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
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
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
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
How do w solv girig problm? Problm Dscriptio Mthmticl Modl Solutio o Mthmticl Modl Usig th Solutio 5
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
Why us Numricl Mthods? To solv problms tht r itrctbl!
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
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
Solutio o Nolir Equtios Som simpl qutios c b solvd lyticlly: 4 3 0 Alytic solutio d roots 3 4 ± 4 4 3 My othr qutios hv o lyticl solutio: 9 5 0 No lytic solutio 0
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
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.
Curv Fittig : Rgrssio Giv st o dt: 0 y 0.5 0.3.3 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
Curv Fittig : Itrpoltio Giv st o dt: i 0 y i 0.5 0.3 5.3 Fid polyomil P whos grph psss through ll tbultd poits. y i P i i i is i th tbl 4
Itgrtio Som uctios c b itgrtd lyticlly: 3 But my uctios hv o lyticlsolutios : 0 d d? 3 9 4 5
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
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
Rprstig Rl Numbrs You r milir with th dciml systm: 3.45 3 0 0 0 0 4 0 5 0 Dciml Systm: Bs 0, Digits 0,,,9 Stdrd Rprsttios: ± 3. 4 5 sig itgr rctio prt prt 8
Normlizd Flotig Poit Rprsttio Normlizd Flotig Poit Rprsttio: ± sig d 0, d. 3 4 0 ± 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
Biry Systm Biry Systm: Bs, Digits {0,} ± sig. 3 4 mtiss ± sigd pot.0 0 3 0.65 0 0
Fct Numbrs tht hv iit psio i o umbrig systm my hv iiit psio i othr umbrig systm:. 0.000000 000... You c vr rprst. ctly i biry systm.
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
Sigiict Digits Sigiict digits r thos digits tht c b usd with coidc. Sigl-Prcisio: 7 Sigiict Digits.75494 0-38 to 3.4083 0 38 Doubl-Prcisio: 5 Sigiict Digits.50738 0-308 to.797693 0 308 3
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
Clcultor Empl Suppos you wt to comput: 3.578 *.39 usig clcultor with two-digit rctios 3.57 *.3 7.60 Tru swr: 7.65334 5
Sigiict Digits -Empl 48.9 6
Accurcy d Prcisio Accurcyis rltd to th closss to th tru vlu. Prcisiois rltd to th closss to othr stimtd vlus. 7
8
Roudig d Choppig Roudig: Rplc th umbr by th rst mchi umbr Roud-o Error Choppig: Throw ll tr digits. Tructio Error 9
Roudig d Choppig 30
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
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
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
Loss o Sigiict Digits Subtrctio o two rltivly clos umbrs c ld to loss o sigiict digits or sigiicc Empl: Suppos 7 sigiict digits 0.34567, y 0.34566 y 0.000000 -> sigiict digit 34
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, 4.8484, b 4c 0.000000 4c 34560 0.00009 c / b b 4c 0.00009
36 Tylor Sris 0 0 3 3 '! : writ c w covrg, sris th I!... 3!! : bout o psio Tylor sris Th Sris Tylor or
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 : 0 3... 3! w c writ : 0! 0 37
38 Mcluri Sris Empl. Th sris covrgs or... 3!!! 0! 0 0 0 ' ' 0 3 0 0 < or o psio sris Mcluri Obti
Tylor Sris 3 Empl.5 p 0.5.5 0.5 0 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 39
Mcluri Sris Empl Obti Mcluri sris psio o si : ' 3 si si cos si cos 0 0! 0 '0 Th sris covrgs or 3 < 0 0 0 3 3!. 0 5 5! 7 7!... 40
4 3-3 /3! 5 /5! 0 si - - 3 /3! - -3-4 -4-3 - - 0 3 4 4
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
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
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
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
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 05 3 0. or by 46
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 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 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
Altrtig Sris Thorm Cosidr th ltrtig sris : S I lim d 3 3 0 4 L 4 L th Th sris covrgs d S S S : : Prtil sum sum o First omittd trm th irst trms 50
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
5 Empl 3 Tylor Sris...! 0.5...! 0.5 4 0.5 0.5! 0.5 0.5 4 0.5 4 '0.5 ' 0.5 0 0.5, o psio Tylor sris Obti
53 Empl 3 Error Trm! m! 0.5! 0.5 0.5! 3 [0.5,] Error Error Error Error ξ ξ ξ ξ