Číselné výpočty

ResearchFields

Accordingtothetypesofmathematics,theresearchfieldsofnumericaloperationsincludenumericalapproximation,numericaldifferentiationandnumericalintegration,numericalalgebra,optimizationmethods,numericalsolutionsofordinarydifferentialequations,andintegralsEquationnumericalsolution,partialdifferentialequationnumericalsolution,computationalgeometry,computationalprobabilityandstatistics,etc.Withthewidespreadapplicationanddevelopmentofcomputers,manyproblemsinthefieldofcomputing,suchascomputationalphysics,computationalmechanics,computationalchemistry,computationaleconomics,etc.,canbeattributedtonumericalcomputationalproblems.

Důležité vlastnosti

Číselné výpočty mají následujících pět důležitých charakteristik:

1.Theresultsofnumericalcalculationsarediscreteandmusthaveerrors.Thisisanumericalvalue.Themaincharacteristicsofthedifferencebetweenthecalculationmethodandtheanalyticalmethod.

2.Payattentiontothestabilityofcalculation.Controllingthegrowthmomentumoferrorsandensuringthestabilityofthecalculationprocessisoneofthecoretasksofnumericalcalculationmethods.

3.Payattentiontofastcalculationspeedandhighcalculationaccuracyareimportantfeaturesofnumericalcalculation.

4. Pozor na konstruktivní důkaz.

5.Číselné výpočty používají k provádění chybných výpočtů hlavně myšlenku​​konečné aproximace.

Numericalintegration

Numericalintegrationisanumericalmethodtofindtheapproximatevalueofadefiniteintegral,thatis,thediscreteorweightedaverageapproximatevalueofafinitenumberofsamplingvalues​​oftheintegrandreplacesthevalueofthedefiniteintegral.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.