ResearchFields
Accordingtothetypesofmathematics,theresearchfieldsofnumericaloperationsincludenumericalapproximation,numericaldifferentiationandnumericalintegration,numericalalgebra,optimizationmethods,numericalsolutionsofordinarydifferentialequations,andintegralsEquationnumericalsolution,partialdifferentialequationnumericalsolution,computationalgeometry,computationalprobabilityandstatistics,etc.Withthewidespreadapplicationanddevelopmentofcomputers,manyproblemsinthefieldofcomputing,suchascomputationalphysics,computationalmechanics,computationalchemistry,computationaleconomics,etc.,canbeattributedtonumericalcomputationalproblems.
Importantfeatures
Numericalcalculationshavethefollowingfiveimportantcharacteristics:
1.Theresultsofnumericalcalculationsarediscreteandmusthaveerrors.Thisisanumericalvalue.Themaincharacteristicsofthedifferencebetweenthecalculationmethodandtheanalyticalmethod.
2.Payattentiontothestabilityofcalculation.Controllingthegrowthmomentumoferrorsandensuringthestabilityofthecalculationprocessisoneofthecoretasksofnumericalcalculationmethods.
3.Payattentiontofastcalculationspeedandhighcalculationaccuracyareimportantfeaturesofnumericalcalculation.
4.Payattentiontoconstructiveproof.
5.Numericalcalculationsmainlyusetheideaoffiniteapproximationtoperformerrorcalculations.
Numericalintegration
Numericalintegrationisanumericalmethodtofindtheapproximatevalueofadefiniteintegral,thatis,thediscreteorweightedaverageapproximatevalueofafinitenumberofsamplingvaluesoftheintegrandreplacesthevalueofthedefiniteintegral.Whencalculatingthedefiniteintegralofafunction,inmostcases,theoriginalfunctionoftheintegrandisdifficulttoexpresswithelementaryfunctions.Therefore,therearefewopportunitiestocalculatethedefiniteintegralwiththehelpoftheNewton-Leibnizformulaofcalculus..Inaddition,theintegrandfunctioninmanypracticalproblemsisoftenatabularfunctionorotherformsofdiscontinuousfunctions.Thedefiniteintegralofthistypeoffunctioncannotbesolvedbytheindefiniteintegralmethod.Fortheabovereasons,thetheoryandmethodofnumericalintegrationhasalwaysbeenthebasicsubjectofcomputationalmathematicsresearch.Mathematicsmasterswhohavemadeoutstandingcontributionstocalculus,suchasI.Newton,L.Euler,C.F.Gauss,etc.havealsomadetheirowncontributionsinthefieldofnumericalintegrationandlaiditstheoreticalfoundation.
ConstructingNumericalIntegration
Themostcommonmethodforconstructingnumericalintegrationformulaistoreplacetheintegrandfunctionwithn-thorderinterpolationpolynomialontheintegrationinterval.TheresultingquadratureformulaiscalledinterpolationTypequadratureformula.Especiallywhenthenodesareequallyspaced,itiscalledtheNewton-Coatesformula.Forexample,thetrapezoidalformulaandtheparabolicformulaarethemostbasicapproximateformulas.Buttheiraccuracyispoor.Romberg'salgorithmisamethodofweightedaverageoftheapproximatevalueofthetrapezoidalformulaintheprocessofsuccessivelydividingtheintervalintohalftoobtainamoreaccurateintegralapproximatevalue.Ithasconciseformulas,accuratecalculationresults,convenientuse,goodstability,etc.Advantages,soRomberg'squadratureformulashouldbeusedinthecaseofequidistance.Whencalculatingwithunequaldistancenodes,theGaussianquadratureformulaiscommonlyused.Underthesamenumberofnodes,theaccuracyishigh,thestabilityisgood,andtheinfiniteintegralcanbecalculated.Numericalintegrationisalsoanimportantbasisforthenumericalsolutionofdifferentialequations.Manyimportantformulascanbederivedusingnumericalintegralequations.