LE230: Numerical Technique In Electrical Engineering

Transcription

1 LE30: Numricl Tchiqu I Elctricl Egirig Lctur : Itroductio to Numricl Mthods Wht r umricl mthods d why do w d thm? Cours outli. Numbr Rprsttio Flotig poit umbr Errors i umricl lysis Tylor Thorm

2 My dvic I you do t lt tchr ow t wht lvl you r by sig qustio, or rvlig your igorc you will ot lr or grow. You c t prtd or log, or you will vtully b oud out. Admissio o igorc is ot th irst stp i our ductio. Stv Covy Sv Hbits o Highly Ectiv Popl

3 Cours Objctivs Udrstd umricl tchiqus, i.., mig d sigiicc. Study umricl mthods, i.., Algorithms tht r usd to obti umricl solutios o mthmticl problm. Apply umricl mthods or solvig girig problms. 3

4 Epcttios I this cours, hopully you ll lr Fudmtls o umricl mthods Bsic umricl mthods,.g., solvig systm o qutios, umricl itgrtio, tc. Implmttio o umricl mthods Bsic Progrmmig Applictio o umricl mthods 4

5 How do w solv girig problm? Problm Dscriptio Mthmticl Modl Solutio o Mthmticl Modl Usig th Solutio 5

6 Why us Numricl Mthods? To solv problms tht cot b solvd lyticlly i.., ctly or lyticl solutio is diicult to obti or ot prcticl. π u du

7 Why us Numricl Mthods? To solv problms tht r itrctbl!

8 Wht do w d? Bsic Nds i th Numricl Mthods: Prcticl: C b computd i rsobl mout o tim. Accurt: Good pproimt to th tru vlu, Iormtio bout th pproimtio rror Bouds, rror ordr,. 8

9 Outlis o th Cours Tylor Thorm Numbr Rprsttio Solutio o olir Equtios Solutio o lir Equtios Rgrssio d Itrpoltio Numricl Dirtitio Numricl Itgrtio Solutio o ordiry dirtil qutios ODE Solutio o Prtil dirtil qutios PDE Eigvlu Problm Grph Thory d Applictios 9

10 Solutio o Nolir Equtios Som simpl qutios c b solvd lyticlly: Alytic solutio d roots 3 4 ± My othr qutios hv o lyticl solutio: No lytic solutio 0

11 Solutio o Systms o Lir Equtios 000 qutios i 000 uows. w hv Wht to do i 3, 5 3, 3 W c solv it s : 5 3

12 Crmr s Rul is Not Prcticl Crmr's Rul c b usd to solv th systm: 3 5, 3 5 But Crmr's Rul is ot prcticl or lrg problms. To solv N qutios with N uows, w d N NN! multiplictios. To solv 30 by 30 systm,.3 0 A supr computr ds mor th 0 35 multiplictios r dd. 0 yrs to comput this.

13 Curv Fittig : Rgrssio Giv st o dt: 0 y Slct curv tht bst its th dt. O choic is to id th curv so tht th sum o th squr o th rror is miimizd. 3

14 Curv Fittig : Itrpoltio Giv st o dt: i 0 y i Fid polyomil P whos grph psss through ll tbultd poits. y i P i i i is i th tbl 4

15 Itgrtio Som uctios c b itgrtd lyticlly: 3 But my uctios hv o lyticlsolutios : 0 d d?

16 Solutio o Ordiry Dirtil Equtios A solutio to th dirtil qutio : && t 3& t 3 t 0 & 0 ; 0 0 is uctio t tht stisis th qutios. * Alyticlsolutios r vilblor spcil css oly. 6

17 Solutio o Prtil Dirtil Equtios Prtil Dirtil Equtios r mor diicult to solv th ordiry dirtil qutios: u t u 0 u0, t u, t 0, u,0 si π 7

18 Rprstig Rl Numbrs You r milir with th dciml systm: Dciml Systm: Bs 0, Digits 0,,,9 Stdrd Rprsttios: ± sig itgr rctio prt prt 8

19 Normlizd Flotig Poit Rprsttio Normlizd Flotig Poit Rprsttio: ± sig d 0, d ± mtiss pot rctio ± : sigd pot Scitiic Nottio: Ectly o o-zro digit pprs bor dciml poit. Advtg: Eicit i rprstig vry smll or vry lrg umbrs. 9

20 Biry Systm Biry Systm: Bs, Digits {0,} ± sig. 3 4 mtiss ± sigd pot

21 Fct Numbrs tht hv iit psio i o umbrig systm my hv iiit psio i othr umbrig systm: You c vr rprst. ctly i biry systm.

22 IEEE 754 Flotig-Poit Stdrd Sigl Prcisio 3-bit rprsttio -bit Sig 8-bit Epot 3-bit Frctio S Epot 8 Frctio 3 Doubl Prcisio 64-bit rprsttio -bit Sig -bit Epot 5-bit Frctio S Epot Frctio 5 cotiud

23 Sigiict Digits Sigiict digits r thos digits tht c b usd with coidc. Sigl-Prcisio: 7 Sigiict Digits to Doubl-Prcisio: 5 Sigiict Digits to

24 Rmrs Numbrs tht c b ctly rprstd r clld mchi umbrs. Dirc btw mchi umbrs is ot uiorm Sum o mchi umbrs is ot cssrily mchi umbr 4

25 Clcultor Empl Suppos you wt to comput: *.39 usig clcultor with two-digit rctios 3.57 * Tru swr:

26 Sigiict Digits -Empl

27 Accurcy d Prcisio Accurcyis rltd to th closss to th tru vlu. Prcisiois rltd to th closss to othr stimtd vlus. 7

28 8

29 Roudig d Choppig Roudig: Rplc th umbr by th rst mchi umbr Roud-o Error Choppig: Throw ll tr digits. Tructio Error 9

30 Roudig d Choppig 30

31 Error Diitios Tru Error C b computd i th tru vlu is ow: Absolut Tru Error E t tru vlu pproimtio Absolut Prct Rltiv Error ε t tru vlu pproimtio tru vlu *00 3

32 Error Diitios Estimtd Error Wh th tru vlu is ot ow: Estimtd Absolut Error E currt stimt prvious stimt Estimtd Absolut Prct Rltiv Error ε currt stimt prvious stimt currt stimt *00 3

33 Nottio W sy tht th stimt is corrct to dciml digits i: Error 0 W sy tht th stimt is corrct to dciml digits roudd i: Error 0 33

34 Loss o Sigiict Digits Subtrctio o two rltivly clos umbrs c ld to loss o sigiict digits or sigiicc Empl: Suppos 7 sigiict digits , y y > sigiict digit 34

35 b Loss o Sigiict Digits Empl Cosidr th ollowig qudrtic qutio: b± b b c 0;, I Empl:, b., c. d ssum 7 sigiict digits: b 4c C us b >> 4c, b b 4c 34565>> 4c.08, , b 4c c c / b b 4c

36 36 Tylor Sris '! : writ c w covrg, sris th I!... 3!! : bout o psio Tylor sris Th Sris Tylor or

37 Mcluri Sris Mcluri sris is spcil cs o Tylor sris with th ctr o psio 0. Th I 0 th Mcluri sris ' 0 0! sris covrg, psio o 3 : ! w c writ : 0! 0 37

38 38 Mcluri Sris Empl. Th sris covrgs or... 3!!! 0! ' ' < or o psio sris Mcluri Obti

39 Tylor Sris 3 Empl.5 p

40 Mcluri Sris Empl Obti Mcluri sris psio o si : ' 3 si si cos si cos 0 0! 0 '0 Th sris covrgs or 3 < ! ! 7 7!... 40

41 4 3-3 /3! 5 /5! 0 si /3!

42 Covrgc o Tylor Sris Th Tylor sris covrgs st w trms r dd wh is r th poit o psio. I - is lrg, th mor trms r dd to gt good pproimtio. 4

43 Tylor s Thorm I uctio posssss drivtivs o ordrs o itrvl cotiig d th th vlu o,,..., is giv by : whr : R 0!! ξ d R ξ is trms Tructd Tylor Sris btw Rmidr d. 43

44 Tylor s Thorm W c pply Tylor's thorm or : with th poit o psio 0 i <. I, th th uctio d its drivtivs r ot did. Tylor Thorm is ot pplicbl. 44

45 Error Trm To gt id bout th pproimtio w c driv uppr boud o : R! ξ or ll vlus o ξ btw d. rror, 45

46 Error Trm -Empl How lrg is th rror i w rplcd th irst 4 trms 3o its Tylor sris psio t 0 wh 0.? R R ξ! 0.! ξ 0. R 8.468E or by 46

47 Altrtiv orm o Tylor s Thorm Lt o itrvl h hv drivtivs o 0 cotiig! h d R ordrs h,,..., th : h stp siz R ξ! h whr ξ is btw d h 47

48 48 Tylor s Thorm Altrtiv orms. d btw is!!,. d btw is!! 0 0 h whr h h h h whr ξ ξ ξ ξ

49 49 M Vlu Thorm ', 0, Us Tylor's Thorm or : Proo ', th thr ists, drivtiv is did o th op itrvl d its ], closd itrvl[ cotiuous uctio o is I b ξ b b h b b ξ b ξ b b

50 Altrtig Sris Thorm Cosidr th ltrtig sris : S I lim d L 4 L th Th sris covrgs d S S S : : Prtil sum sum o First omittd trm th irst trms 50

51 Altrtig Sris Empl si c b computd usig : si 3! This is covrgt ltrtig sris sic : Th : 3 si si 3! 3! 4 L 5! 5! d 7! lim 0 5! 7! L 5

52 5 Empl 3 Tylor Sris...! ! ! '0.5 ' , o psio Tylor sris Obti

53 53 Empl 3 Error Trm! m! 0.5! ! 3 [0.5,] Error Error Error Error ξ ξ ξ ξ