: Numerical Analysis Topic 2: Solution of Nonlinear Equations Lectures 5-11:
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1 764: Numercal Analyss Topc : Soluton o Nonlnear Equatons Lectures 5-: UIN Malang Read Chapters 5 and 6 o the tetbook 764_Topc
2 Lecture 5 Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton o Methods Analytcal Solutons Graphcal Methods Numercal Methods Bracketng Methods Open Methods Convergence Notatons Readng Assgnment: Sectons 5. and _Topc
3 Root Fndng Problems Many problems n Scence and Engneerng are epressed as: Gven a nd the value contnuous r uncton such that r, These problems are called root ndng problems. 764_Topc 3
4 Roots o Equatons A number r that satses an equaton s called a root o the equaton. The has our.e., equaton : 4 roots:, , , and The equaton has twosmple roots and and a repeated root 3 wth multplcty. 764_Topc 4
5 Zeros o a Functon Let be a real-valued uncton o a real varable. Any number r or whch r= s called a zero o the uncton. Eamples: and 3 are zeros o the uncton = _Topc 5
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 764_Topc 6
7 Smple Zeros has twosmple zeros one at and one at 764_Topc 7
8 Multple Zeros has double zeros zero wth mulplcty at 764_Topc 8
9 Multple Zeros 3 3 has a zero wth mulplcty 3 at 764_Topc 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. I a uncton has a zero at =r wth multplcty m then the uncton and ts rst m- dervatves are zero at =r and the m th dervatve at r s not zero. 764_Topc
11 Roots o Equatons & Zeros o Functon Gven theequaton : 3 Moveall terms to one sde o the equaton : 3 Dene as : The zeros o are thesame as the rootso theequaton Whch are, 3, 3, and 764_Topc
12 Soluton Methods Several ways to solve nonlnear equatons are possble: Analytcal Solutons Possble or specal equatons only Graphcal Solutons Useul or provdng ntal guesses or other methods Numercal Solutons Open methods Bracketng methods 764_Topc
13 Analytcal Methods Analytcal Solutons are avalable or specal equatons only. Analytcal soluton o : a b c roots b b a 4ac No analytcal soluton s avalable or : e 764_Topc 3
14 Graphcal Methods Graphcal methods are useul to provde an ntal guess to be used by other methods. Solve e The root [,] e Root root.6 764_Topc 4
15 Numercal Methods Many methods are avalable to solve nonlnear equatons: Bsecton Method Newton s Method Secant Method False poston Method Muller s Method Barstow s Method Fed pont teratons. These wll be covered n _Topc 5
16 Bracketng Methods In bracketng methods, the method starts wth an nterval that contans the root and a procedure s used to obtan a smaller nterval contanng the root. Eamples o bracketng methods: Bsecton method False poston method 764_Topc 6
17 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 obtaned. Open methods are usually more ecent than bracketng methods. They may not converge to a root. 764_Topc 7
18 Convergence Notaton A sequence,,..., n,... s sad toconverge to every there ests N such that : to n n N 764_Topc 8
19 764_Topc 9 Convergence Notaton C P C C p n n n n n n : order Convergenc e o Quadratc Convergenc e : Lnear Convergenc e :. to converge,...,, Let
20 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. 764_Topc
21 Lectures 6-7 Bsecton Method The Bsecton Algorthm Convergence Analyss o Bsecton Method Eamples Readng Assgnment: Sectons 5. and _Topc
22 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 by dvdng the nterval nto two equal parts, perorms a smple test and based on the result o the test, hal o the nterval s thrown away. The procedure s repeated untl the desred nterval sze s obtaned. 764_Topc
23 Intermedate Value Theorem Let be dened on the nterval [a,b]. Intermedate value theorem: a uncton s contnuous and a and b have derent sgns then the uncton has at least one zero n the nterval [a,b]. a a b b 764_Topc 3
24 Eamples I a and b have the same sgn, the uncton may have an even number o real zeros or no real zeros n the nterval [a, b]. Bsecton method can not be used n these cases. a The uncton has our real zeros b a The uncton has no real zeros b 764_Topc 4
25 Two More Eamples I a and b have derent sgns, the uncton has at least one real zero. a b Bsecton method can be used to nd one o the zeros. The uncton has one real zero a b The uncton has three real zeros 764_Topc 5
26 Bsecton Method I the uncton s contnuous on [a,b] and a and b have derent sgns, Bsecton method obtans 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. 764_Topc 6
27 Bsecton Method Assumptons: Gven an nterval [a,b] s contnuous on [a,b] a and b have opposte sgns. These assumptons ensure the estence o at least one zero n the nterval [a,b] and the bsecton method can be used to obtan a smaller nterval that contans the zero. 764_Topc 7
28 Bsecton Algorthm Assumptons: s contnuous on [a,b] a b < Algorthm: Loop. Compute the md pont c=a+b/. Evaluate c 3. I a c < then new nterval [a, c] I a c > then new nterval [c, b] End loop a a c b b 764_Topc 8
29 Bsecton Method b a a a 764_Topc 9
30 Eample _Topc 3
31 Flow Chart o Bsecton Method Start: Gven a,b and ε u = a ; v = b c = a+b / ; w = c no yes s u w < no s b-a /<ε yes Stop b=c; v= w a=c; u= w 764_Topc 3
32 764_Topc 3 Eample Answer: [,]? nterval n the 3 : o zero a nd to method Bsecton you use Can 3 used be not can method Bsecton satsed not are Assumptons 3 3 and * [,] on contnuous s
33 764_Topc 33 Eample Answer: [,]? nterval n the 3 : o zero a nd to method Bsecton you use Can 3 used be can method Bsecton satsed are Assumptons - * and [,] on contnuous s
34 Best Estmate and Error Level Bsecton method obtans an nterval that s guaranteed to contan a zero o the uncton. Questons: What s the best estmate o the zero o? What s the error level n the obtaned estmate? 764_Topc 34
35 Best Estmate and Error Level The best 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 b a Error b a 764_Topc 35
36 Stoppng Crtera Two common stoppng crtera. Stop ater a ed number o teratons. Stop when the absolute error s less than a speced value How are these crtera related? 764_Topc 36
37 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 b a n n 764_Topc 37
38 Convergence Analyss Gven, a, b, and How many teratons are needed such that : - r where r s the zero o and s the bsecton estmate.e., c k? log b a log n log 764_Topc 38
39 Convergence Analyss Alternatve Form log b a log n log wdth o ntal nterval n log log desred error b a 764_Topc 39
40 Eample a 6, b 7,.5 How many teratons are needed such that : - r? n log b a log log log log.5 log.9658 n 764_Topc 4
41 Eample Use Bsecton method to nd a root o the equaton = cos wth absolute 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? 764_Topc 4
42 764_Topc 4
43 Bsecton Method Intal Interval a= b =.784 a =.5 c=.7 b=.9 Error <. 764_Topc 43
44 Bsecton Method Error < Error <.5 764_Topc 44
45 Bsecton Method Error < Error <.5 764_Topc 45
46 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 <.5 764_Topc 46
47 A Matlab Program o Bsecton Method a=.5; b=.9; u=a-cosa; v=b-cosb; or =:5 c=a+b/ c=c-cosc u*c< b=c ; v=c; else a=c; u=c; end end c =.7 c = c =.8 c =.33 c =.75 c =.83 c =.75 c = _Topc 47
48 Eample Fnd the root o: 3 3 n the nterval :[,] * s contnuous *, a b Bsecton method can be used to nd the root 764_Topc 48
49 Eample Iteraton a b c= a+b c b-a E E E _Topc 49
50 Bsecton Method Advantages Smple and easy to mplement One uncton evaluaton per teraton The sze o the nterval contanng the zero s reduced by 5% ater each teraton The number o teratons can be determned a pror No knowledge o the dervatve s needed The uncton does not have to be derentable Dsadvantage Slow to converge Good ntermedate appromatons may be dscarded 764_Topc 5
51 Lecture 8-9 Newton-Raphson Method Assumptons Interpretaton Eamples Convergence Analyss 764_Topc 5
52 Newton-Raphson Method Also known as Newton s Method Gven an ntal guess o the root, Newton-Raphson method uses normaton about the uncton and ts dervatve at that pont to nd a better guess o the root. Assumptons: s contnuous and the rst dervatve s known An ntal guess such that s gven 764_Topc 5
53 Newton 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 _Topc 53
54 Dervaton o Newton s Method Gven: Queston Taylor Therorem : Fnd h h : ' A new guess an ntal How do we obtan a such that o guess the root : o h h the root o. better estmate ' h ' Newton Raphson Formula? 764_Topc 54
55 764_Topc 55 Newton s Method end :n or Assumputon Gven ' ', ', END STOP CONTINUE X PRINT X FP X F X X I DO X X X X FP X X X F PROGRAM FORTRAN C *, /,5 4 6* ** 3* ** 3* **3
56 764_Topc 56 Newton s Method end n or Assumputon Gven ' : ', ', end X FP X F X X or X PROGRAM MATLAB / :5 4 % X X FP X FP FP uncton X X F X F F uncton 6* ^ 3* ] [ ^ 3* ^3 ] [ F.m FP.m
57 764_Topc 57 Eample ' Iteraton 3: ' Iteraton : ' : Iteraton 4 3 ' 4, 3 the uncton zero o Fnd a 3 3
58 Eample k Iteraton k k k k+ k+ k _Topc 58
59 Convergence Analyss Theorem : Let where, such that C r ma ' and -r mn -r -r. I '' ' '' 'r k k -r be contnuous at r -r then thereests C 764_Topc 59
60 Convergence Analyss Remarks When the guess s close enough to a smple root o the uncton then Newton s method s guaranteed to converge quadratcally. Quadratc convergence means that the number o correct dgts s nearly doubled at each teraton. 764_Topc 6
61 Problems wth Newton s Method I the ntal guess o the root s ar rom the root the method may not converge. Newton s method converges lnearly near multple zeros { r = r = }. In such a case, moded algorthms can be used to regan the quadratc convergence. 764_Topc 6
62 Multple Roots 3 has three zeros at has zeros at two - 764_Topc 6
63 Problems wth Newton s Method - Runaway - The estmates o the root s gong away rom the root. 764_Topc 63
64 Problems wth Newton s Method - Flat Spot - The value o s zero, the algorthm als. I s very small then wll be very ar rom. 764_Topc 64
65 Problems wth Newton s Method - Cycle - = 3 = 5 = = 4 The algorthm cycles between two values and 764_Topc 65
66 764_Topc 66 Newton s Method or Systems o Non Lnear Equatons ',,...,,..., ' ' the root o o guess an ntal : X F X F X F X F X X Iteraton s Newton F X Gven k k k k
67 764_Topc 67 Eample Solve the ollowng system o equatons:, guess Intal 5 5 y y y. y, 5 5 ', 5 5 X y F y y. y F
68 764_Topc 68 Soluton Usng Newton s Method ', : Iteraton ', : Iteraton X F F. X y F. y y. y F
69 764_Topc 69 Eample Try ths Solve the ollowng system o equatons:, guess Intal y y y y, 4 ', X y F y y y F
70 764_Topc 7 Eample Soluton X k Iteraton
71 Lectures Secant Method Secant Method Eamples Convergence Analyss 764_Topc 7
72 Newton s Method Revew Assumptons :, ', Newton' s Method new estmate : Problem : ' ' s not avalable, or dcult toobtan analytcally. ' are avalable, 764_Topc 7
73 764_Topc 73 Secant Method ' ponts: twontal are ' and h h
74 764_Topc 74 Secant Method New estmate Secant Method: ponts ntal Two Assumptons : that such and
75 764_Topc 75 Secant Method.5
76 764_Topc 76 Secant Method - Flowchart ;,, Stop NO Yes
77 764_Topc 77 Moded Secant Method the method may dverge. not selected properly, I? How to select Problem : ' needed : s guess only one ntal moded Secant method, ths In
78 Eample 5 Fnd the roots o : Intal ponts 5 and wth error _Topc 78
79 Eample _Topc 79
80 Convergence Analyss The rate o convergence o the Secant method s super lnear: r r C,.6 r : root : estmate o the root at the th teraton. It s better than Bsecton method but not as good as Newton s method. 764_Topc 8
81 Lectures Comparson o Root Fndng Methods Advantages/dsadvantages Eamples 764_Topc 8
82 Summary Method Pros Cons Bsecton Newton Secant - Easy, Relable, Convergent - One uncton evaluaton per teraton - No knowledge o dervatve s needed - Fast near the root - Two uncton evaluatons per teraton - Fast slower than Newton - One uncton evaluaton per teraton - No knowledge o dervatve s needed - Slow - Needs an nterval [a,b] contanng the root,.e., ab< - May dverge - Needs dervatve and an ntal guess such that s nonzero - May dverge - Needs two ntal ponts guess, such that - s nonzero 764_Topc 8
83 764_Topc 83 Eample.5 ponts ntal Two : o root the nd to method Secant Use 6 and
84 Soluton k k k _Topc 84
85 764_Topc 85 Eample.. or., or teratons, three ater Stop. pont : ntal the Use : o root a nd to Method Newton's Use 3 k k k
86 Fve Iteratons o the Soluton k k k k ERROR _Topc 86
87 764_Topc 87 Eample.. or., or teratons, three ater Stop. pont : ntal Use the : o root a nd to Method Newton's Use k k k e
88 764_Topc 88 Eample ' ' ', : o root a nd to Method Newton's Use k k k k k e e
89 Eample Estmates o the root o: -cos=..6 Intal guess correct dgt correct dgts correct dgts correct dgts 764_Topc 89
90 Eample In estmatng the root o: -cos=, to get more than 3 correct dgts: 4 teratons o Newton =.8 43 teratons o Bsecton method ntal nterval [.6,.8] 5 teratons o Secant method =.6, =.8 764_Topc 9
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