MAT641- Numerical Analysis (Master course) D. Samy MZIOU

Transcription

1 Syllbus: MAT64- Numericl Alysis Mster course D. Smy MZIOU Review: Clculus lier Algebr Numericl Alysis MATLAB sotwre will be used itesively i this course There will be regulr homewor ssigmets, usully computtiol, but with lots o reedom. Submit the solutios o time preerbly erly, preerbly s PDF give LTe editor try!. Alwys submit your codes.

2 TOPIC Itroductio to Numericl Methods Errors d Numbers represettio

3 Lecture Itroductio to Numericl Methods Wht is umericl Alysis? Wht re NUMERICAL METHODS? Why do we eed them? 3

4 Wht is umericl lysis Wrog deiitio: Numericl lysis is the study o roudig errors Trehese deiitio: Numericl lysis is the study o lgorithms or the problems o cotiuous mthemtics Atiso deiitio: Numericl lysis is the re o mthemtics d computer sciece tht cretes, lyzes, d implemets lgorithms or solvig umericlly the problems o cotiuous mthemtics. 4 Numericl lysis - Scholrpedi

5 Numericl Methods: Numericl Methods Algorithmic methods used to obti umericl solutios o cotiuous mthemticl problem. Although there re my ids o umericl methods, they hve oe commo chrcteristic: they ivribly ivolve lrge umbers o tedious rithmetic clcultios. It is little woder tht with the developmet o st, eiciet digitl computers, the role o umericl methods i egieerig problem solvig hs icresed drmticlly i recet yers. Why do we eed them?. No lyticl solutio eists,. A lyticl solutio is diicult to obti sometimes impossible d is ot prcticl. 3. Grphicl methods re ot precise d useless i high order 5 dimesio >3

6 Alyticl vs. Numericl Methods Fid the itersectio o y = + 3 y = + Fid the itersectio o y = y = cos

7 Wht do we eed? Bsic Needs i the Numericl Methods: Prcticl: C be computed i resoble mout o time. Accurte: Good pproimte to the true vlue, Iormtio bout the pproimtio error Bouds, error order,. Wht is good Numericl Methods: How good is our pproimtio? Error Alysis How eiciet is our method? Algorithm desig, Covergece rte Does our methods lwys wor? Covergece 7

8 Outlies o the Course Tylor Theorem Number Represettio Solutio o olier Equtios Iterpoltio Numericl Dieretitio Numericl Itegrtio Solutio o lier Equtios Lest Squres curve ittig Solutio o ordiry dieretil equtios Solutio o Prtil dieretil equtios 8

9 Solutio o Nolier Equtios Some simple equtios c be solved lyticlly: Alyticsolutio roots d My other equtios hve o lyticl solutio: 9 5 e 0 No ly tic solutio 9

10 Methods or Solvig Nolier Equtios o Bisectio Method o Newto-Rphso Method o Sect Method o Bret s method o Aite s method & Muller method 0

11 Solutio o Systems o Lier Equtios i 000 uows. 000 equtios we hve Wht to do i 3, 5 3, 3 We c solve it s : 5 3

12 Crmer s Rule is Not Prcticl Crmer' s Rule c be used tosolve thesystem: 3 3 5, 5 But Crmer' s Rule is ot prcticl or lrge problems. Tosolve N equtios with N uows, we eed N N N! multiplictios. Tosolve 30 by 30 system, multiplictios re eeded. A super computer eeds more th 0 0 yers to compute this.

13 Methods or Solvig Systems o Lier Equtios o Nive Gussi Elimitio o Gussi Elimitio with Scled Prtil Pivotig o Algorithm or Tri-digol Equtios o Jcobi, Guss-Seidel & SOR methods o Cojugte grdiet method 3

14 Curve Fittig Give set o dt: 0 y Select curve tht best its the dt. Oe choice is to id the curve so tht the sum o the squre o the error is miimized. 4

15 Give set o dt: i 0 y i Iterpoltio Fid polyomil P whose grph psses through ll tbulted poits. yi P i i i is i the tble 5

16 Methods or Curve Fittig o Lest Squres o Lier Regressio o Nolier Lest Squres Problems o Iterpoltio o Newto Polyomil Iterpoltio o Lgrge Iterpoltio 6

17 Itegrtio Some uctios c be itegrted lyticlly: 3 But my uctios 0 d e d? hve o lyticl solutios : 7

18 Methods or Numericl Itegrtio o Upper d Lower Sums o Trpezoid Method o Romberg Method o Guss Qudrture 8

19 Solutio o Ordiry Dieretil Equtios A solutio to thedieretil equtio : t 3 t 3 t 0 0 ; 0 0 is uctio t tht stisies the equtios. * Alyticlsolutios re vilble specil cses oly. or 9

20 Solutio o Prtil Dieretil Equtios Prtil Dieretil Equtios re more diicult to solve th ordiry dieretil equtios: u u0, t u t u, t 0 0, u,0 si 0

21 Methods or ODEs o Implicit d Eplicit Euler schemes o Tylor d Ruge - Kutt methods o Multistep methods

22 Lecture Number Represettio d Accurcy Number Represettio Normlized Flotig Poit Represettio Sigiict Digits Accurcy d Precisio Roudig d Choppig

23 Represetig Rel Numbers You re milir with the deciml system: Deciml System: Bse = 0, Digits 0,,,9 Stdrd Represettios: sig itegrl rctio prt prt 3

24 Normlized Flotig Poit Represettio Normlized Flotig Poit Represettio: sig d. 3 mtiss 4 0 epoet d 0, : siged epoet Scietiic Nottio: Ectly oe o-zero digit ppers beore deciml poit. Advtge: Eiciet i represetig very smll or very lrge umbers. 4

25 Biry System Biry System: Bse =, Digits {0,} sig. 3 4 mtiss siged epoet

26 Fct Numbers tht hve iite epsio i oe umberig system my hve iiite epsio i other umberig system: You c ever represet. ectly i biry system. 6

27 IEEE 754 Flotig-Poit Stdrd Sigle Precisio 3-bit represettio -bit Sig + 8-bit Epoet + 3-bit Frctio S Epoet 8 Frctio 3 S bit: sig bit. or positive, 0 or egtive Epoet: 8 bits. Bis o +7 Frctio: 3 bits Normlized: Vlue=- S.rctio ep-7 Emple 995=

28 IEEE 754 Flotig-Poit Stdrd Double Precisio 64-bit represettio -bit Sig + -bit Epoet + 5-bit Frctio S Epoet Frctio 5 Cotiued Epoet: bits. Bis o +03 Epoet: bits. Bis o +03 Frctio: 5 bits. 8

29 IEEE 754 FLOATING POINT REPRESENTATION Epoets o ll 0's d 's re reserved or specil umbers. Zero is specil vlue deoted with epoet ield o zero d mtiss ield o zero, d we could hve +0 d re deoted with epoet o ll 's d mtiss ield o ll 0's. NN Not--umber is deoted with epoet o ll 's d o-zero mtiss ield.

30 Sigiict Digits Sigiict digits re those digits tht c be used with coidece. Sigle-Precisio: 7 Sigiict Digits to Double-Precisio: 5 Sigiict Digits to Lrger epoet Wider rge o umbers Loger mtiss Higher precisio 30

31 Remrs Some umbers c ot be represeted ectly i mchie represettio. Mchie umbers cot represet ll rel umbers iiitei.e. Oly limited rge o qutities my be represeted Number too lrger overlow Number too smll too close to 0 uderlow Numbers tht c be ectly represeted re clled mchie umbers. Mchie represettio is ot uique tht s why we ormlize the represettio. Dierece betwee mchie umbers is ot uiorm Sum o mchie umbers is ot ecessrily mchie umber 3

32 Clcultor Emple Suppose you wt to compute: *.39 usig clcultor with two-digit rctios 3.57 *.3 = 7.60 True swer:

33 Sigiict Digits - Emple

34 Accurcy d Precisio Accurcy is relted to the closeess to the true vlue. Precisio is relted to the closeess to other estimted vlues. 34

35 35

36 Roudig d Choppig Oly iite umber o qutities my be represeted roud-o or choppig errors Roudig: Replce the umber by the erest mchie umber. Choppig: Throw ll etr digits. 36

37 Roudig d Choppig There re discrete poits o the umber lies tht c be represeted by our computer. How bout the spce betwee? 37

38 Errors d Sigiict Digits I pid $0 or 7 orges. Wht is uit price o ech orge? $ tht is the ect output rom my computer!! Is there y dierece betwee $ d $.4? Is there y dierece betwee $.4 d $.40?

39 Sigiict igures, or digits The sigiict digits o umber re those tht c be used with coidece. They correspod to the umber o certi digits plus oe estimted digits. = 3.5 sigiict digits 3.45 < 3.55 = sigiict digits <

40 Error Deiitios True Error C be computed i the true vlue is ow: AbsoluteTrue Error E t true vlue pproimtio Absolute Reltive Error t true vlue pproimtio true vlue AbsolutePercet Reltive Error t true vlue pproimtio true vlue *00 40

41 Error Deiitios Estimted Error Whe the true vlue is ot ow: Estimted Absolute Error E curret estimte previous estimte Estimted Absolute Reltive Error curret estimte previous estimte curret estimte Estimted Absolute Percet Reltive Error curret estimte previous estimte curret estimte *00 4

42 Nottio We sy tht the estimte is correct to deciml digits i: Error 0 We sy tht the estimte is correct to deciml digits rouded i: Error 0 4

43 Summry Number Represettio Numbers tht hve iite epsio i oe umberig system my hve iiite epsio i other umberig system. Normlized Flotig Poit Represettio Eiciet i represetig very smll or very lrge umbers, Dierece betwee mchie umbers is ot uiorm, Represettio error depeds o the umber o bits used i the mtiss. 43

44 Lectures 3-4 Tylor Theorem Motivtio Tylor Theorem Emples 44

45 Motivtio We c esily compute epressios lie: 30 4 But, How do you compute 4., si0.6? C we use the deiitio to compute si0.6? Is this prcticl wy? b 0.6 Remr: I this course, ll gles re ssumed to be i rdi uless you re told otherwise. 45

46 Tylor Series '! c write : we d the sum eists coverge, series the I! writig codesed or i... 3!! : bout epsio o Tylor series The Tylor Tylor 46

47 Mcluri Series Mcluri series is specil cse o Tylor series with the ceter o epsio = 0. The I 0 the Mcluri ' 0 0! series coverge, 0! series 0 epsio o we : ! c write : 47

48 Mcluri Series Emple. coverges or The series... 3!!! 0! 0 0 '0 ' e or e e e e 48 e o epsio series Mcluri Obti

49 3.5.5 Tylor Series Emple ep

50 Mcluri Series Emple. coverges or The series... 7! 5! 3!! 0 si 0 cos 0 0 si '0 cos ' 0 0 si : si o epsio series Mcluri Obti

51 4 3-3 /3!+ 5 /5! 0 si /3!

52 Mcluri Series Emple 3. coverges or The series... 6! 4!!! 0 cos 0 0 si 0 cos 0 '0 si ' 0 cos cos : o epsio series Mcluri Obti

53 Mcluri Series Emple 4 coverges or Series... : o Epsio Series Mcluri '0 ' 0 o epsio series Mcluri Obti

54 Emple 4 - Remrs C we pply the series or?? How my terms re eeded to get good pproimtio??? These questios will be swered usig Tylor s Theorem. 54

55 Tylor Series Emple 5... : Epsio Series Tylor 6 6 ' ' t o epsio series Tylor Obti

56 Tylor Series Emple Epsio: Series Tylor, ' 0,,, ', l t l o epsio series Tylor Obti

57 Covergece o Tylor Series The Tylor series coverges st ew terms re eeded whe is er the poit o epsio. I - is lrge the more terms re eeded to get good pproimtio. 57

58 Tylor s Theorem I uctio possesses derivtives o orders o itervl cotiig d the the vlue o,,..., is give by : where : R 0!! d R is + terms Tructed Tylor Series betwee Remider d. 58

59 Tylor s Theorem We c pply Tylor' s theorem or : with the poit o epsio 0 i. I, the the uctio d its derivtives re ot deied. Tylor Theorem is ot pplicble. 59

60 Error Term. d betwee ll or! o : boud upper derive c we error, pproimtio bout the ide get To o vlues R 60

61 R R e Error Term - Emple How lrge is the error i we replced the irst 4 terms 3o its Tylor series epsio t 0 whe 0.?! 0. e! or 0. R 8.468E 05 3 e 0. e by 6

62 Let Altertive orm o Tylor s Theorem o itervl h hve derivtives o 0 cotiig! h d R orders h,,..., the : h step size R! h where is betwee d h 6

63 Tylor s Theorem Altertive orms 63. d betwee is!!,. d betwee is!! 0 0 h where h h h h where

64 I d its the there eists ξ, b Me Vlue Theorem is cotiuous uctio o closed itervl [, b] derivtive is deied o the ope itervl, b b ' ξ b Proo : Use Tylor' s Theorem or b ' ξ b 0,, h b 64

65 Cosider the S I lim Altertig Series Theorem d ltertig series : the The series coverges d S S S : : Prtil sum sum o First omitted term the irst terms 65

66 The : si si Altertig Series Emple si c be computedusig : si 3! This is coverget ltertig series sice : 3 3! 3! 4 5! 5! d 7! lim 0 5! 7! 66

67 o to pproimte Emple 7 Obti the Tylor series epsio e t 0.5the ceter o epsio How lrge c the error be whe e with? terms re used 67

68 Emple 7 Tylor Series...! ! ! '0.5 ' e e e e e e e e e e e e e , o epsio series Tylor Obti e

69 Emple 7 Error Term! m! 0.5! ! 3 [0.5,] e Error e Error e Error Error e 69

70 TOTAL NUMERICAL ERROR The totl umericl error is the summtio o the tructio d roud-o errors. I geerl, the oly wy to miimize roud-o errors is to icrese the umber o sigiict igures o the computer. Further, we hve oted tht roud-o error will icrese due to subtrctive ccelltio or due to icrese i the umber o computtios i lysis. I cotrst, the tructio error c be reduced by decresig the step size. Becuse decrese i step size c led to subtrctive ccelltio or to icrese i computtios, the tructio errors re decresed s the roud-o errors re icresed. The strtegy or decresig oe compoet o the totl error leds to icrese o the other compoet. I computtio, we could coceivbly decrese the step size to miimize tructio errors oly to discover tht i doig so, the roud-o error begis to domite the solutio d the totl error grows! 70

71 7