1. CHAPTER II NUMERICAL APPROXIMATION BY: MARIA FERNANDA VERGARA M. UNIVERSIDAD INDUSTRIAL DE SANTANDER

2. NUMERICAL APPROXIMATION A numericalapproximationis a number X’ thatrepresentsanothernumberwhichitsexactvalueis X. X’ becomes more exactwhenisclosertotheexactvalue of X Isimportanttotakeintoaccountthisnumericalapproximationbecausenumericalsolutions 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?

4. ACCURACY AND PRECISION

5. ERROR DEFINITIONS Numericalerrorsoriginatewhenyouapproximatetorepresentexactmathematicalquantitiesoroperations. Thiserrors can be: Truncationerrorswhichhappenwhenapproximations are usedtorepresentmathemathicalprocedures; and round-off errorswhichhappenwhenyou use numberswithlimitedsignificant figures toexpressexactnumbers. ET=Real Value - Approximation

6. RELATIVE ERROR Relative error is a waytoaccountforthe magnitudes of thequantitiesbeingevaluated True percentrelative error

7. EXAMPLE EXERCISE 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.

8. In real worldapplications, wewillnotknowthe true value. So theprocedureistonormalizethe error usingthebestavaliableestimate of the true value: Usinaniterativeapproachto compute answers, theapproximatedrelative error

9. ROUND-OFF ERRORS Thiskind of errorsoriginatebecausecomputers can retain a finitenumber of significant figures, so numbers as e, π, cannotbeexpressedexactly. “Truncationerrors are thosethatresultfromusinganapproximation in place of anexactmathematicalprocedure.” TRUNCATION ERRORS