(1.3.1)--Chapter1MathematicalModelingandE.ppt

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1、1.3NumericalmethodsandalgorithmsevaluationAnumericalmethodisaprocedurethatcansolveamathmodel(mathproblem)asasequenceofarithmeticandlogicoperationsthatcanberealizedonacomputer;andanalyzetheconvergenceanderrorsfromdataandcomputer.Analgorithmmentionedhereisnotjustasimplemathematicalformula,butalsorefer

2、stotheentireproblem-solvingschemeandstepscomposedofthebasicoperationsandtheorderofoperations.Howtochooseagoodalgorithmisaveryimportantpartinnumericalcomputations.AlgorithmdefinitionForexample,whenevaluatingapolynomial,Using a bad algorithm,not only affects the speed and efficiency of thecomputations

3、,butalsodirectlyaffectstheaccuracyofthecomputedresultsduetotheerrorpropagationandaccumulationsofthedataandoperationerrors,andevenleadstofailureofthecomputation.Badalgorithmsmaycauseerrorstoreachunacceptablelevelsthatmayhaltthecomputationorproduceuseless(calledjunk)results.Problems caused by bad algo

4、rithms are refereed as numericalstabilityproblemofthealgorithm.Belowweuseasimpleexampletoillustratehowalgorithmswillaffecttheresults,sometimesitmayhavehugeimpactontheaccuracyofcomputedresults.Ex puteSolution:IfweusetheapproximationsandWhichonewouldyouchoose?Applyingtheabovefouralgorithmstocomputex,t

5、heresultsareshowninTable1.3.1below.Table1.3.1ComparisonoffourdifferentalgorithmsNo.AlgorithmComputedResults1234WecanseefromTable1.3.1thattheresultscomputedbydifferentformulasandapproximationsaredifferent.Therefore,whilestudyingalgorithms,wemustalsounderstandbasicconceptsoferrorsandtheirpropagationso

6、fbasicoperationsandapproximations;andstudyofnumericalstabilityofalgorithms.Otherwise,areasonablygoodalgorithmmayalsogenerateawrongresult.Tojudgeagoodorbadalgorithm,thecomputationaltimeisakeyindicator.SincetheactualCPUtimedependscomputers,thecomputationaltimeismeasuredbyalloperations counts such as a

7、rithmetic and logic operations.In this way,computationaltimeisdirectlyrelatedtothenumberofoperations.Ofcourse,thecomputational time of an algorithm will be different for same type of problemsbecause the data of each problem is different.Generally,we consider thecomputationaltimeusingtheworst-casesce

8、nariointhediscussion.1.3NumericalmethodsandalgorithmsevaluationPopupQuestions：1.Numericalalgorithmrefersto(d)theentireproblem-solvingplanandsteps;a.Additionsandsubtractions;b.Addition,subtractionandlogicaloperations;c.Multiplicationsanddivisions;d.Basicoperationsandorder.2.Donotconsiderwhichofthefollowingfactorswhenevaluatingandselectingalgorithms?（c）a.Speedandefficiencyofcomputation;b.Theapproximatingpropertyofcomputernumbersystems;c.Errorinmodels;d.Errorpropagationandaccumulations.