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CHAPTER 2: Numerical Approximation

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M
Maria Fernanda

Numerical approximation involves finding a number X' that approximates the value of another number X. While numerical solutions are not exact, the goal is to obtain a solution close to the real solution. Significant figures are important in numerical methods because they designate the reliable digits in a number. Numerical errors originate from approximating exact mathematical quantities and can be truncation errors from approximations or round-off errors from limited significant figures. Relative error provides a way to account for magnitude and the true percent relative error measures accuracy. The Taylor series can be used to approximate functions as polynomials using values and derivatives at other points.

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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 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?
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, π, cannotbeexpressedexactly.“Truncationerrors are thosethatresultfromusinganapproximation in place of anexactmathematicalprocedure.”TRUNCATION ERRORS
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:
BIBLIOGRAPHYCHAPRA, Steven. “Numericalmethodsforengineers”; McGraw Hill.ROCHA, Gustavo. “Métodos Numéricos”.2005

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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 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