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# Chapter 2: Numerical Approximation

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### Chapter 2: Numerical Approximation

1. 1. CHAPTER II<br />NUMERICAL APPROXIMATION<br />BY: MARIA FERNANDA VERGARA M.<br />UNIVERSIDAD INDUSTRIAL DE SANTANDER<br />
2. 2. NUMERICAL APPROXIMATION<br />A numericalapproximationis a number X’ thatrepresentsanothernumberwhichitsexactvalueis X. X’ becomes more exactwhenisclosertotheexactvalue of X<br />Isimportanttotakeintoaccountthisnumericalapproximationbecausenumericalsolutions are notexact, butthemainobjectiveistoget a solutionreallyclosetothe real solution.<br />
3. 3. SIGNIFICANT FIGURES<br />“The concept of a significant figure, ordigit, has beendevelopedtoformallydesignatethereliability of a numericalvalue. Thesignificantdigits of a number are thosethat can beusedwithconfidence. Theycorrespondtothenumber of certaindigits plus oneestimateddigit.”-Numericalmethodsforengineers, CHAPRA-.<br />Whysignificant figures are important in numericalmethods?<br />
4. 4. ACCURACY AND PRECISION<br />
5. 5. ERROR DEFINITIONS<br />Numericalerrorsoriginatewhenyouapproximatetorepresentexactmathematicalquantitiesoroperations. Thiserrors can be: Truncationerrorswhichhappenwhenapproximations are usedtorepresentmathemathicalprocedures; and round-off errorswhichhappenwhenyou use numberswithlimitedsignificant figures toexpressexactnumbers.<br />ET=Real Value - Approximation<br />
6. 6. RELATIVE ERROR<br />Relative error is a waytoaccountforthe magnitudes of thequantitiesbeingevaluated<br />True percentrelative error<br />
7. 7. EXAMPLE EXERCISE<br />Themeasure of a bridge is 9999cm, and themeasure of a rivetis 9 cm, ifthe true values are 10.000cm and 10cm, respectively, compute the true error and the true percentrelative error foreach case.<br />
8. 8. In real worldapplications, wewillnotknowthe true value. So theprocedureistonormalizethe error usingthebestavaliableestimate of the true value:<br />Usinaniterativeapproachto compute answers, theapproximatedrelative error<br />
9. 9. ROUND-OFF ERRORS<br />Thiskind of errorsoriginatebecausecomputers can retain a finitenumber of significant figures, so numbers as e, π, cannotbeexpressedexactly.<br />“Truncationerrors are thosethatresultfromusinganapproximation in place of anexactmathematicalprocedure.”<br />TRUNCATION ERRORS<br />
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