CHAPTER II<br />NUMERICAL APPROXIMATION<br />BY: MARIA FERNANDA VERGARA M.<br />UNIVERSIDAD INDUSTRIAL DE SANTANDER<br />
NUMERICAL APPROXIMATION<br />A numericalapproximationis a number X’ thatrepresentsanothernumberwhichitsexactvalueis X. X’ ...
SIGNIFICANT FIGURES<br />“The concept of  a significant figure, ordigit, has beendevelopedtoformallydesignatethereliabilit...
ACCURACY AND PRECISION<br />
ERROR DEFINITIONS<br />Numericalerrorsoriginatewhenyouapproximatetorepresentexactmathematicalquantitiesoroperations. Thise...
RELATIVE ERROR<br />Relative error is a waytoaccountforthe magnitudes of thequantitiesbeingevaluated<br />True percentrela...
EXAMPLE EXERCISE<br />Themeasure of a bridge is 9999cm, and themeasure of a rivetis 9 cm, ifthe true values are 10.000cm a...
In real worldapplications, wewillnotknowthe true value. So theprocedureistonormalizethe error usingthebestavaliableestimat...
ROUND-OFF ERRORS<br />Thiskind of errorsoriginatebecausecomputers can retain a finitenumber of significant figures, so num...
Upcoming SlideShare
Loading in...5
×

Chapter 2: Numerical Approximation

613

Published on

Published in: Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
613
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
14
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

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 />
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×