《结构工程中的数学方法》课程教学课件（讲稿）Topic_1_Errors

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

1 Errors in numericalanalysisRound-off ErrorsErrorsourcelimited number of digitsof machine numbersRepresentationofmachinenumbers(16-bitcomputer)base-2system(binary),16-bitcomputer(wordlength=16bits)·integerrepresentation28272625213212211210292423222120214sign》range of numbers covered:-((215-1)+1),..., +((215-1)-32,768,..., +32,767.overflowoverflowfloating-point representation》according to standard IEEE754(IEEE=Institute of Electrical and Electronics Engineers)》(number)=(sign)(mantissa)·(base)^(exponent)restriction:mantissa e[1,2]base = 2, mantissa =1+ 2a -2*,a (0,1)k=14Michael Beer, Engineering Mathematics

14 Error source ● limited number of digits of machine numbers Round-off Errors Representation of machine numbers (16-bit computer) sign 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20 ● base-2 system (binary), 16-bit computer (word length = 16 bits) » range of numbers covered: −((215−1)+1), ., +((215−1) −32,768, ., +32,767 ● integer representation overflow overflow » (number) = (sign) · (mantissa) · (base) ^ (exponent) ● floating-point representation { } − = = =+ ⋅ ∈ ∑ p k k k k 1 base 2, mantissa 1 a 2 ,a 0,1 restriction: mantissa ∈ [1, 2) » according to standard IEEE 754 (IEEE = Institute of Electrical and Electronics Engineers) 1 Errors in numerical analysis Michael Beer, Engineering Mathematics

1ErrorsinnumericalanalysisRound-off ErrorsRepresentation of machine numbers (cont'd)floating-point representation (cont'd)signsigned exponentmantissa1 bitrbitsp bits》single precision formatsize: 32 bits; r = 8, p = 23gap between two consecutivemantissa numbers:=2-23~1.192.10-7precision:6-7 decimal digitsrange of numbers covered (approx.): ± 2-126±2+128±1.175·10-38..,±3.403·10+38(in addition to zero)underflowoverflow》doubleprecisionformatsize: 64 bits; r = 11, p = 52; gap: = 2-52~2.220.10-16precision:15-16decimaldigitsrange of numbers covered (approx.): ± 2-1022,±2+1024±2.225·10-308,.,±1.798·10+308(in addition to zero)underflowoverflow》quadprecisionformat(underconstruction inIEEE754r);size:128 bit15Michael Beer,Engineering Mathematics

15 Round-off Errors range of numbers covered (approx.): ± 2−126 , ., ± 2+128 (in addition to zero) ± 1.175·10−38 , ., ± 3.403·10+38 underflow overflow sign signed exponent mantissa 1 bit r bits p bits Representation of machine numbers (cont'd) ● floating-point representation (cont'd) » single precision format size: 32 bits; r = 8, p = 23 gap between two consecutive mantissa numbers: ε = 2−23 ≈ 1.192·10−7 precision: 6−7 decimal digits range of numbers covered (approx.): ± 2−1022 , ., ± 2+1024 (in addition to zero) ± 2.225·10−308 , ., ± 1.798·10+308 underflow overflow » double precision format size: 64 bits; r = 11, p = 52; gap: ε = 2−52 ≈ 2.220·10−16 precision: 15−16 decimal digits » quad precision format (under construction in IEEE 754r); size: 128 bit 1 Errors in numerical analysis Michael Beer, Engineering Mathematics

1 Errors in numericalanalysisRound-offErrorsCharacteristicsofmachinenumbers.discrete finite countable set D of available numerical representations,combinatorialproblem:/D/≤264=1.845.1019》compare:the set R of real numbers is connected,infiniteand uncountable;ID<R》exact representation of a truereal numberis an exceptional case!!!》eachindividualcomputeroperationfl.lisperformedwithapproximated numbersx,andyieldsanapproximated resultYa:y =ya(f[xa1(Xt1), Xa2(Xt2)]) with Xt1,Xt2 eRandXa1,Xa2,YaEID)》transformationfromIR to IDleadsto errorsDefinition:round-off error.difference between thetrue real numberXtand its computerapproximationXarEro = Xt - Xa(xt) with Xt E R and xa(xt) e D; (form of quantisation error)》Erincreaseswithmagnitudeofxt(constantbutincreasingexponent)16Michael Beer, Engineering Mathematics

16 Round-off Errors Characteristics of machine numbers ● discrete finite countable set  of available numerical representations, combinatorial problem: || ≤ 264 ≈ 1.845·1019 » compare: the set  of real numbers is connected, infinite and uncountable; ||«|| 1 Errors in numerical analysis Definition: round-off error ● difference between the true real number xt and its computer approximation xa, Ero = xt − xa(xt) with xt ∈  and xa(xt) ∈ ; (form of quantisation error) » Ero increases with magnitude of xt (constant ε but increasing exponent) » exact representation of a true real number is an exceptional case !!! » each individual computer operation f[.] is performed with approximated numbers xa and yields an approximated result ya: y = ya(f[xa 1(xt 1), xa 2(xt 2)]) with xt 1, xt 2 ∈  and xa 1, xa 2,ya ∈  » transformation from  to  leads to errors Michael Beer, Engineering Mathematics

1Errorsin numericalanalysisRound-offErrorsMachineapproximation of realnumberschopping:decimal digits that arenot covered bythe machine numbersarenottakenintoaccount,theyaresimplyomittedrounding:the last decimal digit d,that is covered by the machinenumbers is roundedin dependence on the value of the consecutive digit dk+i (first digit thatis not covered):》dk=dkifdk+15》dk=dk+1ifdk+1≥5example:machinerepresentationof元》truevalue:元t=0.314159265358979323846264338327950288...101》approximations:choppingrounding元=0.3141592.101元=0.3141593·101single precision(k=7 digits)(ero = 2.08·10-7)(ero =-1.10·10-7)doubleprecision元=0.314159265358979.101元=0.314159265358979.101(k=15 digits)(er。= 1.03·10-15)(ero = 1.03·10-15)17MichaelBeer,EngineeringMathematics

17 Round-off Errors Machine approximation of real numbers ● chopping: decimal digits that are not covered by the machine numbers are not taken into account, they are simply omitted ● rounding: the last decimal digit dk that is covered by the machine numbers is rounded in dependence on the value of the consecutive digit dk+1 (first digit that is not covered): » dk = dk if dk+1 < 5 » dk = dk+1 if dk+1 ≥ 5 1 Errors in numerical analysis chopping rounding single precision π = 0.3141592·101 π = 0.3141593·101 (k=7 digits) (ero = 2.08·10−7) (ero = −1.10·10−7) double precision π = 0.314159265358979·101 π = 0.314159265358979·101 (k=15 digits) (ero = 1.03·10−15) (ero = 1.03·10−15) ● example: machine representation of π » true value: πt = 0.314159265358979323846264338327950288. ·101 » approximations: Michael Beer, Engineering Mathematics

1 Errors in numericalanalysisRound-off ErrorsComputationswithmachinenumbersproblematiccases》operationsthatincreasefasttheabsolute value oftheexponent=underflow/overflowerrorsexample:(0.45·10-6)^(0.83·102)~1.64679·10-5270</1.64679.10-527|<2.225-10-308(doubleprecision)=underflow》large number of consecutive/recursive operations=erroraccumulationexample:Xi+1 = Xj (1-200·cos(y)), i = 0,..., n,Xo = 1recursive operation:supposethaty=元/2= Xn=1Vnsingleprecision,dk=7:n=100:Xn=0.1013555.101,ero=-0.013n=1000:x,=0.1144119.101,er。=-0.144!!!18Michael Beer, Engineering Mathematics

18 Round-off Errors Computations with machine numbers 1 Errors in numerical analysis ● problematic cases » operations that increase fast the absolute value of the exponent ⇒ underflow / overflow errors example: (0.45·10−6)^(0.83·102) ≈ 1.64679·10−527 0<|1.64679·10−527|< 2.225·10−308 (double precision) ⇒ underflow » large number of consecutive / recursive operations ⇒ error accumulation example: recursive operation: xi+1 = xi · (1−200·cos(y)), i = 0, ., n, x0 = 1 suppose that y = π/2 ⇒ xn = 1 "n single precision, dk = 7: n = 100: xn = 0.1013555·101, ero = −0.013 n = 1000: xn = 0.1144119·101, ero = −0.144 !!! Michael Beer, Engineering Mathematics