Numerical Methods Solution of Nonlinear Equations

Transcription

1 umercal Methods Soluton o onlnear Equatons

2 Lecture Soluton o onlnear Equatons Root Fndng Prolems Dentons Classcaton o Methods Analytcal Solutons Graphcal Methods umercal Methods Bracketng Methods Open Methods Convergence otatons

3 Root Fndng Prolems Many prolems n Scence and Engneerng are epressed as: Gven a contnuous uncton, nd the value r such that r These prolems are called root ndng prolems. 3

4 Roots o Equatons A numer r that satses an equaton s called a root o the equaton. The equaton :.e., has our roots :, 3, 3, and The equaton has two smple roots and and a repeated root 3 wth multplcty. 4

5 Zeros o a Functon Let e a real-valued uncton o a real varale. Any numer r or whch r s called a zero o the uncton. Eamples: and 3 are zeros o the uncton

6 Graphcal Interpretaton o Zeros The real zeros o a uncton are the values o at whch the graph o the uncton crosses or touches the -as. Real zeros o 6

7 Smple Zeros has two smple zeros one at and one at 7

8 Multple Zeros has doule zeros zero wth mulplcty at 8

9 Multple Zeros 3 3 has a zero wth mulplcty 3 at 9

10 Facts Any n th order polynomal has eactly n zeros countng real and comple zeros wth ther multplctes. Any polynomal wth an odd order has at least one real zero.

11 Roots o Equatons & Zeros o Functon Gven theequaton : 4 3 Move all terms to one sde o the equaton : 4 3 Dene as : The zeros o are the same as the rootso theequaton Whch are, 3, 3, and

12 Roots o Equatons & Zeros o Functon Gven theequaton : 3 3 Dene as : 3 3 The zeros o are the same as the roots o theequaton Whch are, , and

13 Soluton Methods Several ways to solve nonlnear equatons are possle: Analytcal Solutons Possle or specal equatons only Graphcal Solutons Useul or provdng ntal guesses or other methods umercal Solutons Open methods Bracketng methods 3

14 Analytcal Methods Analytcal Solutons are avalale or specal equatons only. Analytcal soluton o : a c roots ± 4ac a o analytcal soluton s avalale or : e 4

15 Graphcal Methods Graphcal methods are useul to provde an ntal guess to e used y other methods. Solve e The root [,] e Root root.6 5

16 umercal Methods Many methods are avalale to solve nonlnear equatons: Bsecton Method ewton s Method Secant Method False poston Method Muller s Method Barstow s Method Fed pont teratons. These wll e covered n ths lecture 6

17 Bracketng Methods In racketng methods, the method starts wth an nterval that contans the root and a procedure s used to otan a smaller nterval contanng the root. Eamples o racketng methods: Bsecton method False poston method 7

18 Open Methods In the open methods, the method starts wth one or more ntal guess ponts. In each teraton, a new guess o the root s otaned. Open methods are usually more ecent than racketng methods. They may not converge to a root. 8

19 Convergence otaton A sequence,,..., n,... s sad to converge to every ε > there ests such that : to n < ε n > 9

20 Convergence otaton C n n Lnear Convergence :. to converge,...,, Let C P C p n n n n : order Convergence o Quadratc Convergence :

21 Speed o Convergence We can compare derent methods n terms o ther convergence rate. Quadratc convergence s aster than lnear convergence. A method wth convergence order q converges aster than a method wth convergence order p q>p. Methods o convergence order p> are sad to have super lnear convergence.

22 Lectures 6-7 Bsecton Method The Bsecton Algorthm Convergence Analyss o Bsecton Method Eamples

23 Introducton The Bsecton method s one o the smplest methods to nd a zero o a nonlnear uncton. It s also called nterval halvng method. To use the Bsecton method, one needs an ntal nterval that s known to contan a zero o the uncton. The method systematcally reduces the nterval. It does ths y dvdng the nterval nto two equal parts, perorms a smple test and ased on the result o the test, hal o the nterval s thrown away. The procedure s repeated untl the desred nterval sze s otaned. 3

24 Intermedate Value Theorem Let e dened on the nterval [a,]. Intermedate value theorem: a uncton s contnuous and a and have derent sgns then the uncton has at least one zero n the nterval [a,]. a a 4

25 Eamples I a and have the same sgn, the uncton may have an even numer o real zeros or no real zeros n the nterval [a, ]. Bsecton method can not e used n these cases. a The uncton has our real zeros a The uncton has no real zeros 5

26 Two More Eamples I a and have derent sgns, the uncton has at least one real zero. a Bsecton method can e used to nd one o the zeros. The uncton has one real zero a The uncton has three real zeros 6

27 Bsecton Method I the uncton s contnuous on [a,] and a and have derent sgns, Bsecton method otans a new nterval that s hal o the current nterval and the sgn o the uncton at the end ponts o the nterval are derent. Ths allows us to repeat the Bsecton procedure to urther reduce the sze o the nterval. 7

28 Bsecton Method Assumptons: Gven an nterval [a,] s contnuous on [a,] a and have opposte sgns. These assumptons ensure the estence o at least one zero n the nterval [a,] and the secton method can e used to otan a smaller nterval that contans the zero. 8

29 Bsecton Algorthm Assumptons: s contnuous on [a,] a < Algorthm: Loop. Compute the md pont ca/. Evaluate c 3. I a c < then new nterval [a, c] I a c > then new nterval [c, ] End loop a c a 9

30 Bsecton Method a a a 3

31 Eample

32 Flow Chart o Bsecton Method Start: Gven a, and ε u a ; v c a / ; w c no yes s u w < no s -a /<ε yes Stop c; v w ac; u w 3

33 Eample Can you use Bsecton method to nd a zero o : 3 3 n the nterval [,]? Answer: s contnuous on [,] and * 3 3 > Assumptons are not satsed Bsecton method can not e used 33

34 Eample Answer: [,]? nterval n the 3 : o zero a nd to method Bsecton use you Can 3 34 used can e method Bsecton satsed are Assumptons - * and on [,] contnuous s <

35 Best Estmate and Error Level Bsecton method otans an nterval that s guaranteed to contan a zero o the uncton. Questons: What s the est estmate o the zero o? What s the error level n the otaned estmate? 35

36 Best Estmate and Error Level The est estmate o the zero o the uncton ater the rst teraton o the Bsecton method s the md pont o the ntal nterval: Estmate o the zero : r a Error a 36

37 Stoppng Crtera Two common stoppng crtera. Stop ater a ed numer o teratons. Stop when the asolute error s less than a speced value How are these crtera related? 37

38 Stoppng Crtera c n : s the mdpont o the nterval at the n th teraton c n s usually used as the estmate o the root. r : s the zero o the uncton. Ater n teratons : error r -c n E n a a n n 38

39 Convergence Analyss Gven, a,, and ε How many teratons are needed such that : - r ε where r s the zero o and s the secton estmate.e., ck? log a log ε n log 39

40 Convergence Analyss Alternatve Form log a log ε n log wdth o ntal nterval n log log desred error a ε 4

41 Eample? : such that needed are many teratons How.5 7, 6, - r a ε ε ε log log.5 log log log log n a n ε

42 Eample Use Bsecton method to nd a root o the equaton cos wth asolute error <. assume the ntal nterval [.5,.9] Queston : What s? Queston : Are the assumptons satsed? Queston 3: How many teratons are needed? Queston 4: How to compute the new estmate? 4

43 CISE3_Topc 43

44 Bsecton Method Intal Interval a a.5 c.7.9 Error <. 44

45 Bsecton Method Error < Error <.5 45

46 Bsecton Method Error < Error <.5 46

47 Summary Intal nterval contanng the root: [.5,.9] Ater 5 teratons: Interval contanng the root: [.75,.75] Best estmate o the root s.7375 Error <.5 47

48 A Matla Program o Bsecton Method a.5;.9; ua-cosa; v-cos; or :5 ca/ cc-cosc u*c< c ; vc; else ac; uc; end end c.7 c c.8 c.33 c.75 c.83 c.75 c

49 Eample Fnd the root o: 3 3 n the nterval:[,] * s contnuous *, a < Bsecton method can e used to nd the root 49

50 Eample Iteraton a c a c -a E E E

51 Bsecton Method Advantages Smple and easy to mplement One uncton evaluaton per teraton The sze o the nterval contanng the zero s reduced y 5% ater each teraton The numer o teratons can e determned a pror o knowledge o the dervatve s needed The uncton does not have to e derentale Dsadvantage Slow to converge Good ntermedate appromatons may e dscarded 5

52 Bsecton MethodHomework Wrte a computer code that nds the real roots o nth degree polynomals y usng Bsecton Methods ote: C or C programmng languages should e preerred, and the code has also to calculate the teraton numer requred! 5

53 Lecture 3 ewton-raphson Method Assumptons Interpretaton Eamples Convergence Analyss 53

54 ewton-raphson Method Also known as ewton s Method Gven an ntal guess o the root, ewton-raphson method uses normaton aout the uncton and ts dervatve at that pont to nd a etter guess o the root. Assumptons: s contnuous and the rst dervatve s known An ntal guess such that s gven 54

55 ewton Raphson Method -Graphcal Depcton - I the ntal guess at the root s, then a tangent to the uncton o that s s etrapolated down to the -as to provde an estmate o the root at. 55

56 Dervaton o ewton s Method ' Taylor Therorem :? etter estmate How do we otan a : the root o o guess an ntal : h h Queston Gven 56 ' : the root o A new guess '. such that Fnd ' Taylor Therorem : h h h h h ewton Raphson Formula

57 ewton s Method :n or Assumputon Gven ', ', X X X X FP X X X F PROGRAM FORTRA C 4 6* ** 3* ** 3* **3 57 end :n or ' ED STOP COTIUE X PRIT X FP X F X X I DO *, /,5

58 ewton s Method n or Assumputon Gven : ', ', X FP FP uncton X X F X F F uncton ] [ ^ 3* ^3 ] [ F.m FP.m 58 end n or ' : end X FP X F X X or X PROGRAM MATLAB / :5 4 % X X FP * 6 ^ * 3 FP.m

59 Eample Iteraton : 4 3 ' 4, 3 the uncton zero o Fnd a ' Iteraton 3: ' Iteraton : ' Iteraton : 3

60 Eample k Iteraton k k k k k k

61 Convergence Analyss Theorem : Let where, r such that C ' and ma mn -r δ -r -r δ. I '' ' δ '' 'r k k -r e contnuous -r at then there ests δ > C r 6

62 Convergence Analyss Remarks When the guess s close enough to a smple root o the uncton then ewton s method s guaranteed to converge quadratcally. Quadratc convergence means that the numer o correct dgts s nearly douled at each teraton. 6

63 Prolems wth ewton s Method I the ntal guess o the root s ar rom the root the method may not converge. ewton s method converges lnearly near multple zeros { r r }. In such a case, moded algorthms can e used to regan the quadratc convergence. 63

64 Multple Roots 3 has three zeros at has zeros at two - 64

65 Prolems wth ewton s Method -Runaway - The estmates o the root s gong away rom the root. 65

66 Prolems wth ewton s Method -Flat Spot - The value o s zero, the algorthm als. I s very small then wll e very ar rom. 66

67 Prolems wth ewton s Method -Cycle The algorthm cycles etween two values and 67

68 ewton s Method or Systems o on Lnear Equatons [ ] ' ' root o the o guess an ntal : X F X F X X Iteraton s ewton F X Gven k k k k 68 M M M ',,...,,..., ' X F X F X F X F X X k k k k

69 Eample Solve the ollowng system o equatons:, guess Intal 5 5 y y y. y 69, guess Intal y, 5 5 ', 5 5 X y F y y. y F

70 Soluton Usng ewton s Method ', : Iteraton. X y F. y y. y F ', : Iteraton.5 6 X F F X

71 Eample Try ths Solve the ollowng system o equatons:, Intal guess y y y y 7, guess Intal y, 4 ', X y F y y y F

72 Eample Soluton X Iteraton X k

73 ewton s MethodHomework Wrte a computer code that nds the real roots o nth degree polynomals y usng ewton s Methods ote: C or C programmng languages should e preerred! 73

74 Lectures Secant Method Secant Method Eamples Convergence Analyss 74

75 ewton s Method Revew Assumptons :, ', ' ewton' s Method new estmate: Prolem : ' s not avalale, or dcult to otan analytcally. ' are avalale, 75

76 Secant Method ponts : are two ntal ' and h h 76 '

77 Secant Method ponts ntal Two Assumptons : that such and 77 ew estmate Secant Method : that such

78 Secant Method.5 78

79 Secant Method -Flowchart ;,, 79 ; < ε Stop O Yes

80 Moded Secant Method ' needed : guess s only one ntal moded Secant method, ths In δ δ 8 the method may dverge. properly, not selected I? How to select Prolem : δ δ δ δ δ

81 Eample 5 Fnd the roots o : 5 3 Intal ponts and wth error <

82 Eample

83 Convergence Analyss The rate o convergence o the Secant method s super lnear: r r α C, α.6 r : root : estmate o the root at the th teraton. It s etter than Bsecton method ut not as good as ewton s method. 83

84 Secant MethodHomework Wrte a computer code that nds the real roots o nth degree polynomals y usng Secant Methods ote: C or C programmng languages should e preerred! 84

85 Lectures Comparson o Root Fndng Methods Advantages/dsadvantages Eamples 85

86 Summary Method Pros Bsecton - Easy, Relale, Convergent - One uncton evaluaton per teraton ewton - o knowledge o dervatve s needed - Fast near the root - Two uncton evaluatons per teraton Cons - Slow - eeds an nterval [a,] contanng the root,.e., a< - May dverge - eeds dervatve and an ntal guess such that s nonzero Secant - Fast slower than ewton - One uncton evaluaton per teraton - o knowledge o dervatve s needed - May dverge - eeds two ntal ponts guess, such that - s nonzero 86

87 Eample : root o the nd to Secant method Use ponts ntal Two and

88 Soluton k k k

89 Eample. : pont ntal the Use : root o a nd to Method ewton's Use or., or teratons, three ater Stop. : pont ntal the Use < < k k k

90 Fve Iteratons o the Soluton k k k k ERROR

91 Eample Use ewton's Method to nd a root o : e Use the ntal pont :. Stop ater three teratons, or k k <., or k <.. 9

92 Eample ' ', : root o a nd to Method ewton's Use k e e ' ' k k k k k

93 Eample Estmates o the root o: -cos..6 Intal guess correct dgt correct dgts correct dgts correct dgts 93

94 Eample In estmatng the root o: -cos, to get more than 3 correct dgts: 4 teratons o ewton.8 43 teratons o Bsecton method ntal nterval [.6,.8] 5 teratons o Secant method.6,.8 94

95 Multple Roots In dervng ewton s method we assumed that at zero F α. In the case o a multple root, the dervatve o the uncton vanshes too. So we need to mody our ormula. F 3 F

96 Multple Roots Suppose that at a there s a zero o multplcty n. F A a n F n An a nf a a nf F We need to terate!!! 96

97 Multple Roots We need to develop some test as to how we may e certan aout the multplcty a n n F F An n a a nf F a n F F F F or a root o multplcty should all e equal as we approach the root at a!!! a n CISE3_Topc 97

98 Multple Roots F a root a near -.5, a root a near.7, a root a 3 near Usng ewton s method: a a.6566 a

99 Multple Roots To consder possle doule roots: at -.4 F F F F F F F.3897 s a doule root!!! 99

100 Multple Roots An ntal guess o -.6 F F F F.685 F F.689 F A urther teraton estalshes the value o the doule root:.683

101 Multple RootsHomework Wrte a computer code that nds the all roots o 7th degree polynomal gven elow. F ote: C or C programmng languages should e preerred!

102 ested Multplcaton The process o nested multplcaton s an ecent method o evaluatng a polynomal and ts dervatves at a partcular pont. Consder; F a a a3 a4 a5 a6 a7 to evaluate F wll requre; 6543 multplcatons as well as 6 addtons!!

103 ested Multplcaton On the other hand we wrte the equaton only 6 multplcatons and 6 addtons are requred! a a a a a a a F Suppose we wsh to evaluate the equaton at 3 Q F Q F Q Q Q F

104 ested Multplcaton Assume the ollowng representaton or Q F Orgnal Comparng oth equatons, we have 4... F... a a a a F a a a a...

105 ested Multplcaton Coecents o are ound to evaluate F a a a a F The general orm: a 5

106 ested Multplcaton F a We need to evaluate F at.5 a 6 * a 6 6.5* a a a a a F.5 8 F

107 ested MultplcatonHomework Wrte a computer code that calculates the speced values o nth degree polynomals ote: C or C programmng languages should e preerred! 7

108 Comple RootsLn-BarstowMethod n n n n a a a a F remander Q s r F 3... s r F r s r F Remander o ths dvson s wrtten as or convenence r

109 s -s...- r -r... s s s r r r F Comple Roots 9 r s r s r r r r F

110 Comple Roots F r r 3 s r - s r r s... F n n n3 a a a3... an an Orgnal Comparng oth equatons, we have a a r a3 3 r s an n rn sn a r s n n n n a a r 3 a3 r s n an rn sn a r s n n n n

111 Comple Roots F r s 3 r... n an rn sn n an rn sn All and, r and s are unknown! Consequently, we may calculate the or an ntal guess or r and s We also want and n n to e zero. So we need to systematcally change r and s to orce n and n to e zero.

112 Comple Roots Suppose that true values o r and s are R and S respectvely. Then epandng and n a two varales n n Taylor seres around r,s and settng the remander terms to zero, we have...,, s S s r R r s r S R n n n n...,, s S s r R r s r S R n n n n

113 Comple Roots We may calculate the needed partal dervatves rom a a r 3 a3 r s n an rn sn a r s n n n n r n r r r r n r C 3 rc C rc sc C n n n3 n 4 3 rc sc C 3 n rc n sc n C n 3

114 Comple Roots s s 3 C s Smlarly, the rest are calculated as : n n n n n C sc rc s 3 n n n n n C sc rc s 4 C rc s

115 ,, s S s r R r s r S R n n n n,, s S s r R r s r S R n n n n Comple Roots 5 s S C r R C n n n s S C r R C n n n The equatons ecome We regard the two equaton as two equatons n two unknowns R and S whch we solve to otan mproved values or r and s.

116 Comple Roots Lets dene δr R r δs S s Cn r C r C C n δ n n n δ n δs s Ater ndng the values o δr and δs, we update δ and take R r δ r S s δs r R s S The process s then terated 6

117 Comple Roots Ater ndng the values o r and s, F r s Q remander The roots can e determned as r s,, m r m 4ac a r 4s 7

118 Comple Roots Eample F Eample: nd comple roots o F Soluton: Root Easy! Root Root Root

119 Comple Roots Eample a a a3 4 a4 5 a5 7 Suppose ntal value o r and s s. a a r 3 a3 r s 4 a4 r3 s 5 a5 r4 s 3.* 4.*.9.9.* * 4.9.* * 4.4.*

120 Comple Roots Eample C C rc C 3 3 rc sc C 4 4 rc3 sc C C.9.*.8 C3 4.9.*.8.* 4.7 C4 4.4.* 4.7.*

121 Comple Roots Eample n 5 Cn r C r C C n δ n n nδ n δs s δ C3δr Cδs C r C s 4 5 4δ 3δ δr. 8δs δr 4. 7δs δr δs

122 Comple Roots Eample R r δ r S s δs r R s S R..747 S..586 R.3747 S.486 Restart the process agan wth the new values untl r.3747 s.486 δr < error δs < error

123 Comple Roots Eample Ater 5 teratons; δr δs.54 R r S s , r m r 4s The comple roots are

124 Comple Roots Homework Wrte a computer code that calculates comple roots o nth degree polynomals ote: C or C programmng languages has to e preerred! 4

125 Gratutes Ths Lecture note has een prepared y modyng the notes o CISE3 rom KFU! 5