,the termsR Æ ÇH ;K;a;d;\; R Æ Ê¾ for i�× Ï H 3 ;d;a; 3 Ï ¾ ×v+ form a generatingsystemfor thevec-tor space� e{¾ M They arein fact alsolinearly independentsincethe only way toproducethezeroclassby summingup linearcombinationsof

R Æ�ÇH ;\;a;d;U; R ÆÑÊ¾ is bychoosingall theircoefficientsto bezero;sothe

R Æ�ÇH ;/;a;d;/; R ÆÑÊ¾ actuallyform abasisof � e ¾ , calledthediffer ential basis, for reasonsthatwill becomeclearlater. Wenow want to studyhow many basiselementsexist andthusfind the dimensionof � e ¾ . To this end,we representeachO -tupleof numbers

Ï H � MaMdM � Ï ¾ thatsatisfyi{× Ï H 3 ;a;d; 3 Ï ¾ ×Ø+ by thefollowing uniquesequenceof onesandzeros:� P � P � MdMaM � PÙ Ú/Û ÜÆ Ç times

�2i�� P � P � MaMdM � PÙ Ú[Û ÜÆnÝ times

�ci�� MdMaM �ci�� P � P � MdMaM � PÙ Ú/Û ÜÆ Ê times

�2i�� P � P � MdMaM � P � MÙ Ú[Û Ü� F Æ Ç F ÍÄÍÄÍ F Æ Ê times

(2.45)

Apparently, thevalueP

appearsatotalof + times,andthevalue i appearsO times,for a total lengthof + 3 O�M Thetotal numberof suchvectors,which is thenequalto the numberof basisvectors,is given by the numberof ways that i can bedistributedin the + 3 O slots;this numberis givenbyÞ ��+�� O � jdlnm� dim � e ¾ � | + 3 OO ~ � �`+ 3 O � Ð+ Ð O Ð M (2.46)

For later use,we alsonotethat a similar argumentshows that the basisvectorsbelongingto thepreciseorder + canbedescribedby thesequence� P � P � MaMdM � PÙ Ú[Û ÜÆ�Ç times

�ci�� P � P � MdMaM � PÙ Ú/Û ÜÆ Ý times

�2i�� MaMdM �2i�� P � P � MaMaM � PÙ Ú[Û ÜÆÑÊ times

�[� (2.47)

which haslength + 3 O � P and O � P zeros,yielding thefollowing totalnumberof possibilities: ß ��+�� O �^� | + 3 O � PO � P ~ � Þ �`+�� O � P � M (2.48)


TABLE ITHE DIMENSION OF àbá°â AS A FUNCTION OF ã AND ä , FOR ã�å`ä�æ�ç/å2è�è�è.å.ç.é1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10 11

2 3 6 10 15 21 28 36 45 55 66

3 4 10 20 35 56 84 120 165 220 286

4 5 15 35 70 126 210 330 495 715 1,001

5 6 21 56 126 252 462 792 1,287 2,002 3,003

6 7 28 84 210 462 924 1,716 3,003 5,005 8,008

7 8 36 120 330 792 1,716 3,432 6,435 11,440 19,448

8 9 45 165 495 1,287 3,003 6,435 12,870 24,310 43,758

9 10 55 220 715 2,002 5,005 11,440 24,310 48,620 92,378

10 11 66 286 1,001 3,003 8,008 19,448 43,758 92,378 184,756

Sincethenumberof termsup to order + equalsthenumberof termsup to order+ � P plusthenumberof termsof exactorder +�� we haveÞ ��+�� O �^� Þ �`+ � P � O � 3 Þ ��+�� O � P �[� (2.49)

whichalsofollowsdirectly from thepropertiesof binomialcoefficients.Table I gives the dimensionof � e ¾ as a function of + and O , for +�� O �P � MaMdM � P i . Note that the table clearly shows the recursive relation (Eq. 2.49).

Moreover, the table is symmetricabout the diagonal,which is a direct resultof the fact that the dimensionof � e ¾ , Þ �`+�� O � , is symmetricin + and O , i.e.,Þ �`+�� O �^� Þ � O �2+V� M

Since� e{¾ is finite dimensional,it cannotbeafield.Otherwise,thiswouldcon-tradictZermelo’s Theorem,which assertsthat theonly finite dimensionalvectorspacesover � which arefields are � andthe quaternions(in which multiplica-tion is not commutative). This will be confirmedagainlater by the existenceofnilpotentelements(theinfinitesimals),which canhaveno multiplicative inverse.

Similar to the structureH e H È � eN¾ canalsobe ordered(Berz 1990b).Given aDA number, we look for thetermsthathavethelowestsumof exponentsof the

R thatoccurs.Fromall combinationsof theseexponents,we find theoneswith thehighestexponentin

R H , andfrom thesetheoneswith thehighestexponentofR W ,

andsoon.Theresultingcombinationis calledtheleadingterm andits coefficientthe leading coefficient. For example,if a DA vectorhasa constantpart,thentheconstantpart is the leadingterm; if theconstantpartvanishesbut thecoefficientofR H is nonzero,then

R H will betheleadingterm.We now saytheDA numberispositive if its leadingcoefficient is positive.


We concludethat if � and � arepositive, thensoareboth � 3 � and � ; � . Theleadingtermof � 3 � is eithertheleadingtermof � or theleadingtermof � , andif theleadingtermsareequal,theleadingcoefficientscannotaddup to zerosincethey arebothpositive.Theleadingtermof � ; � is theproductof theleadingtermof � andthe leadingterm of � , andthe leadingcoefficient is the productof theleadingcoefficientsof � and � .

For � z�ê� in � e{¾ , we sayë �g�D� if � � � is positive�g�D� otherwiseM (2.50)

Then,for any � , � in � e{¾ , exactly oneof each ��ì� , �í�C� and ��C� holds.Moreover, for any �� Å in � e{¾ , �� and �� Å entailsthat �� Å . Therefore,theorderis total; alternatively, we saythat � e{¾ is totally ordered.Moreover, theorderis compatiblewith additionandmultiplication,i.e., for all ��c�� Å in � e{¾ wehave that ë �g�� I�î� 3 Å �Ø� 3 Å�g�� Å �Øi �I�î� ; Å �� ; Å M (2.51)

We call theorderlexicographic becausewhencomparingtwo numbers� and� , we look at the leadingcoefficient of � � � . This is equivalent to startingbycomparingthe leftmost termsof � and � , and thenmoving right until we havefounda termin which thetwo numbersdisagree.We alsointroducetheabsolutevaluevia ï � ï � ë � if �gðØi� � else

M (2.52)

Note that theabsolutevaluemapsfrom � e ¾ into itself, andnot into � asin thenormintroducedlater.

Observethatfor all ]N�ñi in � , andfor all Á in p P � MaMdM � O s ,i�� R Â �¿] M (2.53)

Therefore,allR Â

areinfinitely smallor infinitesimal.Now we canorderthebasisdefinedin Eq. (2.44)asfollows:

ß H � P � ß W � R H �ß�ò � R W � ;d;a; �

ß ¾ Ì H � R ¾ � ;a;a;� ßíó FLH � ßíó � R �¾£ô (2.54)

andan arbitraryelement�� ¤ , ¥ � � e{¾ cannow be representedasan orderedÞ-tuple �{�7� Å H � Å W � MaMaM � Å

ó �/� (2.55)


wherethe Å ’saretheexpansioncoefficientsof � in thebasisp ß H � ß W � MdMaM �ß ó s

andaregivenby Eq. (2.43).Addition andmultiplicationrules(2.33),(2.34)and(2.35) in � e ¾ canbe reformulatedas follows. For ��õ�`� H �2� W � MaMaM �2�

ó � , �� H �c� W � MdMaM ��ó � in � e ¾ , andfor

Tin � , wehave:� 3 �w�_�`� H 3 � H �c� W 3 � W � MdMaM �c�

ó 3 � ó � (2.56)T ; �N�_� T ; � H � T ; � W � MdMaM � T ; �ó � (2.57)� ; �w�_� Å H � Å W � MdMaM � Å

ó �/� (2.58)

where Å Â � Ëö@÷ Í ö�ø.ùGö#ú �u ; � ÆUM (2.59)

It follows that an element� of � e{¾ is an infinitesimal if andonly if its firstcomponent� H � the so-calledreal part , vanishes.To concludethe section,wealsoobserve that the total number &y��+�� O � of multiplicationsof the form � ; � necessaryto computeall coefficients Å Â of theproductis givenby&y��+�� O �^� Þ ��+�� O �^� | + 3 � O� O ~ � ��+ 3 � O � Ð+ Ð � � O � Ð � (2.60)

sincethenumberof eachcombination

ß ; ß Æ of monomialsin � e{¾ in Eq.(2.59)correspondsuniquelyto onemonomialin � e W ¾ M

2.2.4 The Tower of Ideals, Nilpotency, and Fixed Points

To any element¤ , ¥ � � e ¾ , wedefinethedepth û � ¤ , ¥ � asû � ¤ , ¥ �^� ë Orderof first nonvanishingderivativeof, ¤ , ¥ z�:i+ 3QP ¤ , ¥ �:i M (2.61)

In particular, any function,

thatdoesnot vanishat theorigin has û � ¤ , ¥ �w��i M Ina similar way, on theset � egü¾ thatdescribesvectorfunctions

�, �� , H � MdMaM � , ü �from � ¾ to � ü � wedefineû �2� ¤ , H ¥ � MaMdM � ¤ , ü ¥ �2�� ý ²!³H Î Â Î ü û � ¤ , Â ¥ � (2.62)

We observethatfor any ��#� � e ¾ , wehaveû � � ; �[�^��ý ²´³ � û � �u� 3 û � �[�[�.+ 3QP �and û � � 3 �[��ð�ý ²´³ � û � �u�[� û � �[�2� M (2.63)


In fact,exceptthat û never exceeds+ 3þP � it behaveslike a valuation. We nowdefinethesetsÿ Â as ÿ Â � p �y� � e{¾ ï û � �K�0ð Á s M (2.64)

We apparentlyhavetherelation� e ¾ � ÿ Z�� ÿ H � MaMdM � ÿ � � ÿ � Ì H � p iKs M (2.65)

All ÿ Â ’s areideals, i.e., for �� ÿ Â andany �� e ¾ � we have � ; �� ÿ Â � whichreadilyfollows from Eq.(2.63).Becauseof Eq. (2.65),the ÿ Â ’s areoftenreferredto asthe“tower of ideals.” TheIdeal ÿ H apparentlyis just thecollectionof thoseelementsthatareinfinitely smallor zeroin absolutevalue.An interestingpropertyof theelementsin ÿ H (andhencealsothosein ÿ W � ÿ

ò � MaMaM ) is thatthey arenilpotent.Following themultiplication rulesof thedepth,we seethat if �� ÿ H andhenceû �`�K�0ð P � û � � � Ì H�� ðþ�`+ 3QP � ; û � �u�wðñ+ 3DP � (2.66)

andsoindeed � � Ì H �:i M (2.67)

In particular, all differentialgeneratorsR Â

arenilpotent.On the otherhand,allelementsin � e{¾�� ÿ H arenotnilpotent,but theirpowersactuallystayin � e{¾�� ÿ H ôindeed,they all have û � �K��Di�� whichalsoentailsthat û � � Â �^�Qi M

We now introducea norm of thevector � in � e{¾ M Let � have thecomponents�`� H � MdMaM �c� ó � in thedifferentialbasis;thenwesetï´ï � ï´ï � Þ ��+�� O � ; ý��¶ ù H ÈÄÉÄÉÄÉ È ó ï � ï M (2.68)

We seethat ï´ï � ï´ï �Di if andonly if �{�Qiï´ï T ; � ï´ï � ï T ï ; ï!ï � ï!ï for all realT M

By considering

ï �u 3 �[ ï × ï �u ï 3 ï �/ ï � we seethatï´ï � 3 � ï!ï × ï!ï � ï!ï 3 ï´ï � ï!ï (2.69)

andhence

ï´ï\ï´ïis indeedanorm.Let Å �v� Å H � MaMaM � Å ó ��:� ; � M Fromthemultipli-

cationrule,werememberthateachof the Å ’s is madeby summingoverproductsof � and � M Sincetherecannotbe more contributionsto the sum as

Þ ��+�� O �/�


we have Å × Þ ��+�� O � ; ý��¶w� ; ý��¶w� . This entailsthatÞ ��+�� O � ; ý��¶ ï Å ï ×Þ �`+�� O � W ý��¶I� �\n� ; ý�h¶£�r�/r� andhence

ï´ï � ; � ï´ï × ï´ï � ï´ï ; ï!ï � ï´ï � (2.70)

sothenormsatisfiesa Schwartz inequality.Weproceedto thestudyof operatorsandproveapowerful fixed-pointtheorem,

which will simplify many of thesubsequentarguments.Let beanoperatorontheset

ß » � egü¾ M Thenwe saythat is contracting on

ßif for any

�� w� ßwith

�� z� ��A� û � � �� a�.�� û � �� [� (2.71)

Therefore,in practicaltermsthismeansthatafterapplicationof � thederivativesin�� and

�� agreeto ahigherorderthanthey did beforeapplicationof gMFor example,considertheset

ß � ÿ H andtheoperator thatsatisfies� �`�K�� WhM Then, �`�K� � �r�[��:� W � � W �7� � 3 �[� ; �`� � �[�[� andthus û � � �K� � � �[�2�°�ý ²!³ � û �`� 3 �a� 3 û � � � �[�[�.+ 3�P �� û � � � �[� , since� 3 �w� ÿ H andso û �`� 3 �a��ñi MWe alsoseethat if we have two operators H and W that arecontractingon

ß, thentheirsumis contracting on

ßbecause,accordingto Eq. (2.63),û �.� H � �� 3 W � ��K�2� � � H � ��/� 3 W � ��K�.�2�ðØý ²!³ � û � H � ��K� � H � ��[�2�/� û � W � �� W � ��a�.�.��Øý ²!³ � û � �� /�[� û � �� [�.�^� û � �� /� M (2.72)

Furthermore,if theset

ßon which theoperatorsact is a subsetof ÿ H , thentheir

productis alsocontracting.Another importantcaseof a contractingoperatoris the antiderivation

E FLHÂ MIndeed,assumeû � �� [�� thenthefirst nonvanishingderivativesof

�� areof order � M However, afterintegrationwith respectto any variableÁ � thederivativesof�� areof order � 3DP � andthus,usingthelinearity of theantiderivation,û � E FLHÂ �� E FIHÂ ��/�� û � E FLHÂ � �� /�2�0� û � �� /� (2.73)

andthustheantiderivationoperatoris contracting.This will proveeminentlyim-portantin thestudyof differentialequationsfollowing.

Contractingoperatorshave a very importantproperty—they satisfy a fixed-point theorem. Let be a contractingoperatoron

ß » � e ¾ that maps

ßinto

ß M Then hasa unique fixed point �� ß that satisfiesthe fixed-pointproblem �� `�K� M (2.74)

Moreover, let �\Z beany elementin

ß M Thenthesequence� Â � �`� Â FIH � for Á � P � � � MdMaM (2.75)


convergesin finitely many steps [in fact, at most �`+ 3CP � steps]to the fixedpoint � . Thefixed-pointtheoremis of greatpracticalusefulnesssinceit ensurestheexistenceof asolutionand,moreover, allows its exactdeterminationin averysimpleway in finitely many steps.

The proof of the fixed-pointtheoremis rathersimple.Considerthe sequence� Â Ì H � �`� Â � M We have thatû � � Â Ì W � � Â Ì H �� û � � � � Â Ì H � � � �`� Â �2�� û �`� Â Ì H � � Â � MBecauseû assumesonly integer values,we thuseven have û � � Â Ì W � � Â Ì H �yðû �`� Â Ì H � � Â � 3QP � andhenceby inductionit follows thatû � � � Ì W � � � Ì H ��ð û �`� � Ì H � � � � 3QP ð MaMdMð û �`� H � �bZ�� 3 ��+ 3DP ��ðñ+ 3DP MBecauseof the definition (Eq. 2.61)of û , this entailsthat � � Ì W �C� � Ì H � or be-cause� � Ì W � � � � Ì H �/� � � Ì H � �`� � Ì H � MTherefore,the fixed point hasindeedbeenreachedexactly after �`+ 3_P � steps.The fixed point is alsounique; if both � and �� weredistinct fixed points,thenû �`� � ��d�� û � �`�K� � �`��A�.�0� û � � � ��a�[� which is a contradiction.

The fixed-point theorem,and also its proof, hasmany similarities to its fa-mous counterpartdue to Banachon Cauchy-completenormedspaces.In Ba-nach’s version,contractivity is measurednot discretelyvia û but continuouslyvia the norm,andit is sufficient that thereexists a real Y with i�× Y�� P with

ï � � H � � �`� W � ï �ìY ; ï � H � � W ï M Thenthe resultingsequenceis shown to beCauchy, andits limit is thesolution.In theDA versionof thefixed-pointtheorem,convergencehappensveryconvenientlyafterfinitely many steps.

Similar to thecaseof theBanachfixed-pointtheorem,theDA fixed-pointtheo-remalsohasmany usefulapplications.In particular, it allowsfor aratherstraight-forward solutionof ODEsandPDEsby utilizing operatorsthat containderiva-tions,aswill beshown later.

For example,we want to utilize thefixed-pointtheoremto prove thequotientrule of differentiation:Given a function

,andits partial derivativesof order +��

that satisfies, �ì\� z�9i M Thenwe want to determineall the partialderivativesof

order + of the functionP < , M In DA terminology, this is equivalentto finding a

multiplicative inverseof theelement� � ¤ , ¥ � ÿ Z » � e ¾ M Let � Z � , �ìb� M Thenwe have û �`� � � Z �#�:i M Let �#� ¤ P < , ¥ � andasdemanded� ; �t� P M This canberewrittenas �`� � � Z � ; � 3 � Z ; �� P or�� P� Z ;hp P � �`� � �bZ�� ; �As M


Definingtheoperator by �r�[� � p P � �`� � �\ZA� ; �As�<h�\Z�� we havea fixed-pointproblem.Let Å H z� Å W � � e ¾ M Thenwehaveû � � Å H � � � Å W �.�^� û | � � � Z�\Z ; � Å H � Å W � ~�Qý ²!³ | û | � � �\Z�\Z ~ 3 û � Å H � Å W �/�2+ 3QP ~� û � Å H � Å W �/�andhencethe operatoris contracting.Thus,a uniquefixed point exists, and itcanbereachedin + 3�P stepsfrom any startingvalue.Thismethodhasparticularadvantagesfor thequestionof efficientcomputation of multiplicative inverses.

In a similar but somewhat more involved example,we derive the root rule,which allows the determinationof the derivativesof � , from thoseof

, M Wedemand

, � ib�#�êi M Let � � ¤ , ¥ �b�t� ¤ � , ¥nM Then � W �7� M Setting � Z � , � ib� andwriting ��ì� Z 3�� and �g� � � Z 3 �� we have û � ��y�9i and û � ��[�g�9i M Thecondition � W �:� canthenberewrittenas��w� �� W� � � Z � � ��a� M (2.76)

Theoperator is contractingon theset

ß � p �� ï û � ��d��7iKs ô if Å H and Å W arein

ß � û � � Å H � � � Å W �2�� û | P� � �\Z � Å WW � Å W H � ~ � û � Å W � Å H � 3 û � Å W 3 Å H ��ý ²´³ � û � Å W � Å H �[�.+ 3DP � M (2.77)

Hence,the operatoris contracting,andthe root canbe obtainedby iteration infinitely many steps.Again, themethodis usefulfor thepracticalcomputationofroots.

2.3 ADVANCED METHODS

2.3.1 Composition and Inversion

Let us considera function - on � ¾ in O variablesthat is at least + timesdiffer-entiable.Given thesetof derivatives ¤ � ¥ � up to order + of a function � from� ¾ to � ¾ � we want to determinethe derivativesup to order + of the composi-tion -�� Q-£� � � , andhencetheclass ¤ -�� ¥ � M Accordingto thechainrule,the + th derivativesof -��#� at the origin canbe calculatedfrom the respectivederivativesof � at theorigin andthederivativesof upto order + of - at thepoint

ADVANCED METHODS 101�� i\� M This impliesthatthemereknowledgeof ¤ - ¥ � is sufficient for this purposeif �� i\�@� �i ô on theotherhand,if this is not thecase,theknowledgeof ¤ - ¥ � isnotsufficientbecausethisdoesnotprovideinformationaboutthederivativesof -at �� iU� M

Sincethederivativesof - areonly requiredto order+�� for thepracticalpurposesof calculatingtheclass ¤ -��w� ¥ � it is actuallypossibleto replace- by its Taylorpolynomial �� around�� ib� since�� and - havethesamederivatives;therefore,¤ -�� ¥ � � ¤ �� ¥ � M (2.78)

However, sincethepolynomialevaluationrequiresonly additionsandmultiplica-tions,andbothadditionsandmultiplicationscommutewith thebracketoperationby virtue of Eqs.(2.33)and(2.34),i.e., ¤ � 3 � ¥ � ¤ � ¥ 3 ¤ � ¥ and ¤ � ; � ¥ � ¤ � ¥ ; ¤ � ¥ �we have ¤ -�� ¥ � � �� ¤ � ¥ � � M (2.79)

Thisgreatlysimplifiesthepracticalcomputation of thecompositionsinceit nowmerelyrequirestheevaluationof theTaylorpolynomialof - ontheclasses¤ � ¥ � �i.e.,purelyDA arithmetic.

It now alsofollows readilyhow to handlecompositionsof multidimensionalfunctions;replacing- by the function � mappingfrom � ¾ to � ¾ , anddenotingtheTaylor polynomialof � around�� i\� by

�� we have¤ � �� ¥ � � �� ¤ � ¥ � � M (2.80)

A particularlyimportantcaseis thesituationin which �� i\�^� �i ô in thiscase,theTaylor polynomial

�� of � around �� iU� is completelyspecifiedby the class¤ �Ø¥ � M This entailsthatwe candefinea compositionoperationin the ideal ÿ ¾H »� e ¾¾ via ¤ �Ø¥ � � ¤ � ¥ � jalnm� ¤ � �� ¥ � M (2.81)

Now we addressthequestionof thedeterminationof derivativesof inversefunc-tions.Let usassumewe aregivena map � from � ¾ to � ¾ with �� iU�� i thatis + timesdifferentiable,and let us assumethat its linearization

ßaroundthe

origin is invertible.Then,accordingto the implicit function theorem,thereis aneighborhoodaround

�i wherethefunctionis invertible,andsothereis a function� FLHsuchthat �� FIH �� M Moreover, � FLH

is differentiableto order + .Thegoal is to find the derivativesof � FIH

from thoseof � in anautomatedway. In theDA representation,this seeminglydifficult problemcanbesolvedbyanelegantandclosedalgorithm.


Webegin by splitting themap � into its linearpart

ßandits purelynonlinear

part � � sothatwehave � � ß 3 ��M (2.82)

Composingtheinverse� FIHwith � , weobtain��°� FIH ��ß �°� FIH �� FLH� FIH � ß FIH �t� � � � �� FIH � M (2.83)

The resultingequationapparentlyis a fixed-point problem. Moving to equiva-lenceclasses,weseethattheright-handsideis contracting,since! ß FLH �t� ¤ � ¥ � � ¤ �Ø¥ � � ¤ "X¥ � �$# � ! ß FIH �t� ¤ � ¥ � � ¤ �Ø¥ � � ¤ %�¥ � �&#� ß FLH �t� ¤ �Ø¥ � � ¤ %�¥ � � ¤ �Ø¥ � � ¤ "�¥ � � M (2.84)

However, notethat ¤ �Ø¥ � � ÿ ¾W , andso for any ¤ "X¥ � and ¤ %°¥ � � ÿ ¾H � if ¤ "X¥ � and¤ %�¥ � agreeto order Á � then ¤ �Ø¥ � � ¤ "X¥ � and ¤ �Ø¥ � � ¤ %°¥ � agreeto order Á 3_Pbecauseevery termin ¤ "X¥ and ¤ %°¥ is multiplied by at leastonenilpotentelement(namely, othercomponentsfrom ¤ "X¥ and ¤ %°¥ � M Accordingto thefixed-pointtheo-rem,this ensurestheexistenceof a unique fixed point, which canbereachedatmost + iterativesteps.Therefore,wehavedevelopedanalgorithmto determineallderivativesof upto order + of � FLH

in finitely many stepsthroughmereiteration.Becauseit requiresonly iterationof a relatively simpleoperator, it is particularlyusefulfor computation.

2.3.2 Important Elementary Functions

In this section,we introduceimportantfunctionson the DA � e{¾ M In particular,this includesthefunctionstypically availableintrinsically in a computerenviron-ment,suchas µa¶K· , ±2²!³ , and ' º)( . As discussedin Section2.3.1,for any function -thatis at least+ timesdifferentiable,thechainrule in principleallowsthecompu-tationof thederivativesof order + of -�� , attheorigin,andhence¤ -�� , ¥ � from themereknowledgeof thederivativesof

, � andhence¤ , ¥ � aswell asthederivativesof - at

, � �iU� M This allowsusto define-L� ¤ , ¥ � jalnm� ¤ -L� , � ¥nM (2.85)

ADVANCED METHODS 103

In particular, wethuscanobtaintheexponential,thetrigonometricfunctions,anda varietyof otherfunctionsasµ[¶K·V� ¤ , ¥ � jalnm� ¤ µa¶u·G� , � ¥ � ' º)( � ¤ , ¥ � jalnm� ¤ ' º)( � , � ¥ � (2.86)±.²´³ � ¤ , ¥ � jalnm� ¤ ±2²!³ � , � ¥ � ¹aº ± � ¤ , ¥ � jdlnm� ¤ ¹[º ± � , � ¥ � (2.87)� ¤ , ¥ jalnm� ¤ � , ¥ for

, � ib� positive, (2.88)µ�* ¹ MWhile this methodformally settlesthe issueof the introductionof important

functionson � e ¾ in a rathercompleteandstraightforwardway, it is not usefulfor computation of thepreviousfunctionsin practicein anefficient andgeneralway. In thefollowing, we illustratehow thiscanbeachieved.

We observe thatby thevery natureof thedefinitionof theimportantfunctionsvia Eq.(2.85),many of theoriginal propertiesof thefunctionstranslateto � e{¾ MFor example,functionalrelationshipssuchasaddition theoremscontinueto holdin � e{¾ . To illustratethis fact,weshow theadditiontheoremfor thesin function:±.²´³ � ¤ , ¥ 3 ¤ - ¥ �^� ¤ ±.²´³ � , 3 -K� ¥ � ¤ ±2²!³ � , � ¹aº ± ��-K� 3 ¹aº ± � , � ±2²!³ ��-K� ¥� ¤ ±.²´³ � , � ¹[º ± �x-�� ¥ 3 ¤ ¹[º ± � , � ±.²´³ ��-K� ¥� ¤ ±.²´³ � , � ¥Ñ¤ ¹aº ± �x-�� ¥ 3 ¤ ¹aº ± � , � ¥Ñ¤ ±.²´³ ��-K� ¥� ±2²!³ � ¤ , ¥ � ¹aº ± � ¤ - ¥ � 3 ¹[º ± � ¤ , ¥ � ±.²´³ � ¤ - ¥ �[� (2.89)

wherewehavemadeuseof thedefinitionsof thefunctionson � e ¾ aswell asthefact that ¤ � * � ¥ � ¤ � ¥ * ¤ � ¥ for all elementaryoperationssuchas

3 � � , and ' .Obviously, otherfunctionalrulescanbederivedin averyanalogousway.

Theadditiontheoremsfor elementaryfunctionsallow usto deriveconvenientcomputational tools using only elementaryoperationson � e{¾ M For this pur-pose,one rewrites the expressionin sucha way that elementaryfunctionsactonly aroundpointsfor which thecoefficientsof their Taylor expansionareread-ily available.As discussedin Section2.3.1,accordingto Eq. (2.81) this allowssubstitutionof thefunctionitself by its + th orderTaylor polynomial,whichcom-putationallyrequiresonly additionsandmultiplications.For example,considertheelementaryfunction -{� ±.²´³ . Wesplit theargumentfunction

,in theclass¤ , ¥

into two parts,its constantpart � Z � , � ib� andits differential(nilpotent)part�,,

sothat ¤ , ¥ � ¤ � Z ¥ 3 ¤ �, ¥nM (2.90)


We thenconclude±2²´³ � ¤ , ¥ �^� ±2²!³ � ¤ � Z ¥ 3 ¤ �, ¥ �^� ±2²´³ � ¤ � Z ¥ � ¹[º ± � ¤ �, ¥ � 3 ¹aº ± � ¤ � Z ¥ � ±.²´³ � ¤ �, ¥ �� ¤ ±.²´³ �`� Z � ¥ Â � � ¡Ë ù Z � � P � ¤ �, ¥fW � � o � Ð 3 ¤ ¹[º ± �`� Z � ¥ Â � � ¡Ë ù Z � � P � ¤ �, ¥xW Ì H� � o 3QP � Ð � (2.91)

where Á ��+V� is thelargestintegerthatdoesnotexceed+G< � MThestrategy for otherfunctionsis similar; for example,we useµ[¶K·V� �\Z 3 �a�^�Dµa¶u·I� �\Z�� ; µ[¶K·V� �a� (2.92)� � Z 3 �w�ê� � Z ;,+ P°3 ��\Z (for � Z �ñi\� (2.93)' º)( � �\Z 3 �a�^� ' º)( �`�\ZA� 3 ' º)( | P°3 �� Z ~ (for �\Z��Øi\� (2.94)P�\Z 3 � � P�\Z ; PP°3 �a<h�bZ (for � Z �Øi\� (2.95)

andtherespectiveTaylorexpansionsaroundi orP.

It is now possibleto obtain derivatives of any order of a functional depen-dency that is given in termsof elementaryoperationsandelementaryfunctionsby merelyevaluatingin theDA framework andobserving¤ , ��S H � MdMaM �2S ¾ � ¥ � , � ¤ S H ¥ � MaMdM � ¤ S OU¥ �� , � R H � MdMaM � R ¾ � M (2.96)

2.3.3 Power Series on � e{¾We saythatasequence�`� Â � convergesin � e{¾ if thereexistsanumber� in � e{¾suchthat �.-[� Â � �/-d� convergesto zeroin � . Thatis,given 0 �Øi in � , thereexistsapositive integer 1 suchthat -[� Â � �/-#� 0 for all Á � 1 .

With thepreviouslydefinednorm, � e ¾ is aBanachspace, i.e.,anormedspacewhich is Cauchycomplete.A sequence� � Â � in � e ¾ is saidto be Cauchyif foreach0 ��i in � thereexistsapositiveinteger 1 suchthat -[� Â � �324-@� 0 wheneverÁ and � areboth ð 1 . For theproof of theCauchycompletenessof � e ¾ , weneedto show thateveryCauchysequencein � e{¾ convergesin � e{¾ .

Let �`� Â � beaCauchysequencein � e{¾ andlet 0 �ñi in � begiven.Thenthereexistsa positive integer 1 ��i suchthat-[� Â � � 2 -t� 0 for all Á �5�Gð 1�M (2.97)

Accordingto thedefinitionof thenorm,it follows from Eq.(2.97)thatï � Â È � � 2 È ï � 0 for all Á �.��ð 1 andfor all o � P � � � MaMdM � Þ � (2.98)


where � Â È and � 2 È denotethe o th componentsof � Â and � 2 , respectively. Thus,the sequence� � Â È r� is a Cauchysequencein � for each o . Since � is Cauchycomplete,thesequence�`� Â È n� convergesto somerealnumber�/ for eacho . Let�w�v�r� H �� W � MaMaM �c�

ó � M (2.99)

Weclaimthat � � Â � convergesto � in � e ¾ . Let 0 �ñi in � begiven.Foreacho thereexistsa positive integer Á suchthat

ï � Â È � �/ ï � 0 < Þ for all Á ð Á M Let 1 �ý��¶ p Á � o � P � � � MaMdM � Þ s . Then

ï � Â È � �/ ï � 0 < Þ for all o � P � � � MaMdM � Þ andfor all Á ð 1 . Therefore, -[� Â � �6-t� 0 for all Á ð 1 � (2.100)

andhence�`� Â � convergesto � in � e ¾ asclaimed.One particularly important classof sequencesis that of power series. Letp � Â s �Â ù Z be a sequenceof real numbers.Then 7 �Â ù Z � Â98 Â � 8 � � e{¾ is called

a power seriesin8

(centeredat i ). We saythata power series7 �Â ù Z � Â98 Âcon-

vergesin � eN¾ if thesequenceof partial sums,: ü � 7 ü Â ù Z � Â;8 Â, convergesin� e{¾ .

Now let 7 �Â ù Z � Â<8 Âbe a power seriesin the real number

8 � with real coeffi-cientsandwith real radiusof convergence= . We show that theseriesconvergesin � e{¾ if

ï > � 8 � ï � = , where

> � 8 � denotestherealpartof8. Fromwhatwe have

donesofar, it sufficesto show thatthesequence� : ü � is Cauchyin � e{¾ , where: ü � üËÂ ù Z � Â;8 Â M (2.101)

Let ? ��+ begiven.Write8 � 8 Z 3@�8

, where8 Z is theconstantpartof

8 � andhence

8 � 8 Z � ÿ H M Then,üËÂ ù Z � Â � 8 Z 3A�8 � Â � �ËÂ ù Z � Â � 8 Z 3��8 � Â 3 üËÂ ù � Ì H � Â � 8 Z 3��8 � Â� �ËÂ ù Z � Â � 8 Z 3��8 � Â 3 üËÂ ù � Ì H � Â ÂË ù Z Á Ðo Ð � Á � o � Ð 8 Â F Z �8 � �ËÂ ù Z � Â � 8 Z 3��8 � Â3 üËÂ ù � Ì H � Â ÁtMaMaM � Á � +V� 8 ÂZ �Ë ù Z | �8 8 Z � Á � + � P � Ðo Ð � Á � o � Ð ~ �

(2.102)


whereusehasbeenmadeof thefactthat�8 �Qi for all o �ñ+ . Now let �� ? �Ø+

begiven.Then,: 2 � : ü � 2ËÂ ù ü Ì H � Â ; Á ; MaMdM ; � Á � +V� 8 ÂZ �Ë ù Z �8 8 Z � Á � + � P � Ðo Ð � Á � o � Ð M (2.103)

Therefore,- : 2 � : ü -w�CBBBBB 2ËÂ ù ü Ì H � Â ; ÁtMdMaM � Á � +V� 8 ÂZ �Ë ù Z | �8 8 Z � Á � + � P � Ðo Ð � Á � o � Ð ~ BBBBB× 2ËÂ ù ü Ì H -[� Â ; ÁtMdMaM � Á � +V� 8 ÂZ - ; BBBBB �Ë ù Z �8 8 Z � Á � + � P � Ðo Ð � Á � o � Ð BBBBB×ED 2ËÂ ù ü Ì Hï � Â ; Á#MaMaM � Á � +V� 8 ÂZ ï F ; BBBBB �Ë ù Z �8 8 Z o Ð ��+ 3DP � o � Ð BBBBB M

(2.104)

Since Á ; MdMaM ; � Á � +V��× Á � Ì H ,'´² ýÂ�G � � Á ; MaMaM ; � Á � +V�.� H4H Â × '!² ýÂ&G � � Á � Ì H � H.H Â � P M (2.105)

Therefore,the first factorconvergesto i if

ï 8 Z ï � = . Sincethe secondfactorisfinite, - : 2 � : ü - convergesto i . Thus,thesequence� : ü � is Cauchyin � e{¾ for

ï 8 Z ï � = . Therefore,theseriesconvergesif

ï 8 Z ï � = .Now thatwe have seenthat the power seriesactuallyconverges,the question

arisesasto whethertheDA vectorof thepower seriesappliedto a function,

isthesameasthepower seriesappliedto theDA vector ¤ , ¥ , which we just provedto converge.Of course,if thepowerserieshasonly finitely many terms,theques-tion is settledbecausethe elementaryoperationsof additionandmultiplicationcommutewith the classoperations,andwe have that ¤ , 3 - ¥ � ¤ , ¥ 3 ¤ - ¥ , etc.However, in thecasein which thereare infinitely many operations,commuta-tivity is not ensured. The answerto the questioncanbeobtainedwith the helpof thederivativeconvergencetheorem.

Let �g�D� in � andlet p ,hÂ s �Â ù Z beasequenceof real-valuedfunctionsthataredifferentiableon ¤ ��c� ¥ . Supposethat p , Â s �Â ù Z convergesuniformly on ¤ ��c� ¥ to adifferentiablefunction

,andthat p ,£¢Â s �Â ù Z convergesuniformly on ¤ �� ¥ to some

function - . Then -£�`S��^� , ¢ �`S�� for all S�� ¤ �� ¥ .For the proof, let SD� ¤ ��c� ¥ and 0 � i in � be given.Thereexists a positive

integer 1 suchthatï , ¢Â �`S�� -L��S£� ï � 0 <;I for all Á ð 1 andfor S�� ¤ �� ¥ � (2.106)


from which wehaveï , ¢Â Ç ��S�� , ¢Â Ý ��S£� ï � 0 < � for all S � ¤ ��c� ¥ and Á H � Á W ð 1�M (2.107)

Let Á Z�ð 1 begiven.Thereexists J ÂLK ��i suchthatMMMM ,hÂ K � T � � ,UÂ K ��S£�T � S � , ¢ÂLK �`S�� MMMM � 0 <;Iwhenever T � ¤ ��c� ¥ � T z�DS and

ï T � S ï � J ÂLK M (2.108)

By themeanvaluetheorem,for any integer ? ð 1 andfor anyT z�QS in ¤ ��c� ¥ ,ï � , ü � ,hÂ K �[�`S�� , ü � ,UÂ K �[� T � ï � ï � , ü � ,hÂ K � ¢ � Å � ï ; ï S � T ï (2.109)

for some Å betweenS andT. Using Eq. (2.107)andthe fact that Å � ¤ �� ¥ and? � Á Z�ð 1 , ï � , ü � ,hÂ K � ¢ � Å � ï � 0 < � M (2.110)

CombiningEqs.(2.110)and(2.109),wegetï � , ü � , ÂLK �a��S�� , ü � , ÂLK �a� T � ï � 0 < � ; ï S � T ï � (2.111)

which werewrite asMMMM , ü � T � � , ü ��S��T � S � , ÂLK � T � � , ÂLK ��S��T � S MMMM � 0 < � M (2.112)

Letting ? ON in Eq. (2.112),we obtainthatMMMM , � T � � , ��S£�T � S � , ÂLK � T � � , ÂLK ��S£�T � S MMMM � 0 < � M (2.113)

UsingEqs.(2.106),(2.108),and(2.113),wefinally obtainthatMMMM , � T � � , �`S��T � S � -L��S�� MMMM × MMMM , � T � � , �`S��T � S � , ÂLK � T � � , ÂLK ��S£�T � S MMMM3 MMMM , ÂLK � T � � , ÂLK ��S£�T � S � , ¢ÂLK �`S�� MMMM 3 MM , ¢ÂLK �`S�� -L��S£� MM� 0 < � 3 0 <;I 3 0 <;I�� 0AM (2.114)

SinceEq. (2.114)holdswheneverT � ¤ �� ¥ and i � ï T � S ï � J ÂLK

, we concludethat

,£¢ ��S��°��-L��S£� . Thisfinishestheproof of thetheorem.


We next try to generalizethe previous theoremto higher order derivatives:Let p ,hÂ s �Â ù H beasequenceof real-valuedfunctionsthatare + timesdifferentiableon ¤ ��c� ¥ . Supposethat p ,hÂ s �Â ù H convergesuniformly on ¤ �� ¥ to an + timesdif-

ferentiablefunction,

andthat p , � Â ¡Â s �Â ù H convergesuniformly on ¤ �� ¥ to somefunction - Â , Á � P � � � MaMaM �2+ . Then, - Â �`S��w� , � Â ¡ �`S�� for all S�� ¤ ��c� ¥ andfor allÁ � P � � � MaMdM �.+ .

Theproof of thestatementfollows by inductionon + . It is true for +�� P be-causeit directly follows from theprevioustheorem.Assumeit is truefor +í� Ï ,andshow thatit will betruefor +�� Ï 3�P . Therefore,if p , Â s �Â ù H is asequenceofreal-valuedfunctionsthatare

Ïtimesdifferentiableon ¤ �� ¥ , p , Â s �Â ù H converges

uniformly on ¤ ��c� ¥ to aÏ

timesdifferentiablefunction,

, and p , � Â ¡Â s �Â ù H convergesuniformly on ¤ ��c� ¥ to somefunction - Â , Á � P � � � MdMaM � Ï , then - Â ��S�� , � Â ¡ ��S��for all SØ� ¤ �� ¥ andfor all Á � P � � � MaMaM � Ï . Now let p ,hÂ s �Â ù H bea sequenceofreal-valuedfunctionsthatare � Ï 3êP � timesdifferentiableon ¤ ��c� ¥ . Supposethatp , Â s �Â ù H convergesuniformly on ¤ �� ¥ to a � Ï 3�P � timesdifferentiablefunction,

and that p , � Â ¡Â s �Â ù H convergesuniformly on ¤ �� ¥ to somefunction - Â , Á �P � � � MaMaM � Ï 3�P . Then - Â �`S�� , � Â ¡ �`S�� for all S�� ¤ �� ¥ andfor all Á � P � � � MaMdM � Ïby the inductionhypothesis.Furthermore,applyingthe previous theoremto thesequencep , � Æ ¡Â s �Â ù H , weobtainthat - Æ Ì H �`S�� , � Æ Ì H.¡ ��S£� for all S�� ¤ �� ¥ . There-fore, - Â �`S��0� , � Â ¡ �`S�� for all S�� ¤ ��c� ¥ andfor all Á � P � � � MdMaM � Ï 3:P . Thus,thestatementis truefor +�� Ï 3QP . Therefore,it is truefor all + .

The two previous resultsare readily extendableto sequencesof real-valuedfunctionson � ¾ , in which derivativesareto bereplacedby partialderivatives.

We cannow apply the result to the questionof convergenceof power seriesof DA vectorsto the DA vectorof the power series.As shown previously, thepowerseriesof DA vectorsconvergesin theDA normwithin theclassicalradiusof convergence.This entailsthat all its components,andhencethe derivatives,convergeuniformly to thevaluesof thelimit DA vector. Hence,accordingto theprevioustheorem,this limit containsthederivativesof thelimit function.

2.3.4 ODE and PDE Solvers

Historically, themainimpetusfor thedevelopmentof differentialalgebraictech-niqueswerequestionsof quadratureandthesolutionof ordinarydifferentialequa-tions. In our case,the direct availability of the derivations

E andtheir inversesE FLH allows to deviseefficientnumericalintegratorsof any order.Onesuchmethodis basedon the commonrewriting of the ODE as a fixed-

point problemby useof�� : � ��u�� 3QPSRR ÷ �, � �� T ¢ � RbT ¢ M (2.115)


Utilizing the operationE FLH¾ Ì H for the integral, the problemreadily translatesinto

thedifferentialalgebra� e ¾ Ì H , wherebesidesthe O positionvariables,thereis anadditionalvariable

T ¢.

As shown in Eq.(2.73),theoperationE FLH¾ Ì H is contracting.Sinceû � �, � �� H � T ¢ � � �, � �� W � T ¢ �.�0ð û � �� H � �� W �[� (2.116)

it follows that û � : � �� H � � : � �� W �.�w� û � �� H � �� W �/� (2.117)

andhencetheoperator: on � eN¾ is contracting.Thus,thereexistsauniquefixedpoint, which moreover canbe reachedin at most + 3�P stepsfrom any startingcondition.Thus,iterating�� H � �i (2.118)�� Â Ì H � : � �� Â � for Á � P � � � MdMaM �.+ 3QP (2.119)

yieldstheexactexpansionof�� in the O variablesandthetime

Tto order + M The

resultingpolynomialcanbeusedasan + th ordernumericalintegratorby insertinganexplicit timestep.

It is interestingthat the computational expensefor this integratoris actuallysignificantly lower than that of a conventionalintegrator. In the caseof a con-ventionalintegrator, the right-handsidehasto be evaluatedrepeatedly, and thenumberof theseevaluationsis at leastequalto thetimesteporderandsometimesnearlytwiceasmuch,dependingonthescheme.TheDA integratorrequires+ 3�Piterationsand hence + 3 P evaluationsof the right-handside,but sincein the� Á � P � ststeponly the Á th orderwill bedeterminedexactly, it is sufficient to per-form this steponly atorder Á . Sincethecomputationalexpenseof DA operationsincreasesquickly with order, by farthemostimportantcontributionto timecomesfrom the last step.In typical cases,the last stepconsumesmorethantwo-thirdsof thetotal time, andtheDA integratoris aboutoneorder of magnitude fasterthanaconventionalintegrator.

For ODEs that are simultaneouslyautonomousand origin preserving, i.e.,time independentandadmitting

�� T �k� �i asa solution,anotherevenmoreeffi-cientmethodcanbeapplied.For agivenfunctiononphasespace-#� �� T �£� accord-ing to Eq. (1.7), the time derivative alongthesolutionof any function - is givenby RRbT -t� �� T �°� �, ; �T - 3 EE T -N�VUXW� - MTheoperatorU W� , thevector field of theODE,alsoallowscomputationof higherderivativesvia

R � < RUT � -t�SU � W� - . Thisapproachis well known (ConteanddeBoor


1980)andin factis evensometimesusedin practiceto deriveanalyticallow-orderintegrationformulasfor certainfunctions

�,. Thelimitation is thatunless

�,is very

simple,it is usuallyimpossibleto computetherepeatedactionof U W� analytically,andfor this reasonthis approachhasnot beenvery usefulin practice.However,usingthedifferentialalgebras� e ¾ , andin particularthe

E Âoperatorswhich dis-

tinguishthemfrom ordinaryalgebras,it is possibleto performtherequiredopera-tionseasily. To thisend,onejustevaluates

�,in � e{¾ andusesthe

EKÂs to compute

thehigherderivativesof - .If the Taylor expansionof the solution in time

Tis possible,we have for the

final coordinates��

�� Ë ù H T ; U W�o Ð �w� (2.120)

where� is theidentity function.If - and

,arenotexplicitly timedependent,thetimederivativepartvanishesfor

all iterationsU � W� M Furthermore,if�,is origin preserving,thentheorderthatis lost

by aplain applicationofEuÂ

in thegradientis restoredthroughthemultiplication;for the derivativesup to order + of

, Â ; EKÂ -£� only the derivativesof up to order+ � P ofEKÂ - areimportant.Thismeansthatwith oneevaluationof

�, � thepowersof the operatorUXW� canbe evaluatedto any orderdirectly within DA for any + .Using it for -%�v� Â � thecomponentsof thevector

�� we obtainan integrator ofadjustableorder.

While for generalfunctions,it is not a priori clearwhetherthe seriesin theso-calledpropagatorconverges,within theframework of � e ¾ this is alwaysthecase,aswe will now show. Becauseof the Cauchycompletenessof � e ¾ withrespectto thenormon � eN¾ � it is sufficient to show thatÂ Ì 2Ë ù Â U W�o Ð ¤ - ¥ i for Á YN andfor all � M (2.121)

Usingthedefinitionof thenorm(Eq.2.68),thetriangleinequality(Eq.2.69),andtheSchwarzinequality(Eq.2.70),we first observe thatï!ï UXW� ¤ - ¥ ï!ï � BBBBB ¾ËÂ ù H , Â ; EKÂ ¤ - ¥ BBBBB × ¾ËÂ ù H - , Â ; EKÂ ¤ - ¥ -× ¾ËÂ ù H

ï´ï ,hÂ ï!ï ; ï!ï ¤ - ¥ ï!ï × � ; ï´ï ¤ - ¥ ï!ï � (2.122)


wherethefactor � � 7 � Â ù H ï!ï , Â ï´ï is independentof ¤ - ¥rM Thus,BBBBBÂ Ì 2Ë ù Â U W�o Ð ¤ - ¥ BBBBB ×

Â Ì 2Ë ù Â BBBBB U W�o Ð ¤ - ¥ BBBBB ×Z- ¤ - ¥ - ; Â Ì 2Ë ù Â � o Ð M (2.123)

However, sincethereal-numberexponentialseriesconvergesfor any argument� �thepartialsum 7 Â Ì 2 ù Â � < o Ð tendsto zeroas Á ON for all choicesof � M

This methodis the computationally most advantageousapproach for au-tonomous,origin-preservingODEs.In practice,oneevaluationof theright-handside

�,is required,andeachnew order requiresmerelyonederivation andone

multiplication per dimension.For complicatedfunctions�, � the actualcomputa-

tionalexpensefor eachnew orderis only asmallmultipleof thecostof evaluationof�, � andhencethismethodis mostefficientathighordersandlargetimesteps.In

practicalusein thecodeCOSYINFINITY, stepsizesarefixedandordersadjusteddynamically, typically falling in a rangebetween

�Uänd §Ui M

To conclude,we addressthequestionof thesolution of PDEs. Similar to thefixed point ODE solver, it is alsopossibleto iteratively solve PDEs in finitelymany stepsby rephrasingthem in termsof a fixed-pointproblem.The detailsdependon thePDE at hand,but the key ideais to eliminatedifferentiationwithrespectto one variableby integration.For example,considerthe generalPDE� H E < E S �`� W E < E S�[k� 3 � H E < E\8 �r� W E < E�8 [k� 3 Å H E < E �� Å W E < E �3[��þi�� where � �� and Å arefunctionsof S , 8

, and � M ThePDEis rewrittenas[v�][ ï ^ ù Z 3_P ^ P� W` EE�8 [ ï ^ ù Z� P ^ | � H� H EE S | � W EE S [ ~ 3 Å H� H EE � | Å W EE � [ ~@~a MThe equationis now in fixed-pointform. Now assumethe derivativesof [ andE [°< E\8

with respectto S and � areknown in theplane8 �Qi M Thentheright-hand

side is contracting,and the variousordersin8

canbe iteratively calculatedbymereiteration.

2.3.5 The Levi-Civita Field

We discussin thissectionaveryusefulextensionof DAs, afield thatwasdiscov-eredfirst by Levi-Civita, uponwhich conceptsof analysishave recentlybeende-veloped(Berz1994).Theresultsobtainedareverysimilarto theonesin nonstan-dard analysis; however, thenumbersystemsrequiredherecanbeconstructeddi-rectly anddescribedon acomputer, whereastheonesin nonstandardanalysisareexceedinglylarge,nonconstructive(in thestrict sensethat theaxiomof choiceisusedandin a practicalsense),andrequirequitea machineryof formal logic for


their formulation.For adetailedstudyof thisfield andits calculus,referto ((Berz1994),(1996)).We startwith thestudyof thealgebraicstructureof thefield.

First,weintroduceafamily of specialsubsetsof therationalnumbers.A subset

ßof the rational numbersb is said to be left-finite if below every (rational)

boundthereareonly finitely many elementsof

ß. We denotethe family of all

left-finite subsetsof b by c . Elementsof c satisfythefollowing properties.Let

ß � Þ � c begiven,thenß z� d��I� ß

hasaminimum (2.124)e » ß �I� e � c (2.125)ßgf Þ � c and

ßgh Þ � c (2.126)ß 3 Þ � p S 3 8 ï S � ß � 8 � Þ s�� c¦M (2.127)S�� ß 3 Þ �I�ji only finitely many �`��[�� ßlk Þ

with S �Q� 3 � M(2.128)

We now definethenew setof numbersm :m � p , � b � suchthatsupp� , �w� c sb� (2.129)

where

supp� , �^� p Yk� b � , � Yh� z�DiKs (2.130)

Thus, m is thesetof all real-valuedfunctionson b with left-finite support.Fromnow on,we denoteelementsof m by SV� 8 � MaMdM , andtheir valuesat a givensupportpoint Y�� b by S ¤ Y ¥ � 8 ¤ Y ¥ � MaMdM . Thiswill savesomeconfusionwhenwetalk aboutfunctionson m .

According to Eq. (2.124),supp(S ) hasa minimum if S z�i . We defineforS � m û �`S��°� ë min(supp(S )) if S z�:iN if S �:i M (2.131)

Comparingtwo elementsS and8

of m , wesaySon 8qp � û ��S£�°� û � 8 � M (2.132)S¦½ 8qp � û ��S£�°� û � 8 � and S ¤ û ��S£� ¥ � 8 ¤ û � 8 � ¥rM (2.133)S � 8qp �îS ¤ Y ¥ � 8 ¤ Y ¥ for all Y�×ñ] M (2.134)

Apparently, theorderis lexicographicin thesensethatthefunctionsS and8

arecomparedfrom left to right. At this point, thesedefinitionsmay look somewhat


arbitrary, but after having introducedorderon m , we will seethat û describes“ordersof infinite largenessor smallness”,therelation“ n ” correspondsto agree-mentof orderof magnitude,whereastherelation“ ½ ” correspondsto agreementup to aninfinitely smallrelativeerror.

We defineaddition on m componentwise:For SG� 8 � m , and YN� b ,��S 3 8 � ¤ Y ¥ � S ¤ Y ¥ 3 8 ¤ Y ¥nM (2.135)

Wenotethatthesupportof S 3 8is containedin theunionof thesupportsof S and8

andis thusitself left-finite by Eq. (2.126)and(2.125).Therefore,S 3 8 � m ,andthus m is closedundertheadditiondefinedpreviously.

Multiplication is definedasfollows:For SV� 8 � m , and Y�� b ,��S ; 8 � ¤ Y ¥ � Ërts È rvu;wyx Èrts Ì rvu ù r S ¤ Y&z ¥ ; 8 ¤ Y ^ ¥rM (2.136)

Note that supp�`S ; 8 � » supp��S�� 3 supp� 8 � ; henceaccordingto Eq. (2.127),supp��S ; 8 � is left-finite. Thus S ; 8 � m . Note also that left-finitenessof thesupportsof m numbersis essentialfor thedefinitionof multiplication;accordingto Eq. (2.128),for any given Y�� b , only finitely many termscontribute to thesumin thedefinitionof theproduct.

With the addition and multiplication definedpreviously, � m � 3 � ; � is a field.The only nontrivial stepin the proof of the previous statementis the proof oftheexistenceof multiplicative inversesof nonzeroelements.For this,we needtoinvokethefixed-pointtheorem.

To show thatthenew field is anextensionof thefield of realnumbers,defineamap - � � �£ m by -L��S£� ¤ Y ¥ � ë S if Y��Qii otherwise

M (2.137)

Then- “embeds” � into m ( - isone-to-one,-L��S 3 8 ��-£�`S�� 3 -L� 8 � and-L��S ; 8 �^�-L��S�� ; -L� 8 �.� .Regardedasa vectorspaceover � , m is infinite dimensional. With themulti-

plicationdefinedpreviously, m is analgebra.Furthermore,if wedefineanopera-tion

E�� m �� m by � E S�� ¤ Y ¥ � � Y 3QP � ; S ¤ Y 3DP ¥ � (2.138)

thenE

is aderivationand � m � 3 � ; � E � adiffer ential algebrabecauseE

caneasilybeshown to satisfy E �`S 3 8 �^� E S 3 E�8


and E ��S ; 8 �^�v� E S£� ; 8 3 S ; � E�8 �/� for all SV� 8 � m�M (2.139)

We definean orderon m by first introducinga setof positive elements,m Ì .Let S�� m benonzero;then,S�� m Ì p �îS ¤ û ��S£� ¥ �Øi M (2.140)

For S and8

in m , S � 8�p �S � 8 � m ÌS � 8�p � 8 �ØS M (2.141)

Then, the order is total andcompatiblewith additionandmultiplication in theusualway.

Having introducedanorderon m , wedefinethenumberR

asfollows:R ¤ Y ¥ � ` Pif Y�� Pi if Y z� P M (2.142)

Then,accordingto theorderdefinedpreviously,i�� R �] for all positive ]N� ��M (2.143)

Hence,R

is infinitely small.Moreover, it is easyto show thatfor allT � b �R R ¤ Y ¥ � ` P

if Y�� Ti if Y z� T M (2.144)

It followsdirectly from thepreviousequationthatR R is infinitely smallp � T �ØiR R is infinitely largep � T �Øi M (2.145)

Having infinitely small andinfinitely largenumbers,m is nonarchimedean.We introduceanabsolutevalueon m in thenaturalway:ï S ï � ë S if S�ð�i� S if S��i M (2.146)m is Cauchy complete with respectto this absolutevalue, i.e., every Cauchy

sequencein m convergesin m .


Let ��S � � bea Cauchysequencein m ; let YN� b begivenandlet + besuchthatthe termsof the Cauchysequencedo not differ by morethan 0 � R r Ì H from +on.Define S ¤ Y ¥ � �:S � ¤ Y ¥ . Then �`S � � convergesto S . Sincethelimit S agreeswithanelementof the sequenceto the left of any Y , its supportis left-finite andthusS�� m .

We returnto the fixed-point theorem, which provesto be a powerful mathe-maticaltool for thedetailedstudyof m : Let

, � m m bea functiondefinedonan interval

ßaroundtheorigin suchthat

, � ß � » ß , andlet,

becontractingwith aninfinitely smallcontractionfactor, then

,hasauniquefixedpoint in

ß.

Theproofis verysimilarto theBanachspacecase.Onebeginswith anarbitraryelement S�Z and defines S Ì H � , ��S � . The resultingsequenceis Cauchyandconvergesby theCauchy completenessof m . Then,S¦� '´² ý � G � S � is thefixedpoint of

,.

Now lete � m benonzeroandshow that

ehasamultiplicative inversein m .

Writee ��S Z ; R ; � P�3 S�� ,whereS Z is real, S is infinitely small,and ]�� û � e � .

Since S Z ; R hasthe inverseS FLHZ ; R F£ , it sufficesto find an inverseto � P@3 S�� .Write theinverseas � P°3 8 � . Thenwe haveP �v� P°3 S£� ; � P°3 8 �� P°3 S 3 8 3 S ; 8 MThus, 8 � � S � S ; 8 M (2.147)

This is a fixed-pointproblemfor8

with the function, � 8 �¦� � S � S ; 8 and

ß � p �� m � û �`�u�N�_i�s . Since S is infinitely small,8

is infinitely small byEq. (2.147).Thusany solution

8to Eq. (2.147)must lie in

ß. Also, since S is

infinitely small,,

is contractingwith an infinitely small contractionfactor. Thefixed-pointtheoremthenassertstheexistenceof a uniquefixedpoint of

,in

ßandthusin m .

Thefixed-pointtheoremcanalsobeusedto provetheexistenceof rootsof pos-itivenumbersin m andto provethatthestructure{ obtainedfrom m by adjoiningtheimaginaryunit is algebraicallyclosed(Berz1994).

Thealgebraicpropertiesof m allow for adirectintroductionof importantfunc-tions, suchaspolynomials,rationalfunctions,roots,andany combinationthereof.Besidestheseconventionalfunctions,m readilycontainsdelta functions. For ex-ample,thefunction , �`S��^� RR W 3 S W (2.148)

assumesthe infinitely largevalueR FLH

at theorigin, falls off as

ï S ï getslarger, isinfinitely small for any real S , andbecomesevensmallerfor infinitely large (inmagnitude)S .


For thescopeof thisbook,however, it is moreimportantto studytheextensibil-ity of thestandardfunctions,in particularthoserepresentableby powerseries.Forthis purpose,we studytwo kinds of convergence.The first kind of convergenceis associatedwith theabsolutevaluedefinedin Eq. (2.146)andis calledstrongconvergence. The secondkind of convergenceis calledweak convergence; wesaythatthesequence��S � � convergesweaklyto thelimit S�� m if for all Y{� b ,S � ¤ Y ¥ �£ S ¤ Y ¥ as + �£ |N . Therefore,weakconvergenceis componentwise.Itfollowsfrom thedefinitionsthatstrongconvergenceimpliesweakconvergencetothesamelimit. Weakconvergencedoesnotnecessarilyimply strongconvergence,however. It is theweakconvergencethatwill allow usto generalizepower seriesto m .

Let 7 � � S � be a power serieswith real coefficientsandconventionalradiusof convergence= . Then 7 � � S � convergesweakly for all S with

ï > ��S£� ï � = toan elementof m . This allows the automaticgeneralizationof any power serieswithin its radiusof convergenceto thefield m . Therefore,in a simpleway, thereis a very largeclassof functionsreadilyavailable.In particular, this includesalltheconventionalintrinsic functions of a computerenvironment.

Let,

be a function on a subset

ßof m . We say

,is differ entiable with

derivative,£¢ ��S£� at thepoint S�� ß if for any 0 �ñi{� m thereexistsa J �ñi{� m

with J < 0 not infinitely smallsuchthatMMMM , ��S 3ñ¼ S�� , ��S£�¼ S � , ¢ ��S£� MMMM � 0 (2.149)

for all¼ S with S 3�¼ S�� ß and

ï ¼ S ï � J . Therefore,thisdefinitionverymuchresemblestheconventionaldifferentiation,with animportantdifferencebeingtherequirementthat J not be too small.This restriction,which is automaticallysat-isfied in archimedeanstructures(e.g., � ), is crucial to making the conceptofdifferentiationin m practicallyuseful.

The usual rules for sumsand productshold in the sameway as in the realcase,with the only exceptionthat factorsarenot allowed to be infinitely large.Furthermore,it follows that if

,coincideswith a real functionon all realpoints

andis differentiable,thensois thereal function,andthederivativesagreeat anygivenrealpoint up to an infinitely smallerror. This will allow thecomputationof derivativesof real functions using techniquesof m .

A very importantconsequenceof thedefinitionof derivativesis thefundamen-tal result:Derivativesarediffer ential quotientsup to aninfinitely smallerror.

Let¼ S z�ìi be a differential,i.e., an infinitely small number. Choose0 �Ci

infinitely small suchthat

ï ¼ S ï < 0 is alsoinfinitely small.Becauseof differentia-bility, thereis a J �9i with J < 0 at mostfinite suchthat the differencequotientdiffersfrom thederivative by aninfinitely smallerror lessthan 0 for all

¼ S with

ï ¼ S ï � J . However, since J < 0 is at mostfinite and

ï ¼ S ï < 0 is infinitely small,

ï ¼ S ï < J is infinitely small, and in particular

ï ¼ S ï � J . Thus,¼ S yields an in-

finitely smallerrorin thedifferencequotientfrom thederivative.


Thiselegantmethodallows thecompututationof therealderivativeof any realfunctionthathasalreadybeenextendedto thefield m andis differentiablethere.In particular, all realfunctionsthatcanbeexpressedin termsof powerseriescom-binedin finitely many operationscanbeconvenientlydifferentiated.However, italsoworks for many othercasesin which the DA methodsfail. Furthermore,itis of historical interestsinceit retroactively justifies the ideasof the fathersofcalculusof derivativesbeingdifferentialquotients.It is worth pointing out thatthecomputationof thederivativesastherealpartsof differencequotientscorre-spondsto the result in Eq. (2.31),except that division by

Ris impossiblethere,

leadingto thedifferentform of theexpression.Equivalentresultsto theintermediatevaluetheorem,Rolle’s theorem,andTay-

lor’s theoremcanbe proved to hold in m . For more details,see((Berz 1994),(1992a)).

The last result we mention here is the Cauchy point formula. Let, �7 � ù Z �\c�`� � � Z � be a power serieswith real coefficients.Thenthe function is

uniquelydeterminedby its valueatapoint � Z 3 À , whereÀ is anarbitrarynonzeroinfinitely smallnumber. In particular, if wechooseÀ � R , wehave, �`�AZ 3 R �� Ë ù Z � R � (2.150)

from which wereadilyobtaina formulato computeall the �u ’s:�\V� , �`� Z 3 R � ¤ or¥nM (2.151)

This formulaallows thecomputationof derivativesof any functionwhich canbewritten asa power serieswith a nonzeroradiusof convergence;this includesall differentiablefunctionsobtainedin finitely many stepsusingarithmeticandintrinsic functions.

Besidesallowing illuminating theoreticalconclusions,thestrengthof theLevi-Civita numbersis that they canbeusedin practicein a computer envir onment.In this respect,they differ from the nonconstructive structuresin nonstandardanalysis.

Implementationof theLevi-Civita numbersis not asdirectasthatof theDAsdiscussedpreviously since m is infinite dimensional.However, sincethereareonly finitely many supportpointsbelow every bound,it is possibleto pick anysuchboundandstoreall thevaluesof a functionto theleft of it. Therefore,eachLevi-Civita numberis representedby thesevaluesandthevalueof thebound.

Thesumof two suchfunctionscanthenbecomputedfor all valuesto the leftof theminimumof thetwo bounds;therefore,theminimumof theboundsis theboundof the sum.In a similar way, it is possibleto find a boundbelow whichthe productof two suchnumberscanbe computedfrom the boundsof the twonumbers.Theboundto which eachindividual variableis known is carriedalongthroughall arithmetic.

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Chapter 2 Differential Algebraic Techniquesbt.pa.msu.edu/pub/papers/AIEP108book/chap2.pdf · Chapter 2 Differential Algebraic Techniques ... Campbell, and Petzold 1989 ... mination