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LinearAlgebraanditsApplications3323342001139164


wwwelseviercom/locate/laaApartialPad-via-Lanczosmethodforreduced-ordermodelingZhaojunBaiRolandWFreundAbstractTheclassicalLanczosprocesscanbeusedtoefcientlygeneratePadapproximantsofthetransferfunctiono

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1 LinearAlgebraanditsApplications332–334(2
LinearAlgebraanditsApplications332–334(2001)139–164 www.elsevier.com/locate/laaApartialPadé-via-Lanczosmethodforreduced-ordermodelingZhaojunBai,RolandW.Freund AbstractTheclassicalLanczosprocesscanbeusedtoefcientlygeneratePadéapproximantsofthetransferfunctionofagivensingle-inputsingle-outputtime-invariantlineardynamicalsystem. Correspondingauthor.Tel.:+1-908-582-4226;fax:+1-908-582-5857. Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–1641.IntroductionIthaslongbeenknownthatPadéapproximationisausefultoolforgeneratingreduced-ordermodelsoflineardynamicalsystems;see,e.g.,[14]andthereferencesgiventhere.Inrecentyears,therehasbeenrenewedinterestinandextensiveresearchintoPadé-basedreduced-ordermodeling.Theserecentdevelopmentsweremainlytriggeredbythelandmarkpaper[28]thatdemonstratedthepotentialofusingPadéapproximationinthesimulationoflargeelectronicVLSIcircuits.SomeoftherecentresearchinthisareafocusedontheefcientandnumericallystablecomputationofPadé-basedreduced-ordermodels.Inparticular,itisnowwidelyacceptedthatKrylov-subspacemethods,suchastheLanczosalgorithm[26]ortheArnoldiprocess[4]shouldbeemployed,inordertoavoidtheinherentnumericalill-conditioningofgeneratingPadéapproximantsdirectlyfromtheTaylorcoefcientsofthetransferfunctionofthelineardynamicalsystem.Wereferthereaderto[15,16]forrecentsurveysofreduced-ordermodelingtechniquesbasedonKrylovsubspacesandtheiruseincircuitsimulation.Ontheotherhand,itisalsowellknownthat,whenappliedtostablelineardy-namicalsy

2 stems,reduced-ordermodelingtechniquesbas
stems,reduced-ordermodelingtechniquesbasedonPadéapproximationingeneraldonotpreservethestabilityoftheoriginalsystem;see,e.g.,[1,14,31].Forsomeapplications,suchastheuseofPadé-basedreduced-ordermodelsfortheefcientcomputationofthefrequencyresponseoflarge-scalelineardynamicalsys-tems,thepossibleoccurrenceofunstablepolesisnotanissue[12].However,oftenreduced-ordermodelingisusedtoreplacelargelinearsubsystemsofastable,pos-siblynonlinear,systembysmallerapproximatemodels,withthegoaltoreducethecomplexityofthesimulationoftheoverallsystem.Inthiscontext,itiscrucialthatthereduced-ordermodelsofthelinearsubsystemsarestable,inordertoensurestabilityofthesimulationoftheoverallsystem.Incircuitsimulation,reduced-ordermodel-ingisoftenappliedtolargelinearsubsystemsthatrepresentnetworksconsistingofonlyresistors,inductors,andcapacitors.TheseRLCnetworksarestableandpassive,andagain,forthestabilityoftheoverallsimulation,itiscrucialthatreduced-ordermodelspreservethepassivityoftheoriginalRLCnetwork;see,e.g.,[29].Unfortu-nately,exceptforthespecialcasesofRC,RL,andLCnetworks[18,19],Padé-basedreduced-ordermodelsofRLCnetworksarenotpassiveingeneral.ThepurposeofthispaperistoexploretheuseofpartialPadéapproximationfortheconstructionofstable,andpossiblypassive,reduced-ordermodels.TruePadéapproximantsarerationalfunctionsofagivenorderwhereallavailabledegreesoffreedomareusedtomatchtheTaylorexpansionofthefunctiontobeapproximatedinasmanyleadingcoefcientsaspossible;see,e.g.,[8].PartialPadéapproximants[9],on

3 theotherhand,haveanumberofprescribedpole
theotherhand,haveanumberofprescribedpolesandzeros,whileonlytheremainingdegreesoffreedomareusedtomatchtheTaylorexpansionofthefunctiontobeapproximatedinasmanyleadingcoefcientsaspossible.OurmainmotivationforstudyingtheuseofpartialPadéapproximationforreduced-ordermodelingisbasedontheobservationthat,typically,theinstability,andpossiblynonpassivity,of Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164reduced-ordermodelsbasedontruePadéapproximantsisonlyduetoasmallnumberofunstablepolesandzeros.Byprescribingasmallnumberofstablepolesandzeros,whilepreservingasmuchoftheapproximationpropertyofthePadéapproximantaspossible,weobtainanewreduced-ordermodelbasedonpartialPadéapproximation.Often,thisnewmodelisstableandpossiblypassive.Inparticular,wepresentanalgorithmthatgeneratespartialPadéapproximantsviarank-1updatesofthetridiago-nalmatricesgeneratedbytheLanczosprocess.Duetotheuseofonlyrank-1updates,thealgorithmislimitedtopartialPadéapproximantswhosenumber,,ofprescribedpolesandzerosisboundedby,whereistheorderofthetruePadéapproximant.However,westressthat,forallpracticalpurposes,thisdoesnotposealimitationatall.First,weneverencounteredasituationwherethetruePadéapproximanthadmorethanunstablepolesandzeros,whichwouldthenrequiretoprescribepolesandzeros.Second,apartialPadéapproximantwithprescribedpolesandzerosmatchesthetruePadéapproximantinitsrst2Taylorcoefcients.Soevenifwouldoccur,theapproximationpropertyoftheresultingpartialPadéapproximantwouldbetooweaktobeofpractica

4 linterest.Weremarkthatthereissomerelated
linterest.Weremarkthatthereissomerelatedearlierwork;see[1,2,31].However,thetechniquesproposedthereallinvolveexplicitmatchingoftheTaylorcoefcientsofthetransferfunction,andhencetheyareinherentlynumericallyunstable.In[24],itisproposedtouseanimplicitlyrestartedLanczosmethodtoremedythepossibleinstabilityofPadé-basedreduced-ordermodels.However,theimplicitrestartsmod-ifysomeofthedatathatdescribesthegivenlineardynamicalsystem.Consequently,aspointedoutin[15],thereduced-ordermodelgeneratedbythisprocessnolongermatchesleadingTaylorcoefcientsofthetransferfunctionofthegivensystem,andthisisundesirableinsomeapplications,suchascircuitsimulation,wheretheleadingTaylorcoefcientshavesomephysicalmeaning.ForthespecialcaseofRLCnetworks,itisactuallypossibletogenerateprovablystableandpassivereduced-ordermodelsbymeansofprojectionontoKrylovsub-spaces;see[17,27,32].However,thetransferfunctionsoftheseprojectedreduced-ordermodelsmatchonlyhalfasmanyTaylorcoefcientsoftheoriginaltransferfunctionasthecorrespondingPadéapproximantderivedfromthesameKrylovsub-space.Moreover,theseprojectiontechniquesrequireaveryspecificformulationoftheequationsthatcharacterizeagivenRLCnetwork.Forexample,in[5],thereisanexampleofasimpleRLCnetworkforwhichtheprojectedreduced-ordermodelisunstableifanotherformulationofthenetworkequationsisused.Incontrast,Padéap-proximationandalsopartialPadéapproximationyieldidenticalresults,independentofthechosenformulationofthenetworkequations.Finally,wewouldliketostressthatthetechniqu

5 esdescribedinthispaperarenotrestrictedto
esdescribedinthispaperarenotrestrictedtoRLCnetworksandcanbeemployedforreduced-ordermodelingofgeneralsingle-inputsingle-outputtime-invariantlineardynamicalsystems.Theremainderofthepaperisorganizedasfollows.InSection2,wecollectsomefactsabouttransferfunctionsofsingle-inputsingle-outputtime-invariantlineardynamicalsystems.InSection3,webrieflyreviewPadéapproximantsoftransfer Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164functionsandtheircomputationbymeansoftheLanczosprocess.InSections4and5,weshowhowpartialPadéapproximantscanbeobtainedviasuitablerank-1updatesoftheLanczostridiagonalmatrix.InSection6,wepresentastatementoftheoverallcomputationalprocedureforgeneratingpartialPadéapproximantsviatheLanczosprocess.InSection7,wereporttheresultsofnumericalexperimentsfortwocircuitexamples.Finally,inSection8,wemakesomeconcludingremarks.Throughoutthisarticle,weuseboldfaceletterstodenotevectorsandmatrices.identitymatrixisdenotedbyandthezeromatrixbyiftheactualdimensionsofthesematricesareapparentfromthecontext,weomittheseindicesandsimplywrite.Forsquarematrices,wedenotebythesetofalleigenvaluesof.Thesetsofrealandcomplexnumbersaredenoted,respectively.For,ReistherealpartofandImistheimaginarypartof.Weusetodenotetheopenright-halfofthecomplexplane.Finally,denotesthesetofrationalfunctionswithrealnumeratorpolynomialofdegreeatmostandrealdenominatorpolynomialofdegreeatmost2.TransferfunctionsandsomepropertiesInthissection,wecollectsomefactsabouttransferfunct

6 ionsofsingle-inputsin-gle-outputtime-inv
ionsofsingle-inputsin-gle-outputtime-invariantlineardynamicalsystems.2.1.Time-invariantlineardynamicalsystemsWeconsidersingle-inputsingle-outputtime-invariantlineardynamicalsystemsgivenbystate-variabledescriptionsoftheform        where.In(1),thefunction  representstheinputofthesystem,  istheoutput,and isthe-dimensionalvectorofstatevari-ables.Thematricesareallowedtobesingular,andweonlyassumethatthepencilregular,i.e.,thematrixissingularonlyfornitelymanyvaluesof.Notethattherstequationin(1)isasystemofdifferential-algebraicequationsifissingularandasystemofordinarydifferentialequationsifisnonsingular.Theinput–outputbehaviorofthelineardynamicalsystem(1)isdescribedbyitstransferfunction    Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164see,e.g.,[10].InSection3,weconsiderPadéapproximantsoftransferfunctionsofform(2),andtheircomputationviatheLanczosprocess.Tothisend,wewillneedthefollowingrepresentationofintermsofasinglematrix,insteadofthetwomatricesinthedefinitionof.Letbeanyxedexpansionpointsuchthatthematrixisnonsingular.Letwherebeanyformalfactorizationof.Forexample,(3)canbechosenasa‘true’LUfactorizationwherearetriangularmatrices,possiblypermutedduetopivotingorthe‘trivial’factorizationgivenby.Using(3),representation(2)ofcanberewrittenasfollows: where2.2.StabilityIflineardynamicalsystem(1)describesanactualphysicalsystem,suchasafunctioningelectroniccircuit,thenitwillnecessarilybestable.Roughlyspeaking,stabilitymeansthatfor

7 boundedinputs  ,thestate-variablevecto
boundedinputs  ,thestate-variablevector of(1)willremainboundedforalltimes;see,e.g.,[3,Chapter3.7]or[10,Chapter8].Fortime-invariantlineardynamicalsystems(1),stabilitycanbedenedviathetransferfunction.DeÞnition1.Thetransferfunctionofasingle-inputsingle-outputtime-invariantlineardynamicalsystemissaidtobestableinthesenseofLyapunovhasnopolesinandifanypoleofontheimaginaryaxisissimple.Notethatforfunctionsgivenby(4),anypoleisoftheform whereHowever,ingeneral,noteveryofform(5)isapoleof.Indeed,thepolesofaregivenby(5)if,andonlyif,thetriplein(4)isaminimalrealizationof.Here,foragiventransferfunction,arepre-sentation(4)iscalledaminimalrealizationifthestate-spacedimensionminimal.If(4)isaminimalrealization,thenthestabilityofcanbecharacterizedcompletelyintermsof;see,e.g.,[3,Theorem3.7.2].Next,westatethisresult.TheoremA.LetHbeatransferfunctiongivenbyandassumethatisaminimalrealization.Then Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164ThepolesofHaregivenby where ThetransferfunctionHisstableifandonlyifthefollowingtwoconditionsaresatised(i)Reforall (ii)thenoccursonlyinblocksintheJordancanonicalformof2.3.PassivityNext,wedenepassivity,whichisastrongerconditionthanstability.Rough-lyspeaking,asystemispassiveifitdoesnotgenerateenergy.Forexample,anyRLCnetworkispassive.Fortime-invariantlineardynamicalsystems(1),passivityisequivalenttopositiverealnessoftheassociatedtransferfunction;see,e.g.,[3]or[33,Chapter4].Basedonthisequivalence,inthispaper,wewill

8 usethefollowingdefinitionofpassivity.DeÞ
usethefollowingdefinitionofpassivity.DeÞnition2.Thetransferfunctionofasingle-inputsingle-outputtime-invariantlineardynamicalsystemissaidtobepassiveif:(i)hasnopolesin(ii) forall(iii)Re 0forallNotethat,forlineardynamicalsystems(1),condition(ii)isalwayssatisedsincethedatain(1)isassumedtobereal.Furthermore,usingrepresentation(4)of,condition(i)canbecheckedviacomputingtheeigenvaluesofthematrixin(4).Nowassumethatcondition(i)issatised.Then,bytheMaximumModulusTheorem,condition(iii)issatisedonlyif 0forallIn[6],itisshownhow(7)canbecheckedviacomputingtheeigenvaluesofacertainmatrixpencilderivedfromrepresentation(4)ofInthefollowingtheorem,wecollectsomewell-knownnecessaryconditionsforpassivity.TheoremB(Necessaryconditionsforpassivity).IfHispassivethenHisstable.IfHispassivethenHhasnopolesandzerosinandanypossiblepoleorzeroofHontheimaginaryaxisissimple. Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–1643.PadŽapproximationviatheLanczosprocessInthissection,webrieflyreviewtheconceptofPadéapproximationofagiventransferfunctionandthenumericalcomputationofthesePadéapproximantsviatheLanczosprocess.3.1.Padéapproximantsbethetransferfunctiongivenby(4).Notethatisarationalfunction.Moreprecisely,whereisthestate-spacedimensionof(1).Incircuitsimulation,canbeextremelylarge,andthenisreplacedbyanapproximationwithstate-spacedimension.Awidely-usedandoftentheonlyviablechoiceofisPadéapproximation.Expandingtransferfunction(4)about,wehave whereAfunctionissaidtobeanPa

9 déapproximant(abouttheexpansionpoint)if(
déapproximant(abouttheexpansionpoint)if(8)andthecorrespondingexpansionofagreeintherstterms,i.e.,ForanoverviewofPadéapproximants,wereferthereaderto[8].NotethatEq.(9)represents2conditionsforthe2degreesoffreedomthatdescribeanyfunction.Inparticular,(9)denesauniquethPadéap-proximantif,andonlyif,theso-calledmomentmatrixisnonsingularInthispaper,forsimplicity,weassumethat(10)issatisedforall3.2.ComputationviatheLanczosprocessThestandardapproachtocomputingistogeneratethecoefcientsofthenumeratoranddenominatorpolynomialsofviathesolutionofsystemsoflinearequationswithcoefcientmatrix.However,ingeneral,duetothetypicalill-con-ditioningof,thisapproachisfeasibleonlyforverymoderatevaluesof,suchas10;see[12]forexamples.Fortunately,thesenumericaldifcultiescaneasilybeavoidedbyexploitingthewell-knownconnection[23]betweenPadéapproximantsandtheLanczosprocess[26].Next,westatethisconnection.Thestartingpointisrepresentation(4)of.WeapplythenonsymmetricLanczosalgorithmtothematrixfrom(4)andwithfrom(4)asright, Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164respectivelyleft,startingvector.Weremarkthatourassumptiononthenonsingular-ityofthemomentmatrices(10)guaranteesthatnobreakdownsoccurintheLanczosprocess;see,e.g.,[20,23].Aftersteps,theLanczosalgorithmhasgeneratedrightandleftLanczosvectors,suchthat,forallspanspanspanspan0forallConditions(12)and(13)determinetheLanczosvectorsonlyuptoascaling.Weusethescaling1forallTh

10 eLanczosvectorsaregeneratedbymeansofthre
eLanczosvectorsaregeneratedbymeansofthree-termrecurrences.FortherightLanczosvectorsin(11),theserecurrencescanbestatedcompactlyinmatrixformasfollows:isthethunitvectoroflength,andisthetridiagonalmatrixwhere0forallTheleftLanczosvectorsin(11)satisfyanequationsimilarto(14).ThePadé-LanczosconnectionthenstatesthatthethPadéapproximanttothetransferfunctionin(4)isgivenbywheredenotestherstunitvectoroflength.Foraproofof(16),wereferthereaderto[12]or[23].Computingvia(16)hasbeenadvocatedin[11,12,21],andfollowing[11,12],thisapproachisnowknownasthePVL(PadéviaLanczos)method.3.3.PolesandzerosofUsingCramer’srule,representation(16)ofcanberewrittenas I0T Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164whereisthetridiagonalmatrixobtainedbydeletingtherstrowandrstcolumnofthematrixin(15).Representation(17)showsthatthepolesandzerosofcanbeobtainedviatheeigenvaluesof.Moreprecisely,thepolesaregivenby,andthezerosby.Notethatcommonpolesandzeros,,whichthenwouldcancelout,cannotoccurin(17).Thisisaconsequenceofthefactthatallsub-andsuperdiagonalelementsofin(15)arenonzero.Wethushavethefollowingresult.Lemma1.bethenPadéapproximantofH.Thenthepolesandthezerosaregivenby where whereRemark1.TheLanczosprocessisintimatelyconnectedwithformallyorthogonalpolynomials;see,e.g.,[20,23]andthereferencesgiventhere.Moreprecisely,eachpairofrightandleftLanczosvectorsof(11)canbeexpressedaswhereisamonicpolynomialofdegree1and0aresuitablescal-ingfactors.Inviewof(18),bi-orthogonality(18)

11 oftheLanczosvectorsisequivalenttotheform
oftheLanczosvectorsisequivalenttotheformalorthogonality,0forallofthepolynomials.Itiswellknown[23]thatisthecharacteristicpolynomialof.Thus,thedenominatorpolynomialinrepresentation(17)ofthereversewithrespectto,i.e.,Similarly,thenumeratorpolynomialin(17)isthereverseofthecharacteristicpolynomialofwithrespecttoRemark2.UsingtheconnectionwithpolynomialsoutlinedinRemark1,itispos-sibletorelateourresultsonpartialPadéapproximantstotheworkin[22,25]onthecalculationofGaussquadratureswithsomeprescribedknots. Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–1644.PartialPadŽapproximationInthissection,werstdenepartialPadéapproximantsandthenestablishacon-nectiontotheLanczosprocess.Inthefollowing,let,andbegivenintegerswithandletbegivencomplexnumbers.Weassumethatthenumbers(20)arepairwisedistinctandthateachofthesetsisclosedundercomplexconjugation: PD Westressthat,in(19),either0or0isallowed,whichmeansthateithercanbetheemptyset.InanalogytotherepresentationsofthepolesandzerosgiveninLemma1,weusethenumbers(20)todeneprescribedpolesand  respectively.NowconsiderapproximantsoftheformwithprescribedpolesandzerosFunctions(24)have2degreesoffreedom,andtherefore,foragivenexpansionpoint,onewouldexpectthatsuchfunctionscanbeusedtomatchtherst2coefcientsinexpansion(8)oftheoriginaltransferfunction.Thisleadstothefollowingdefinition.Afunctionofform(24)iscalledanpartialPadéapproximant(abouttheexpansionpoint)if(8

12 )andthecorrespondingexpansionofagreeinth
)andthecorrespondingexpansionofagreeintherst2terms,i.e.,WeremarkthatthegeneralconceptofpartialPadéapproximation(notonlyoftransferfunctions)wasintroducedandstudiedbyBrezinski[9].Thespecialcasethatonlypolesareprescribed,i.e.,0isusuallyreferredtoasPadé-typeapprox-imation;see,e.g.,[8]. Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164In[9],itisshownhowtogeneratepartialPadéapproximantsdirectlyfromtheinexpansion(8)of.However,asinthecaseofPadéapproximants,thisapproachsuffersfrominherentill-conditioning.Next,weproposeadifferentapproachforconstructingthepartialPadéapproximantviatheLanczosprocess.Westressthat,inprinciple,itisalsopossibletoconstructusinganypairofbasesfortheKrylovsubspaces(12).However,sinceforthecomputationof,theLanczosvectorsappeartobethebasesofchoice,weseenoreasontoemployanyotherbasesfortheconstructionofbethetridiagonalLanczosmatrix.Recallfrom(16)thatdenesthethPadéapproximant.Letbetheintegerdenedin(19).Weconsiderrank-1updatesofoftheformwheredenotesthethunitvectoroflength.Therefore,(26)meansthatthetridiagonalmatrixdifferonlyinthetrailingentriesoftheirlastcolumns.Forexample,when8andisamatrixoftheformInanalogyto(16),wenowsetThefollowinglemmashowsthatfunction(27)isacandidateforthethpartialPadéapproximant.Lemma2.ForanychoiceofthevectortheassociatedfunctiondenedinsatisesProof.Expandingbothin(27)andin(16)about,weseethat(28)holdstrueifforallThus,itonlyremainstoshow(29).Tothisend,notethat,sinceistridiag

13 onalandistherstunitvector,forall1
onalandistherstunitvector,forall1,thetrailingentriesofthecolumnvectorareguaranteedtobezero.Togetherwith(26),weget Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164Usinginductionon,wededucefrom(30)thatforallSimilarly,forall,thetrailingentriesoftherowvectorareguaranteedtobezero,andtogetherwith(26),wegetUsinginductionon,wededucefrom(32)thatforallBymultiplying(33)and(31),itfollowsthatforall1,whichisjustclaim(29).Inviewof(9),(25),and(28),thefunctiondenedin(27)isindeedanpartialPadéapproximantifithastheprescribedpolesandzeros(23).Notethat,inanalogyto(17),function(27)hastherepresentation whereisthematrixobtainedbydeletingtherstrowandtherstcolumnof.By(34),hastheprescribedpolesandzeros(23)if,andonlyif,thematriceshavetheprescribedeigenvalues,respectively.Therefore,wehaveestablishedthefollowingtheorem.Theorem3.betheLanczostridiagonalmatrixandbeitsrank-update.IfthevectorischosensuchthatthenthefunctionisannpartialPadéapproximantofH.ByTheorem3,computinganthpartialPadéapproximantviatheLanczospro-cessreducestoconstructingthevectorin(26)suchthat(35)issatised.Inthefollowingsection,weshowthatsuchavectorcanbeobtainedbysolvingasuitablelinearsystem.5.Computingthevectorzbethesets(21)ofprescribedeigenvaluesof,respectively. Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164Clearly,isaneigenvalueofif,andonlyif,thereisanassociatedlefteigenvector,i.e.,Using(26),wecanrewrite(36)asfollows:hastheprescribedeigenvaluesif,

14 andonlyif,thecolumnvectorin(26)canbechos
andonlyif,thecolumnvectorin(26)canbechosensuchthat,foreach,thereexistsarowvectorsatisfying(37).ThefollowingpropositiongivesaconditionforsuchavectorProposition4.Foreachbeasolutionofandsetotherwise.MoreoverforsetotherwiseThenhastheprescribedeigenvaluesandonlyifthevectorsatisesthesystemoflinearequationswhereProof.beanyvectorsatisfying(37).First,weshowthatIndeed,if0,then(37)reducesto,andthuswithassociatedlefteigenvector.Conversely,letandassumethat0.By(37),itfollowsthatwhereandthusOntheotherhand,by(15),isanunreducedlowerHessenbergmatrix,whichisalsosingularsince.Thisimplies Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–1641andrankwhichisacontradictionto(43).Thus,0andtheproofof(42)iscomplete.Next,notethat,inviewof(15)and(26),isanunreducedupperHessenbergmatrix.Thisimpliesthatalleigenvaluesofhavegeometricmultiplicity1,andthusthelefteigenvectorisuniquelydeterminedby(36)uptoanonzeronormal-izationfactor.By(42),wecanxthenormalizationofsuchthat0if1otherwise.Withnormalization(44),itfollowsthatasolutionof(37)isgivenby(38)ifandby(39)otherwise.Finally,withthevectors,giv-enby(38)or(39),thematrixhasindeedtheprescribedeigenvaluesif,andonlyif,thevectorsatisesnormalizationconditions(44).However,(44)isjustthelinearsystem(41).Fortheprescribedeigenvalues,wecanproceedinanalogousfashion.Thisleadstothefollowingproposition.Proposition5.Foreachbeasolutionofandsetotherwise.Moreoverforsetotherwise.Thenhastheprescribedeigenvaluesandon

15 lyifthevectorsatisesthesystemoflineareq
lyifthevectorsatisesthesystemoflinearequationswhereProofofProposition5iscompletelyanalogoustothatofProposition4andcanthusbeomitted.Bycombiningthelinearsystems(41)and(48),weobtainequationsforthevector.Furthermore,recallfrom(26)thattherstentriesofarezeroandthatonlythetrailingpartneedstobecomputed.Forthecase1,theresultinglinearsystemforisasfollows: Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164denotesthesubmatrixconsistingofthethcolumntothcolumnofthematrix.Forthecase,thelinearsystemforisasfollows:Notethatboth(49)and(50)aresystemsoflinearequationsfortheentriesofTherefore,boththeselinearsystemshaveuniquesolutions,providedtheircoefcientmatricesarenonsingular.Weremarkthat,thoughthecoefcientmatricesof(49)and(50)arecomplexingeneral,thesolutionvectorisalwaysreal.Thisfollowsfromassumption(22)onthesets.Indeed,let beapairofcomplexconjugateprescribedeigenvaluesof.Itthenfollowsfrom(38)–(40)that Multiplyingthethandstrowof(41)fromtheleftby andusing(51),weobtaintheequivalentrealequationsThismeansthat,in(41)andthusalsoin(49),respectively(50),wecanreplacethepairofrowscorrespondingtoeachpairofcomplexconjugatenumbersinbyanequivalentrealpairofrows.Similarly,onecanturnsystem(48)intoarealsystem.Alltogether,thismeansthatsystem(49),respectively(50),canalwaysbemadereal,andthusthesolutionvectorisreal.Insummary,wehavethefollowingalgorithmforcomputingthevectorin(26).Algorithm1ComputingthevectorINPUT:thLanczostridiagonalmatrix,setssatisfying(19)–(22).OUTPUT:Avectorsuc

16 hthatthematriceshavetheprescribedeigenva
hthatthematriceshavetheprescribedeigenvalues,respectively.(1)Forcomputeasasolutionofelsecomputeby(39).(2)Forcomputeasasolutionof(45),elsecomputeby(46). Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164(3)Computeby(40)and(41).(4)Computeby(47)and(48).(5)IfIfthecoefcientmatrixof(49)issingular:stop.Otherwise,solve(49)forthevectorandsetIfthecoefcientmatrixof(50)issingular:stop.Otherwise,solve(50)forthevectorandsetWeconcludethissectionwithremarksaboutthetwospecialcasesofpartialPadéapproximantswithallpoles,respectivelyallzeros,prescribed.Remark3.ConsiderthecasethatallpolesofthepartialPadéap-proximantareprescribed.Inthiscase,0in(19),andthelinearsystem(50)reducesto.InviewofDefinition1,thisspecialcasecanalwaysbeusedtogenerateareduced-ordertransferfunctionthatisguaranteedtobestable.Indeed,alloneneedstodoisprescribepairsofcomplexconjugatepoleswithRe0forall.Ontheotherhand,theassociatedpartialPadéapproximantthenonlymatches2Taylorcoefcientsoftheoriginaltransferfunction,insteadofthe2coefcientsmatchedbythePadéapproximant.Forexample,suppose2ofthepolesarepairwisedistinctandsatisfyRe0,whiletheothertwopoles,say,violatethestabilityof.Thenwecanchoosetheelementsoftheset(21)asfollows:isanypairofprescribedrealorcomplexconjugatepoleswithstrictlynegativerealpart.Forthischoiceof,Algorithm1generatesavectorsuchthattheassociatedreduced-ordertransferfunction(27),,isguaranteedtobestable.Remark4.0and1,thenallthezerosofareprescribed.Inthiscase

17 ,thelinearsystem(49)reducesto,andtheasso
,thelinearsystem(49)reducesto,andtheassociatedfunction,matches21Taylorcoefcientsoftheoriginaltransferfunction6.PVLalgorithmInthissection,wecombinethePVLmethodsketchedinSection3.2withtheupdateprocedureforobtainingpartialPadéapproximantsdescribedinSections4and Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–1645.TheresultingcomputationalprocedureiscalledthePVLalgorithm.ItconsistsofthebasicPVLalgorithmtogeneratethetruePadéapproximantandadditionalpost-processing,withthegoaltoremoveunstablepoles,andpossiblyunstablezeros,AsketchofthePVLalgorithmisasfollows.Algorithm2SketchofthePVLalgorithmINPUT:Expansionpoint,dataofthefunction OUTPUT:AnthPadéapproximantorpartialPadéapproximant(1)RunstepsoftheLanczosprocessappliedtothematrixwithrightandleftstartingvectorstoobtainthetridiagonalLanczosmatrix,andsetischosensuchthatsomeappropriatestoppingcriterionissatised.(2)Computetheeigenvalues,andfromthesethepolesandzeros 01 (3)Checkthestability,andpossiblypassivity,ofisstable,andpossiblypassive,then:stop.(4)Choose0and01with.Prescribepolessuchthatthenumbers 01 satisfy(20)–(22).(5)UseAlgorithm1tocomputethevector(6)Set(7)Computethepoles,andpossiblyzeros,ofandcheckthestability,andpossi-blypassivity,ofisstable,andpossiblypassive,then:stop.Otherwise,returntoStep(4)andchooseanothersetofprescribedpolesandzeros.Next,wemakesomeremarksonthevariousstepsofAlgorithm2.Remark5.ThePVLalgorithm[11,12]essentiallyconsistsofSteps(1)and(2). Z.Bai,R.W.Freund/Lin

18 earAlgebraanditsApplications332–334(2001
earAlgebraanditsApplications332–334(2001)139–164Remark6.InStep(1),asimplestoppingcriterionistochecktheconvergenceofthedominantpolesof,assuggestedin[12].Amoresophisticatedstoppingcrite-rionbasedondirectlyestimatingtheerror wasrecentlyproposedininRemark7.TotestforstabilityinSteps(3)and(7),onejustneedstocheckifallpolesof,respectively,satisfytheconditionsgiveninpart(b)ofTheo-remA.Remark8.Ifthegivenfunctionispassive,theninSteps(3)and(7)onealsoneedstocheckthepassivityof,respectively.ThenecessaryconditionsforpassivitygiveninTheoremBonlyinvolvethepolesandzeros,andthustheycaneasilybecheckedoncetheeigenvalues,respectively,havebeencomputed.Ifthesenecessaryconditionsarenotsatised,thefunction,respectively,isnotpassive.Otherwise,oneproceedstochecktheeigenvalue-basedcharacterizationofpassivityrecentlyproposedin[6].Morepre-cisely,in[6],itisshownthatthefunctiongivenby(52)ispassiveif,andonlyif,thefollowingthreeconditionsaresatised:(i)isstable.(ii)0foragiven(iii)Thematrixpencilhaseithernorealpositiveeigenvaluesorifanyrealpositiveeigenvaluehasevenmultiplicity.HerethematrixpencilisgivenbyAnalogously,forcheckingthepassivityof,onesimplyneedstoreplaceintheaboveconditions(i)–(iii).Remark9.Thepost-processingSteps(3)to(7)areallperformedonmatricesandtheyinvolveOarithmeticoperations.Sinceitisexpectedthat,theoverallcostofthesepost-processingstepsshouldnotbesignificant.Remark10.WearenotawareofanysystematicchoiceoftheprescribedpolesandzerosinStep(4)thatwouldguaranteestability,andpossib

19 lypassivity,oftheasso-ciatedpartialPadéa
lypassivity,oftheasso-ciatedpartialPadéapproximant.ForthenumericalexperimentsreportedinSection7,wechosetheprescribedpolesandzerosbysuitablereectionsoftheunstablepolesandzeroswithrespecttotheimaginaryaxis,andthisleadtostableandpassivemodels.Remark11.Ingeneral,modiedmatrix(26),,isnolongertridiagonalifHowever,itiseasytoreduceagaintotridiagonalform.Indeed,alloneneedstodoisrunstepsoftheLanczosprocessappliedtothematrixwithrightstarting Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164vectorandleftstartingvector.Again,forsimplicity,weassumethatnobreakdownoccursintheLanczosprocess.Then,aftersteps,thealgorithmhasgeneratedatridiagonalmatrixandtwomatricesrightandleftLanczosvectorssuchthatdiagFurthermore,thematrices,andareallnonsingular.Notethat,bythesecondrelationin(54),wehaveUsing(54)and(55),wecanrewriteformula(53)ofasfollows:Thisshowsthat,in(53),wecansimplyreplacebythetridiagonalmatrixobtainedfromstepsoftheLanczosprocessappliedtoasbothrightandleftstartingvector.7.NumericalexamplesInthissection,wepresenttwocircuitexamplesthatdemonstratetheeffectivenessofthePVLmethod.7.1.ThePEECcircuitOurrstexampleisacircuitresultingfromtheso-calledPEECdiscretization[30]ofanelectromagneticproblem.Thiscircuithasoftenbeenusedasatestproblemforreduced-ordermodelingtechniques.ThecircuitisanRLCnetworkconsistingofonlyinductors,capacitors,andinductivecouplings,andasingleresistivesourcethatdrivesthecircuit.Inthiscase,thetransferfunctionrepresentsthecurrentowingthrough

20 oneoftheinductors.Thecircuitisstablewith
oneoftheinductors.Thecircuitisstablewithallpolesofinthelefthalfofthecomplexplane.However,sincethecircuitismostlyanLCnetwork,mostofthepolesareclosetotheimaginaryaxis.Duetothisproximity,duringthecomputationofareduced-ordermodel,numericalandapproximationerrorscaneasilyproduceunstablepoles.Indeed,runningthePVLalgorithm(withexpansionpoint)for60iterationsproducesanalmostexacttransferfunctionintherelevantfrequencyrange.The Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164 x 109 0 0.004 0.006 0.008 0.01 0.012 0.014 Frequency (GHz) exact PVL 60 iter. Fig.1.ThePEECtransferfunction,exactand60PVLiterations.correspondingcurvesfor areshowninFig.1.However,thePadéapproximantinnotstabledueto15unstablespoles.Moreprecisely,eightofthese15aretrulyunstablepoles,whiletheothersevenarenearlystablepoles,asshowninFig.2.Inordertoproduceastablereduced-ordermodel,weranthePVLAlgorithm2toforceallthe15unstablepoles,iIm,intothelefthalf-plane.Thisisdonebysetting,inStep(4)ofAlgorithm2,0,and whereiImforall15unstablepoles.Fig.2showsthepolesofandthepolesofthemodiedreduced-ordertransferfunctiongeneratedbyPVL.Notethathasallitspolesinthelefthalf-plane,andthusisstable.Fig.3showsthecurves fortherelevantfrequencyrange.InFig.4,weplotboththePVLerrorcurve andthePVLerrorcurve therelevantfrequencyrange.Theseerrorcurvesshowthattheaccuracyofthestablereduced-ordertransferfunctionremainssatisfactory.7.2.ApackagemodelOursecondexampleisacircuitthataroseintheanalysisofa64-pi

21 npackagemodelusedforanRFintegratedcircui
npackagemodelusedforanRFintegratedcircuit.Thepackagemodelisdescribedbyapproxi- Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164 0 5x 109 0 1 2 3 4 5 6 7 8x 1010 Real partImaginary part Fig.2.ThePEECcircuit,PVLpolesandPVLpoles. x 109 0 0.004 0.006 0.008 0.01 0.012 0.014 Frequency (GHz) exact 60 iter. Fig.3.ThePEECtransferfunction,exactand60PVLiterations. Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164 x 109 16 1014 1012 1010 10 Frequency (GHz) PVL 60 iter. error 60 iter. error Fig.4.PVLandPVLerrorcurvesforthePEECtransferfunction.mately4000RLCcircuitelements,resultinginmatricesin(2)ofsizeabout2000.Hereweareinterestedinapproximatingthepassivetransferfunctionthatrepresentstheinputimpedanceofoneofthepinsofthepackage.ThePVLalgorithm(withexpansionpoint)requires80iterationstogenerateaPadéapproximantthatapproximatestheexacttransferfunctionsufcientlywellintherelevantfrequencyrange.Fig.5showsthecorrespondingcurvesfor .However,thePadéapproximantisneitherstablenorpassiveduetotwounstablepolesandfourunstablezerosInordertoproduceastableandpassivereduced-ordermodel,weranthePVLAlgorithm2toforcethetwounstablepoles,iIm,andthefourunstablezeros,iIm,intothelefthalf-plane.Thisisdonebyset-ting,inStep(4)ofAlgorithm2,4,and where1Rei10Im where1Rei10Imforallunstablepolesandzerosof.TheresultingpartialPadéapproximantproducedbyPVLnowhasallitspolesandzerosintheleftcomplexhalf-plane.Thus,isstableandsatisesthenecessaryconditionsforpassivitystated

22 inpart(b)ofTheoremB.Furthermore,usingthe
inpart(b)ofTheoremB.Furthermore,usingtheeigenvalue-basedpassivitytestproposedin[6],weveriedthatisindeedpassive.Fig.6showsthecurves  Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164 10 10 Frequency (Hz)V1int/V1ext Exact PVL 80 iter. Fig.5.Thepackagetransferfunction,exactand80PVLiterations.fortherelevantfrequencyrange.Thesecurvesclearlyillustratethatnoac-curacyhasbeenlostbyenforcingstabilityandpassivitybymeansofthePVLpost-processing.8.ConcludingremarksWehaveintroducedthePVLalgorithmforgeneratingreduced-ordermodelsbasedonpartialPadéapproximationviatheLanczosprocess.ThealgorithmcanbeviewedasavariantofPVLwithaddedpost-processingtoremovepossibleunstablepolesandzerosofthePVLreduced-ordermodel.Therearestilltwoimportantopenproblems.First,wearenotawareofanysys-tematicwayofchoosingtheprescribedpolesandzerosofthepartialPadéapprox-imantssothatstability,andpossiblypassivity,ofthecorrespondingreduced-ordermodelcanalwaysbeguaranteed.WhileweobtainedstableandpassivemodelsbysimplyprescribingreectionsoftheunstablePVLpolesandzeroswithrespecttotheimaginaryaxis,thereisadefiniteneedforautomatingtheselectionoftheprescribedpolesandzeros.Onepossibilityistouseanoptimizationprocedurethatminimizessomesuitablemeasureofdistancetostability,andpossiblypassivity.Suchanop-timizationprocedureconsistsofanouteriterationforthechoiceoftheprescribedpolesandzeros,whilePVLisemployedtogeneratetheassociatedpartialPadéapproximantwithineachouteriteration.Workinthisdirectionisinprog

23 ressandwillbereportedelsewhere. Z.Bai,R.
ressandwillbereportedelsewhere. Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164 10 10 Frequency (Hz)V1int/V1ext Exact 80 iter. Fig.6.Thepackagetransferfunction,exactand80PVLiterations.Second,thispaperonlytreatsthecaseofscalartransferfunctions.In[13],thePVLalgorithmhasbeenextendedtothegeneralcaseofmatrix-valuedtransferfunc-tionsofmulti-inputmulti-outputlineardynamicalsystems.Itisalsopossibletoex-tendPVLtothismoregeneralcase,byusingtheconceptofpartialmatrix-Padéapproximation.Suchanextensionwouldbebeyondthescopeofthispaper,andinstead,thiswillbedoneinsomefuturereport.AcknowledgementsWewouldliketothankPeterFeldmannforprovidingthecircuitexamplesforournumericaltests.Theauthorsaregratefultotherefereesandtheeditorfortheirconstructivecommentsthathelpedustoimprovethepresentationofthepaper.References[1]F.J.AlexandroJr.,StablepartialPadéapproximationsforreduced-ordertransferfunctions,IEEETrans.Automat.Control29(1984)159–162.[2]D.F.Anastasakis,N.Gopal,L.T.Pillage,Enhancingthestabilityofasymptoticwaveformevalu-ationfordigitalinterconnectcircuitapplications,IEEETrans.Computer-AidedDesign13(1994)[3]B.D.O.Anderson,S.Vongpanitlerd,NetworkAnalysisandSynthesis,Prentice-Hall,EnglewoodCliffs,NJ,1973. Z.Bai,R.W.Freund/LinearAlgebraanditsApplications332–334(2001)139–164[4]W.E.Arnoldi,Theprincipleofminimizediterationsinthesolutionofthematrixeigenvalueproblem,Quart.Appl.Math.9(1951)17–29.[5]Z.Bai,P.Feldmann,R.W.Freund,Stableandpassivereduced-ordermodelsbasedonpartial

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