CHAPTER 2: Numerical Approximation
- 1. CHAPTER IINUMERICAL APPROXIMATIONBY: MARIA FERNANDA VERGARA M.UNIVERSIDAD INDUSTRIAL DE SANTANDER
- 2. NUMERICAL APPROXIMATIONA numericalapproximationis a number X’ thatrepresentsanothernumberwhichitsexactvalueis X. X’ becomes more exactwhenisclosertotheexactvalue of XIsimportanttotakeintoaccountthisnumericalapproximationbecausenumericalsolutions are notexact, butthemainobjectiveistoget a solutionreallyclosetothe real solution.
- 3. SIGNIFICANT FIGURES“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-.Whysignificant figures are important in numericalmethods?
- 5. ERROR DEFINITIONSNumericalerrorsoriginatewhenyouapproximatetorepresentexactmathematicalquantitiesoroperations. Thiserrors can be: Truncationerrorswhichhappenwhenapproximations are usedtorepresentmathemathicalprocedures; and round-off errorswhichhappenwhenyou use numberswithlimitedsignificant figures toexpressexactnumbers.ET=Real Value - Approximation
- 6. RELATIVE ERRORRelative error is a waytoaccountforthe magnitudes of thequantitiesbeingevaluatedTrue percentrelative error
- 7. EXAMPLE EXERCISEThemeasure 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.
- 8. In real worldapplications, wewillnotknowthe true value. So theprocedureistonormalizethe error usingthebestavaliableestimate of the true value:Usinaniterativeapproachto compute answers, theapproximatedrelative error
- 9. ROUND-OFF ERRORSThiskind of errorsoriginatebecausecomputers can retain a finitenumber of significant figures, so numbers as e, π, cannotbeexpressedexactly.“Truncationerrors are thosethatresultfromusinganapproximation in place of anexactmathematicalprocedure.”TRUNCATION ERRORS
- 10. THE TAYLOR SERIESThe Taylor series provides a meanstofind a functionvalue in a point, usingthefunctionvalue and itsderivatives in anotherpoint. Thetheoremsaysthatanysmoothfunction can beapproximated as polynomial.Taylor’sTheorem: Ifthefunction f and itsfirst n+1 derivatives are continuous in anintervalcontaining a and x, thenthevalue of thefunction at x isgivenbyWhere:
- 11. BIBLIOGRAPHYCHAPRA, Steven. “Numericalmethodsforengineers”; McGraw Hill.ROCHA, Gustavo. “Métodos Numéricos”.2005