oIntroductionNumerical AnalysisGeneralstatement,There is nothing so wrong with the analysisas believing the answer!"Richard P.FeynmanClassificationof problemsexample:blunders/grosserrors:stronglymisleading resultslinearanalysiscomputationaltargetof astronglyresultsnonlinearproblemuncertainty/imprecision:scatter/spread aroundtheactual value1example:random initialisation ofthecomputationaltargetresultsstarting value in an iterationinaccuracy/bias:systematic deviation fromthe actual valuecomputationalexample:negligence ofa certainminoreffecttargetresultsin anapproximation solutionproblem sources:appropriateness and deficient quality ofmeasurements,data,humandecisionsand numerical processing9MichaelBeer,EngineeringMathematics
Michael Beer, Engineering Mathematics 9 General statement 0 Introduction Numerical Analysis „There is nothing so wrong with the analysis as believing the answer!“ Richard P. Feynman Classification of problems ● blunders / gross errors: strongly misleading results target computational results example: linear analysis of a strongly nonlinear problem target computational results target computational results example: negligence of a certain minor effect in an approximation solution example: random initialisation of the starting value in an iteration problem sources: appropriateness and deficient quality of measurements, data, human decisions and numerical processing ● inaccuracy / bias: systematic deviation from the actual value ● uncertainty / imprecision: scatter / spread around the actual value
IntroductionNumerical AnalysisGoalsexclusionof blunders/gross errorssufficiently good precision and accuracylownumericaleffort》highdiligenceinthenumericalformulation》applicationofsophisticatedconcepts/solutionschemes/algorithmsAlgorithmfinitesequenceofwell-definedinstructionsterminates with results in a defined end-state starting from an initial statenumerical procedureto solvea problemFormsof errors innumerical analysisround-off errors.truncationerrors(discretisationerrors)knowledgeaboutsources,magnitudeandpropagationoferrors10Michael Beer, EngineeringMathematics
10 0 Introduction Numerical Analysis Goals ● exclusion of blunders / gross errors ● sufficiently good precision and accuracy ● low numerical effort » high diligence in the numerical formulation » application of sophisticated concepts / solution schemes / algorithms Algorithm ● finite sequence of well-defined instructions ● terminates with results in a defined end-state starting from an initial state Forms of errors in numerical analysis ● round-off errors ● truncation errors (discretisation errors) knowledge about sources, magnitude and propagation of errors numerical procedure to solve a problem Michael Beer, Engineering Mathematics
1ErrorsinnumericalanalysisErrorDefinitionsErrors with respect to the true valueabsolutetrue errorE,=x,-X,x,=truevalue,X,=approximation》 no information abouttheorder of magnitudeof the errorexample:Et =1cmwith respectto length of a bridge,thicknessof a wallrelative trueerrorEX*0X》misleadinginformationif thetruevalueisclosetozeroexample:displacementx,=0.001m(minorinfluence oftheeffectunderconsideration),x.=0.0005m=et=0.5(50%)！true valuexisfrequently notknowninpractice11MichaelBeer,EngineeringMathematics
11 1 Errors in numerical analysis Error Definitions Errors with respect to the true value ● absolute true error » no information about the order of magnitude of the error, example: Et = 1cm with respect to length of a bridge, thickness of a wall ● relative true error » misleading information if the true value is close to zero example: displacement xt=0.001m (minor influence of the effect under consideration), xa=0.0005m ⇒ et=0.5 (50%) ! = ≠ t t t t E e , x 0 x E x x ; x true value, x approximation t ta t =− = = a true value xt is frequently not known in practice ! Michael Beer, Engineering Mathematics
1ErrorsinnumericalanalysisErrorDefinitionsErrorswithrespecttothebestavailableapproximationrelativeapproximateerrorXXarer O; best available approximation = reference value xa,refX》 misleading information if the reference value is close to zerorelative"improvement"ofanapproximationinaseries/iteration[i-1]0间Xa- X aAeXa +O, i=iteration counter0Xa》misleadinginformationif the current approximationisclosetozero example: × = jf(t)dt, ×=号2f((i- 0.5), f(t) = (t -2.1) - 3.28 , a= 4X,=0.000450100150200250300nEt0.01720.00430.00190.00070.00050.001110.754.782.691.721.1942.97et-0.017-0.042-0.022-0.016-0.019-0.041dea12MichaelBeer,EngineeringMathematics
12 Errors with respect to the best available approximation ● relative approximate error » misleading information if the reference value is close to zero − = ≠ a,ref a a a,ref a,ref a,ref x x e , x 0; best available approximation = reference value x x Error Definitions ● relative "improvement" of an approximation in a series / iteration » misleading information if the current approximation is close to zero − − ∆ = ≠ = [i] [i 1] [i] a a a a [i] a x x e , x 0, i iteration counter x ● example: ( ) = ≈ − ∑ n i 1 a a , x f i 0.5 n n n 50 100 150 200 250 300 Et 0.0172 0.0043 0.0019 0.0011 0.0007 0.0005 et 42.97 10.75 4.78 2.69 1.72 1.19 ∆ea −0.042 −0.022 −0.017 −0.016 −0.019 −0.041 1 Errors in numerical analysis = ∫ ( ) a 0 x f t dt ( ) =− − ( ) 4 , f t t 2.1 3.28 , a 4 = x 0.0004 t = Michael Beer, Engineering Mathematics
Errorsinnumericalanalysis1ErrorDefinitionsErrors with respect to the best available approximation (cont'd)example(cont'd)f(t), xf (t) = (t - 2.1)* - 3.2810x=jf(t)dt0t,a04X,e,E0.0650.01etDeaEt-0.015n80300n = 327,xa = -2.0.10-6XDea4.4.10-7n=328,xa=13MichaelBeer,EngineeringMathematics
13 Errors with respect to the best available approximation (cont'd) ● example (cont'd) Error Definitions 1 Errors in numerical analysis t, a f(t), x 0 4 10 ( ) =− − ( ) 4 f t t 2.1 3.28 = ∫ ( ) a 0 x f t dt 0 n 0.065 0 −0.015 80 300 Et 0.01et x, e, E Δea Δea xa n = 327, xa = −2.0⋅10−6 n = 328, xa = 4.4⋅10−7 Michael Beer, Engineering Mathematics