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1、SIAMJ.SCl.COMPUT.()1995SocietyforIndustrialandAppliedMathematicsVol.16,No.2,pp.257-270,March1995001SOLVABILITYOFGENERALDIFFERENTIALALGEBRAICEQUATIONS*STEPHENL.CAMPBELLANDE.GRIEPENTROG*Abstract.Inthelastfewyearstherehasbeenconsiderableresearchondifferentialalge
2、braicequations(DAEs)f(t,x,x)0wherefx'isidenticallysingular.Mostofthisefforthasfocusedoncomputingasolutionthatisassumedtoexist.Thatis,theDAEisassumedsolvable.Morerecentlytherehavebeenexistenceresultsdevelopedusingdifferentialgeometry.Forcomplexhigherindexsystem
3、sthesecharacterizationscanbehardtoverifyinpractice.Inthispaperthecomputationalverificationofsolvabilityisinvestigated.Thisfirstrequiresdevelopinganalternativesetofsufficientconditionsforsolvabilitywhicharemoreamenabletocomputation.Verificationoftheseconditions
4、usingreadilyavailablenumericalandsymbolicsoftwareisthendiscussed.Anexamplefromroboticswhereclassicalgraphtheoreticalapproachesgiveanincorrectanswerisworkedtoillustratetheusefulnessofthesufficientconditionandthecomputationalapproach.Keywords,leastsquares,differ
5、entialalgebraicequations,solvability,completions,numericalmethodsAMSsubjectclassifications.34A09,34A12,65L05,65H10,65L081.Introduction.Manyphysicalproblemsaremosteasilyinitiallymodeledasanonlin-earimplicitsystemofdifferentialandalgebraicequations(DAEs),(1)f(t,
6、x,x')=0withfx'8f/Sx'identicallysingular[1].Avarietyofnumericalmethodshavebeende-velopedfor(1)rangingfrombackwarddifferentiation(BDF)toimplicitRunge-Kutta(IRK)methods[1],[17],[18],[21].Thesemethodsareonlydirectlysuitableforlowerindexprob-lemsandoftenrequirethat
7、theproblemhavespecialstructure.Althoughmanyimportantapplicationscanbesolvedbythesemethodsthereisaneedformoregeneralapproaches.Onemoregeneralapproachwasproposedin[4],[11],[13].Intheactualimplementationofthisapproach,onlytheoriginalequations(1)needtobedifferenti
8、ated.Allsubsequentcalculationscanbedonenumerically.Thismethodcanalsobeusedtofindinitialconditionsforusewithothermethods.Seealso[12].MostoftheresearchonDAEshasfocusedoncomputingasol
