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Chapter 2: Numerical Approximation

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M
Maria Fernanda

This document discusses numerical approximation and related concepts in computational methods. It defines numerical approximation as a number that represents an exact value. Numerical solutions are not exact but the goal is to get very close to the real solution. It also discusses significant figures, which designate the reliable digits of a numerical value. Accuracy and precision are distinguished, where errors can originate from approximations used to represent mathematical operations and numbers with limited significant figures. Relative error and true percent relative error are introduced as ways to account for the magnitudes of quantities being evaluated. Round-off errors from retaining a finite number of significant figures in computers are also mentioned.

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CHAPTER IINUMERICAL APPROXIMATIONBY: MARIA FERNANDA VERGARA M.UNIVERSIDAD INDUSTRIAL DE SANTANDER
NUMERICAL APPROXIMATIONA numericalapproximationis a number X thatrepresentsanothernumberwhichitsexactvalueis X. X becomes more exactwhenisclosertotheexactvalue of XIsimportanttotakeintoaccountthisnumericalapproximationbecausenumericalsolutions are notexact, butthemainobjectiveistoget a solutionreallyclosetothe real solution.
SIGNIFICANT FIGURESThe 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?
ACCURACY AND PRECISION
ERROR DEFINITIONSNumericalerrorsoriginatewhenyouapproximatetorepresentexactmathematicalquantitiesoroperations. Thiserrors can be: Truncationerrorswhichhappenwhenapproximations are usedtorepresentmathemathicalprocedures; and round-off errorswhichhappenwhenyou use numberswithlimitedsignificant figures toexpressexactnumbers.ET=Real Value - Approximation
RELATIVE ERRORRelative error is a waytoaccountforthe magnitudes of thequantitiesbeingevaluatedTrue percentrelative error
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.
In real worldapplications, wewillnotknowthe true value. So theprocedureistonormalizethe error usingthebestavaliableestimate of the true value:Usinaniterativeapproachto compute answers, theapproximatedrelative error
ROUND-OFF ERRORSThiskind of errorsoriginatebecausecomputers can retain a finitenumber of significant figures, so numbers as e, , cannotbeexpressedexactly.Truncationerrors are thosethatresultfromusinganapproximation in place of anexactmathematicalprocedure.TRUNCATION ERRORS

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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 FIGURESThe 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, , cannotbeexpressedexactly.Truncationerrors are thosethatresultfromusinganapproximation in place of anexactmathematicalprocedure.TRUNCATION ERRORS