Numerical Analysis ECIV 3306 Chapter 6

Transcription

1 The Islamc Unversty o Gaza Faculty o Engneerng Cvl Engneerng Department Numercal Analyss ECIV 3306 Chapter 6 Open Methods & System o Non-lnear Eqs Assocate Pro. Mazen Abualtaye Cvl Engneerng Department, The Islamc Unversty o Gaza

2 PART II: ROOTS OF EQUATIONS Bsecton method Bracketng Methods False Poston Method Roots o Equatons Open Methods System o Nonlnear Equatons Roots o polynomals Smple ed pont teraton Newton Raphson Secant Moded Newton Raphson Muller Method

3 Open Methods Bracketng methods are based on assumng an nterval o the uncton whch brackets the root. The bracketng methods always converge to the root. Open methods are based on ormulas that requre only a sngle startng value o or two startng values that do not necessarly bracket the root. These method sometmes dverge rom the true root.

4 Open Methods- Convergence and Dvergence Concepts + + Dvergng ncrements Convergng ncrements

5 6. Smple Fed-Pont Iteraton Rearrange the uncton =0 so that s on the let sde o the equaton: 0 g g Bracketng methods are convergent. Fed-pont methods may sometme dverge, dependng on the statng pont ntal guess and how the uncton behaves.

6 6. Smple Fed-Pont Iteraton Eamples:. 2. = = g= 2 +3/2 3. = sn = g= sn + 3. = e - - = g= e - g or g or g

7 Smple Fed-Pont Iteraton Convergence = g can be epressed as a par o equatons: y = y 2 = g. component equatons Plot them separately.

8 Smple Fed-Pont Iteraton Convergence to compute a new estmate + as epressed by the teratve ormula g Suppose that the true root: r g r Subtractng rom 2 2 g g 3 r r

9 Smple Fed-Pont Iteraton Convergence Dervatve mean value theorem: I g are contnuous n [a,b] then there est at least one value o = wthn the nterval such that: g ' g b g a b a.e. there est one pont where the slope parallel to the lne jonng a & b

10 Smple Fed-Pont Iteraton Convergence g g r r Let g ' a and b g r r ' r r ' ', t, ' g t ' g r r g g g then g E g E I I g r ' g b g a.0 the error decreases wth each teraton.0 the error ncreases wth each teraton g b a

11 Smple Fed-Pont Iteraton Convergence Fed-pont teraton converges : g slope o the lne When the method converges, the error s roughly proportonal to or less than the error o the prevous step, thereore t s called lnearly convergent.

12 Smple Fed-Pont Iteraton-Convergence

13 Eample 6.: Smple Fed-Pont Iteraton = e - -. s manpulated so that we get =g g = e - 2. Thus, the ormula predctng the =e - - Root new value o s: + = e - 3. Guess 0 = 0 4. The teratons contnues tll the appro. error reaches a certan lmtng value

14 Eample 6.: Smple Fed-Pont Iteraton g = e - Recall true root s g e a % e t %

15 Flow Chart Fed Pont Start Input: o, e s, ma =0 e a =.e s

16 whle e a < e s & >ma False n g 0 Prnt: o, o,e a, = or n =0 Stop True e a n n o 00% 0 = n

17 6.2 The Newton-Raphson Method Most wdely used method. Based on Taylor seres epanson: 0 Rearrangn g, 0 when the value o The root s... 2! 2 Solve or Newton-Raphson ormula

18 6.2 The Newton-Raphson Method A tangent to at the ntal pont s etended tll t meets the -as at the mproved estmate o the root +. The teratons contnues tll the appro. error reaches a certan lmtng value. Slope / Root / + 0 /

19 Eample 6.3: The Newton Raphson Method / e e e e Fnd the root o = e - -= 0 = e - - and ` = -e - -; thus Iter. X + e t % <0-8 Recall true root s

20 Flow Chart Newton Raphson Start Input: o, e s, ma =0 e a =.e s

21 whle e a >e s & <ma False n 0 ' = or n =0 0 0 Prnt: o, o,e a, Stop True e a n n o 00% 0 = n

22 Ptalls o The Newton Raphson Method

23 6.3 The Secant Method The dervatve s replaced by a backward nte dvded derence Thus, the ormula predctng the + s: / / /

24 6.3 The Secant Method Requres two ntal estmates o, e.g, o,. However, because s not requred to change sgns between estmates, t s not classed as a bracketng method. The secant method has the same propertes as Newton s method. Convergence s not guaranteed or all o,,.

25 6.3 Secant Method: Eample Use the Secant method to nd the root o e - -=0; = e - - and - =0, = to get + o the rst teraton usng: Recall true root s Iter e t %

26 Comparson o convergence o False Poston and Secant Methods False Poston Secant Method r u u l u l u Use two estmate l and u Use two estmate and - must changes sgns between l and u X r replaces whchever o the orgnal values yelded a uncton value wth the same sgn as r Always converge s not requred to change sgns between and - X + replace X replace - May be dverge

27 Comparson o convergence o False Poston and Secant Methods Use the alse-poston and secant methods to nd the root o = ln. Start computaton wth l = - =0.5, u = = 5.. False poston method 2. Secant method

28 False Poston and Secant Methods Although the secant method may be dvergent, when t converges t usually does so at a qucker rate than the alse poston method l u -

29 Comparson o the true percent relatve Errors E t or the methods to the determne the root o =e - -

30 Flow Chart Secant Method Start Input: -, 0,e s, ma =0 e a =.e s

31 whle e a >e s & < ma False Prnt:,,e a, = or X + =0 Stop True e a 00% X - = X = +

32 6.3.3 Moded Secant Method Rather than usng two ntal values, an alternatve approach s usng a ractonal perturbaton o the ndependent varable to estmate / s a small perturbaton racton /

33 Moded Secant Method: Eample 6.8 Use moded secant method to nd the root o = e - -, 0 = and = 0.0. Recall true root s

34 6.5 Multple Roots = -3-- = = = Double roots 3 trple roots 3

35 6.5 Multple Roots Multple root corresponds to a pont where a uncton s tangent to the -as. Dcultes - Functon does not change sgn wth double or even number o multple root, thereore, cannot use bracketng methods. - Both and =0, dvson by zero wth Newton s and Secant methods whch may dverge around ths root.

36 Moded Newton-Raphson Method or Multple Roots Another alternatve s ntroduced such new u=/ / ; Gettng the roots o u usng Newton-Raphson technque: ] [ // 2 / / 2 / // / / / / u u u Ths uncton has roots at all the same locatons as the orgnal uncton Derentate u=/ /

37 Usng the Newton-Raphson and Moded Newton-Raphson to evaluate the multple roots o = wth an ntal guess o 0 = // 2 / / / Newton Raphson ormula: Moded Newton Raphson ormula: Eample 6.0 Moded Newton-Raphson Method or Multple Roots

38 Moded Newton Raphson Method: Eample Newton Raphson Moded Newton-Raphson Iter e t % ter e t % Newton Raphson technque s lnearly convergng towards the true value o.0 whle the Moded Newton Raphson s quadratcally convergng. For smple roots, moded Newton Raphson s less ecent and requres more computatonal eort than the standard Newton Raphson method.

39 6.6 Systems o Nonlnear Equatons Roots o a set o smultaneous equatons:, 2,., n =0 2, 2,., n =0.. n, 2,., n =0 The soluton s a set o values that smultaneously get the equatons to zero.

40 6.6 Systems o Nonlnear Equatons Eample: 2 + y = 0 and y + 3y 2 = 57 u,y = 2 + y -0 = 0 v,y = y+ 3y 2-57 = 0 The soluton wll be the value o and y whch makes u,y=0 and v,y=0 These are =2 and y=3 Numercal methods used are etenson o the open methods or solvng sngle equaton; Fed pont teraton and Newton-Raphson.

41 6.6 Systems o Nonlnear Equatons. Use an ntal guess =.5 and y = The teraton ormulae: + =0-2 /y and y + =57-3 y 2 3. Frst teraton, = /3.5 = y= = Second teraton:. Fed Pont Iteraton = / = y= = y = 0 y + 3y 2 = Soluton s dvergng so try another teraton ormula

42 6.6 Systems o Nonlnear Equatons. Usng teraton ormula:. Fed Pont Iteraton + =0- y /2 and y + =[57-y /3 ] /2 Frst guess: =.5 and y= st teraton: = /2 = y=57-3.5/ /2 = nd teraton: = /2 = y= / /2 = y = 0 y + 3y 2 = The approach s convergng to true root, =2 and y=3

43 6.6 Systems o Nonlnear Equatons. Fed Pont Iteraton The sucent condton or convergence or the two-equaton case u,y=0 and v,y=0 are: u v and u y v y

44 MS Ecel: Solver u,y= 2 +y-0 =0 v,y=y+3y 2-57=0