efficient algorithms for the matrix cosine and sinenew

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1、NumericalAlgorithms(2005)40:383400DOI10.1007/s11075-005-8141-0Springer2005Efficientalgorithmsforthematrixcosineandsine∗GarethI.Hargreaves∗∗andNicholasJ.Higham∗∗∗SchoolofMathematics,UniversityofManchester,SackvilleStreet,Manchester,M601QD,UKE-mail:hargreaves@ma.man.ac.uk,http://www.ma.man.ac.

2、uk/~hargreaves/higham@ma.man.ac.uk,http://www.ma.man.ac.uk/~higham/Received4February2005;accepted16April2005CommunicatedbyC.BrezinskiSeveralimprovementsaremadetoanalgorithmofHighamandSmithforcomputingthematrixcosine.Theoriginalalgorithmscalesthematrixbyapowerof2tobringthe∞-normto1orless,eval

3、uatesthe[8/8]Padéapproximant,thenusesthedouble-angleformulacos(2A)=2cos2A−Itorecoverthecosineoftheoriginalmatrix.ThefirstimprovementistophrasetruncationerrorboundsintermsofA21/2insteadofthe(nosmallerandpoten-tiallymuchlargerquantity)A.ThesecondistochoosethedegreeofthePadéapproximanttomini

4、mizethecomputationalcostsubjecttoachievingadesiredtruncationerror.Athirdimprovementistouseanabsolute,ratherthanrelative,errorcriterioninthechoiceofPadéapproximant;thisallowstheuseofhigherdegreeapproximantswithoutworseninganapri-orierrorbound.Ourtheoryandexperimentsshowthateachofthesemodificat

5、ionsbringsareductionincomputationalcost.Moreover,becausethemodificationstendtoreducethenumberofdouble-anglestepstheyusuallyresultinamoreaccuratecomputedcosineinfloat-ingpointarithmetic.Wealsoderiveanalgorithmforcomputingbothcos(A)andsin(A),byadaptingtheideasdevelopedforthecosineandintertwining

6、thecosineandsinedoubleanglerecurrences.Keywords:matrixfunction,matrixcosine,matrixsine,matrixexponential,Taylorse-ries,Padéapproximation,Padéapproximant,double-angleformula,roundingerroranalysis,SchurParlettmethod,MATLABAMSsubjectclassification:65F301.IntroductionThematrixexponential,undoubte

7、dlythemost-studiedmatrixfunction,providesthesolutiony(t)=eAytothefirstorderdifferentialsystemdy/dt=Ay,y(0)=y,00whereA∈Cn×nandy∈Cn.Trigonometricmatrixfunctionsplayasimilarrolein∗NumericalAnalysisReport461,ManchesterCentreforComputationalMathematics,F