The Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 5. Bracketing Methods

Transcription

1 The Islami University of Gaza Faulty of Engineering Civil Engineering Department Numerial Analysis ECIV 3306 Chapter 5 Braketing Methods Assoiate Prof. Mazen Abualtayef Civil Engineering Department, The Islami University of Gaza

2 PART II: ROOTS OF EQUATIONS Bisetion method Braketing Methods False Position Method Roots of Equations Open Methods System of Nonlinear Equations Roots of polynomials Simple fied point iteration Newton Raphson Seant Modified Newton Raphson Muller Method

3 Study Objetives for Part Two

4 ROOTS OF EQUATIONS Root of an equation: is the value of the equation variable whih make the equations = b b a b 0 2a 4a But a 5 b 4 3 d 2 e f 0? sin 0?

5 ROOTS OF EQUATIONS Non-omputer methods: - Closed form solution (not always available) - Graphial solution (inaurate) Numerial systemati methods suitable for omputers

6 Plot the funtion f() f() Graphial Solution roots f()=0 f()=0 The roots eist where f() rosses the -ais

7 Graphial Solution: Eample The parahutist veloity is mg ( 1 e What is the drag oeffiient needed to reah a veloity of f() 40 m/s if m=68.1 kg, t =10 s, g= 9.8 m/s 2 f f v ( ) ( ) t m m ) mg t (1 e ) (1 e v =14.75 Chek: F (14.75) = ~ 0.0 v (=14.75) = ~ 40 m/s ) 40

8 Numerial Systemati Methods I. Braketing Methods f() No roots or even number of roots f() Odd number of roots f( l )=+ve f( l )=+ve roots roots f( u )=+ve l u l u f( u )=-ve

9 Braketing Methods (ont.) Two initial guesses ( l and u ) are required for the root whih braket the root (s). If one root of a real and ontinuous funtion, f()=0, is bounded by values l, u then f( l ).f( u ) <0. (The funtion hanges sign on opposite sides of the root)

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12 Braketing Methods 1. Bisetion Method Generally, if f() is real and ontinuous in the interval l to u and f ( l ).f( u )<0, then there is at least one real root between l and u to this funtion. The bisetion method, whih is alternatively alled binary hopping, interval halving, or Bolzano s method, is one type of inremental searh method in whih the interval is always divided in half. The interval at whih the funtion hanges sign is loated. Then the interval is divided in half with the root lies in the midpoint of the subinterval. This proess is repeated to obtained refined estimates.

13 f() Step 1: Choose lower l and upper u guesses for the root suh that: f( l ).f( u )<0 Step 2: The root estimate is: r = ( l + u )/2 Step 3: Subdivide the interval aording to: If (f( l ).f( r )<0) the root lies in the lower subinterval; set u = r and go to step 2. If (f( l ).f( r )>0) the root lies in the upper subinterval; set l = r and go to step 2. If (f( l ).f( r )=0) the root is r and stop f() l l f( u ) f( u ) r2 r = ( l + u )/2 r1 f( r1 ) f( r2 ) u u f( u ) f( u ) (f( l ).f( r )<0): u = r r = ( l + u )/2

14 Bisetion Method - Termination Criteria e t True X relaive true X X true Error : approimat e 100% Approimate relative Error e e a a X X X n r u u X X n r X X n1 r l l 100% 100% (Bisetion) : For the Bisetion Method e a > e t The omputation is terminated when e a beomes less than a ertain riterion (e a < e s )

15 Bisetion method: Eample The parahutist veloity is What is the drag oeffiient needed to reah a veloity of 40 m/s if m = 68.1 kg, t = 10 s, g= 9.8 m/s 2 f() ) ( t m e 1 mg v 40 e f v e 1 mg f t m ) (. ) ( ) ( ) (.

16 f ( ) (1 e ) Assume l =12 and u =16 f( l )=6.067 and f( u )= The root: r =( l + u )/2= Chek f(12).f(14) = =9.517 >0; the root lies between 14 and 16. f() Set l = 14 and u =16, thus the new root r =(14+ 16)/2= 15 f() (f(12).f(14)>0): l = Chek f(14).f(15) = = <0; the root lies bet. 14 and Set l = 14 and u =15, thus the new root r =(14+ 15)/2= 14.5 and so on... True Value at Xr =

17 Bisetion method: Eample In the previous eample, if the stopping riterion is e t = 0.5%; what is the root? True relaive Error : Approimate relative Error : e t X true X X approimat e true 100% e a X n r X X n r n1 r 100%

18 Bisetion method

19 Flow Chart Bisetion Start Input: l, u, e s, mai False f( l ). f( u )<0 i=0 e a =1.1e s while e a > e s & i <mai False r i i u 2 1 r Print: r, f( r ),e a, i Stop

20 u + l =0 True e a u u l l 100% Test=f( l ). f( r ) Test=0 True e a =0.0 Test<0 True u = r False l = r

21 Braketing Methods 2. False-position Method The bisetion method divides the interval l to u in half not aounting for the magnitudes of f( l ) and f( u ). For eample if f( l ) is loser to zero than f( u ), then it is more likely that the root will be loser to f( l ). The fat that the replaement of the urve by a straight line gives a false position of the root is the origin of the name, method of false position. It is also alled the linear interpolation method.

22 2. False-position Method False position method is an alternative approah where f( l ) and f( u ) are joined by a straight line; the intersetion of whih with the -ais represents and improved estimate of the root. f( r )

23 2. False-position Method -Proedure f( r )

24 2. False-position Method -Proedure

25 False-position Method -Proedure Step 1: Choose lower l and upper u guesses for the root suh that: f( l ).f( u )<0 Step 2: The root estimate is: Step 3: Subdivide the interval aording to: If (f( l ).f( r )<0) the root lies in the lower subinterval; u = r and go to step 2. If (f( l ).f( r )>0) the root lies in the upper subinterval; l = r and go to step 2. If (f( l ).f( r )=0) the root is r and stop

26 The parahutist veloity is What is the drag oeffiient needed to reah a f() False position method: Eample veloity of 40 m/s if m =68.1 kg, t =10 s, g= 9.8 m/s 2 f f ( ) ( ) v mg ( 1 e t m mg (1 e ) (1 e m t v ) ) 40

27 False position method: Eample 1. Assume l = 12 and u =16 f( l )= and f( u )= The root: r = f(12). f( ) = < 0; 3. The root lies between 12 and f() f ( ) 4. Assume l = 12 and u = , f( l )=6.067 and f( u )= f( l ).f( u )<0 5. The new root r = This has an approimate error of 0.79% (1 e )

28 False position method: Eample

29 Flow Chart False Position Start Input: l, 0, e s, mai False f( l ). f( u )<0 i=0 e a =1.1e s while e a > e s & i <mai False Print: r, f( r ),e a, i r i i u 1 f ( )( ) u l u f ( ) f ( ) l u Stop

30 i=1 or r =0 True e a r r0 100% r Test=f( l ). f( r ) Test=0 True e a =0.0 Test<0 l = r r0 = r False True u = r r0 = r

31 False Position Method-Eample 2

32 False Position Method-Eample 2

33 Eel Goal Seek Tool

34 Roots of Polynomials: Using Software Pakages MS Eel:Goal seek f()=-os