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Roundoff Truncotion Errors ond
ll), and
nry 29, t,2001
; the difrd miniiable In ction for
CHAPTER OBJECTIVES
The primaryobjective this chapter to acquaint you with the major sources of is of errorsinvolvedin nurnerical rnethods Speciticobjectives topicscovered and are
:loped at mentsof
' ' ' '
' ' ' t '
Understanding distinction the between and precision accuracy how to quantityerror Learning Learninghow errorcstimates be usedto decidewhen to terminate iterrti','e can an calculation Understanding roundofferrorsoccllrbccause how digitalcornputers havea limitedahility to represent nunrbers Understanding lloating-point why numbers have[irnitson their rangeand precision Recognizing that truncation errorsoccurwhenexactmathematical fbrmulations ilrc represcnlcd approx by irnat ions Knowing how to uscthe Taylorseliesto estimate truncation errors Understanding to write lbrwarcl, how backward and ccntered finite-difTerence approximations first and seconcl of derivatives R e c o g n i z i ntg a te l ' f i r r tls r n i n i m i z e" r u n c r r t i e r r o r s a ns o m e l i m eis c r c a s e h o l on c n roundofT errors
YOU'VE GOT A PROBLEM
jumperTo n Chap I, you developed numerical a model for the velocity of a bungee you had to approximate derivativeof velocity solve the problemwith a computer, the with a llnite difl-erence !]=!!dr Lt u(ti+r)-u(ri) t,+t - ti
R O U N D O FA N D T R U N C A T I OE R R O R S F N
Thus, the resultingsolution is not exact-that is, it has error In addition,the computeryou use to obtain the solutionis also an imperfecttool Beit is the cause is a digitaldevice, computer limitedin its ability to represent magnitudes the precision numbers Consequently, machine the itselfyieldsresults and of thatcontainenor your digital computercauseyour lrboth your mathenlatical So approxirnation and Your problemis: How do you deal with suchunsulting rnodelpredictionto be uncertain you to sonleapproaches concepts and that engineers and certainty' This chapterintroduces with this dilemma scientists to deal use
ERRORS
constantly find themselves havingto arccomplish objectives based i Engineers scientists and goal, it is rarely if ever atperf'ection a laudable is on uncertain information Although liom Newton'ssecond law tainedFor example, despite tact that the rnodeldeveloped the neverin practice predictthejumper'sfall is an excellent it cxactly approxirnation, would would resultin desuchas windsand slightvariations air rcsistance in A varietyof factors prediction thesedeviations systematically If high or low, thenwe viationsftom the are new modclHowever, rzrndomly and might needto devclopa il-theyare distributed tightly grouped prediction, rnightbe considered negligible thenthe deviations andthe aroundthe Numericalapproxirlations introduce similar discrepancies model deemedadequate also i n t o t h ea n a l y s i s quantification, rniniThis chaptercovcrs basictopicsrclated tlreidentification, to and with the quantification mizaticln these errors Ccneralinformation concerned ol'erroris of l-his is fbllowed by Sections 12 and43, dealingwith the two reviewedin this section (due error:rclundof'l'crror to computer and majorfirrmsof numerical approximations) trunWe how strategies reto cationerlor (dueto mathe nratical approximations) alsodescribe roundolT increase Finally,we briefly discuss crrorsnot duce truncation error sometinres Thessincludeblundcrs, with the nurnerical nrethods rnodel directlyconnected themselves errorsand datauncertaintv
I i I
411 Accurocy ond Precision with The errorsassociated with both calculations measuremenls be characterized can arrd and precision Att'unrc'v rcl'ers how closelya corlputedor meato regardto theiraccuracy ref'ers how closelyindividuitl to suredvalueagrees with thetruevaluePrct'1\'iorr computed or measured valuesagreewith sachother graphically Theseconcepts can be illustrated using an analogyfiom targetpractice of The bulletholeson eachtargetin Fig4 l can be thoughtof as the predictions a numer(alsocalledbirri)is represents truthInaccurucy ical technique, whereas bull's-eye the the defirrecl systematic deviationfrom the truthThus, althoughthe shotsin Fig 4lc are as because more tightly grouped than in Fig4la, thc two cases equallybiased arc theyate of both cerrtered the upperleft quadrant thc targetImltrecision(alsocalledurrc:ertaintl), on Therefore,althoughFig 4 lD and on the other hand,ref'ers the magnitudeo1'thescatter to (iecentered thc bull's-eye), latteris more precise on the because d are eqLrally accurate the shotsaretightly grouped
I e a
4I ERRORS
accuracy Increasing
olBeritudes n errol' our rerch unersand
o 0)
s based )ver atrnd law :r'sfall It in delhen we I tightly and the :pancies r c lm i n i 'error is the two s to reors not rlodel
FIGURE 4I tg illust o o o from Anexomple morksmonship r o t i n h e c o n c e p l s l o c c u r o c y n d p r e c i s n
(, (b ( o ) i n o c c u r o t c n d r m p r e c i s e , l o c c u r o t e n c ii n r p r e c i s e c Ji n c c c u r o t o n d p r e c i s e , e o e n - d { r J ) ( ' L r o ' ee r ', J p e c : s e o
("o r\umerrcalmethocls snoulcl sufiicrenrly oe accura("e unorused meet irc requtreor particular problemThey also shouldbe precise nrentsof a enoughfbr adequate design In tlris book, we will use thc collectiveterm errur to represent both the inaccuracy and i m p l c c i s i o nl l 'o u r p r e d i ci t n s t o
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