# IE4004 Errors.pdf - IE4004 NUMERICAL ANALYSIS FOR...

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IE4004NUMERICAL ANALYSISFOR INDUSTRIAL ENGINEERSAssist. Prof. ilayda ÜLKÜ2/4/20202019/2020 Spring
REVIEW OF CALCULUSANDMATHEMATICAL PRELIMINARIES
IntroductionList of certain analytical results to which reference will be madewe deal with real and continuous functions in the range consideredand also they possess as many continuous derivatives as requiredDefinition of limit: letfbe a function defined on a setXof real numbers,fis said to have the limitLatx0and writtenIf, given any real number> 0, there exists a real number> 0 such that|f(x)L| <and0 < |xx0| <Lxfxx=)(lim0
Definition of Continuity: functionfis said to be continuous atx0ifDefinition of convergence: Letbe an infinite sequence of realor complex numbers.The sequence is said to converge to a numberx, if for any> 0, thereexists a positive integerN()such thatn>N()implies| x0x| <The notationmeans that the sequence converges to x)()(lim00xfxfxx==1}{nnxxxnx=lim
Definition of differentiability: iffis a function defined in an openinterval containingx0,fis said to be differentiable atx0, ifexists. When this limit exists it is denotedf´(x0)and is called thederivative offatx0A function that has a derivative at eachnumber in a set is said to be differentiable onX.000)()(limxxxfxfxx
ERROR DEFINITIONS
IntroductionFinal results of computations are generallyapproximationsand involveerrors.Ifãis an approximate value of a quantitya(true value), the error is definedas;error = true value - approximate value=aã(sometimes refered asabsolute error)Relative error,rofãis defined byrelative error = absolute error/true valuer=/aFor example,ã=10.5 is an approximation ofa=10.2,its error is=-0.3
In practice anerror boundis defined by|| Which tells us how far away the computed valueãtheunknownacan at most lie.Similarly for the relative error, an error bound isanumber defined by|r| r
Error AnalysisPossible sources of errors in numerical analysisTruncation errorRound-off errorError in original dataProgramming bugsPropagated error
Truncation ErrorTruncation error refers to the error involved in using a truncated or

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lim, Prof layda LK