Topic 4 Root Finding

Transcription

1 Couse Instucto D. Ramond C. Rump Oice: A 337 Phone: (915) E Mail: cump@utep.edu Topic 4 EE 4386/531 Computational Methods in EE Outline Intoduction Backeting Methods The Bisection Method False Position Method Open Methods Newton Raphson Method Secant Method 1

2 Intoduction 3 What is? What values o does () =? Let a b c We can igue this out algebaicall b b 4ac a But what i e This cannot be solved analticall. We need to use a numeical method. 4

3 Methods Backeting Methods Finding a single oot that alls within a known ange. Ve obust Must know something ahead o time. Open Methods Tial and eo iteative methods Do not need bounds, onl an initial guess. Moe eicient than backeting methods Can be unstable and not ind a solution Roots o Polnomials Algoithm speciic to polnomials Phsics o ou poblem must be it to a polnomial Able to ind all oots. Requies initial bounds L U Requies an initial guess 1 5 Recognizing Numbe o Roots Single Root Sign changes on eithe side o oot. Slope at oot is not zeo. Tiple Root Sign changes on eithe side o oot. Slope at oot is zeo. Cuvatue changes on eithe side o oot. Double Root Sign is same on eithe side o oot. Slope at oot is zeo. Cuvatue is same on eithe side o oot. Quaduple Root Sign is same on eithe side o oot. Slope at oot is zeo. Cuvatue is same on eithe side o oot. 6 3

4 Genealizing Algoithms Root inding algoithms ind all values o such that () =. What i we wish to ind all values o such that () = a? Genealization a a Let g a Now peom standad oot inding on g(). 7 Notes In man cicumstances, a single computation o () ma take hous, das, o weeks! In these cases it is highl desied to minimize the total numbe o computations o (). It is oten woth the investment o a ew hous to wite an awesome code that uns in an hou than waiting das o weeks o the answe to come om a simple code ou wote in a ew minutes. 8 4

5 The Bisection Method 9 Step 1 Pick a lowe and uppe bound, L and U that is known to contain a single oot (). -. L Root we wish to ind. () U 1 5

6 Step Calculate the midpoint between L and U as the ist guess o the oot (). -. L Fist guess U L () U 11 Step 3 Calculate the unction at the midpoint. L () U 1 6

7 Step 4 Adjust the bounds. L () U 13 Step 5 Calculate the new midpoint L Second guess U L ().4 () U

8 Step 6 Calculate the unction at the new midpoint U.8.6 L ().4 () Step 7 Adjust the bounds () L U ()

9 Step 8 Calculate a new midpoint U.8.6 ().4 (). L U L 17 Step 9 Calculate the unction at the new midpoint U.8.6 ().4 (). L

10 Step 1 Adjust the bounds U.8.6 ().4 (). L Step 11 Calculate a new midpoint U.8.6 ().4 (). L U L

11 Step 1 Calculate the unction as the new midpoint U (). () L Step 13 And so on Iteation 1 Iteation () () Iteation Iteation () () Iteation Iteation () () Iteation Iteation () ()

12 Adjusting the Bounds (1 o ) Be caeul about signs when adjusting the bounds. () Move uppe bound Hee the unction is positive and we move the uppe bound. () Move uppe bound Hee the unction is negative and we still move the uppe bound. 3 Adjusting the Bounds ( o ) i. I ( L )( ) <, then the oot is closest to L. Make U =. ii. I ( L )( ) >, then the oot is closest to U. Make L =. iii. I ( ) =, then the oot is ound. Done! % Adjust the Bounds i L*< U = ; U = ; elsei L*> L = ; L = ; else beak; end %oot towad L %oot towad U 4 1

13 When is the Algoithm Finished? i. Calculate the amount has moved om one iteation to the net. new old old ii. At the end o each iteation, check i is less than some theshold. toleance Rule o thumb: I ou want some numbe o digits o pecision, ensue is an ode o magnitude less than the desied pecision. WARNING! Do set ou convegence condition to U L <toleance because this will ail when the same bounda is being adjusted each iteation. 5 Algoithm o Bisection Method 1. Choose lowe and uppe bounds, L and U so that the suound a oot.. Evaluate the unction at the endpoints, ( L ) and ( U ). 3. Calculate midpoint. U L 4. Iteate until conveged a) Evaluate the unction at the midpoint ( ). b) Adjust the bounds. I L, then U I L, then L I, then DONE! c) Update the midpoint. U L d) Detemine i conveged i. Calculate step size new old new ii. Algoithm is conveged i < toleance. 6 13

14 Notes on Bisection Method Most obust oot inde Least eicient oot inde Guaanteed to ind a oot as long as the bounds span a cossing Sometimes good to check sign change o bounds No sign change o even numbe o oots. Sign change odd numbe o oots. 7 False Position Method 8 14

15 Moe Intelligent Midpoint False-Position Method new U L U L U L U L U L U L Bisection Method new U L Since L H, it is a good assumption that the oot is close to L than it is to H. 9 Step 1 Pick a lowe and uppe bound, L and U that is known to contain a single oot (). -. L Root we wish to ind. () U 3 15

16 Step Calculate a bette estimate o the position o the oot (). -. Bette estimate new L () U 31 Step 3 Calculate the unction at the new estimate o (). L () U

17 Step 4 Adjust the bounds () L () U 33 Step 5 Estimate the position o the oot b linea appoimation Second guess new.6.4 (). () L U

18 Deivation o Estimate (1 o 3) The equation o a line given a point (, ) and slope m is m Assuming the unction connecting the bounds is close to linea, the slope is m U U L L 35 Deivation o Estimate ( o 3) Let s choose the lowe bound to be the point in ou line equation. L L U L L U L m We now estimate the position o the oot b setting = and solving o. U L new L L U L new L U L U L L L 36 18

19 Deivation o Estimate (3 o 3) Let s choose the uppe bound to be the point in ou line equation. U U L U U U U L We now estimate the position o the oot b setting = and solving o. U L new U U U U new U U L U L L m 37 Intepetation o Estimate (1 o ) We have two possible equations to estimate the position o the oot. L new L U L U L U new U U L U L Let s aveage these equations. new L U L U L U U L U L U L U L U L U L U L 38 19

20 Intepetation o Estimate ( o ) U L new U L U L U L Tpical bisection method equation A coection tem. 39 When False Position Fails The alse position method can ail o ehibit etemel slow convegence when the unction is highl nonlinea between the bounds. This happens because the estimated oot is a ve poo guess. Slide 4

21 Notes on False Position Method Ve simila to bisection method Calculates a moe intelligent midpoint. Conveges much aste o nea linea unctions. Conveges slowe o unctions with abupt cuves 41 Newton Raphson Method 4 1

22 We Stat with Some Function 43 We Make an Initial Guess o the Position o the Root

23 Evaluate the Function at Calculate the Equation o the Line Tangential to the Point on () m

24 Calculate Whee the Line Cosses the Ais Evaluate the Function at

25 Calculate the Equation o the Line Tangential to the Point on () 1 m 49 Calculate Whee the Line Cosses the Ais

26 Evaluate the Function at And so on Conveged in ive iteations. 5 6

27 Algoithm 1. Deive analtical epessions o () and (), o ()/ ().. Detemine a good initial guess i. 3. Iteate until conveged 1. Calculate i. i i i. Calculate new estimate o oot i+1. i 1 i i 3. I i < toleance, Done! 53 Poo o Unstable Convegence 54 7

28 Notes on Newton Raphson Method Does not equie bounds. Requies a good initial guess. Requies () and () to be analtical. Conveges etemel ast o unctions that ae nea linea. Algoithm vulneable to instabilit Can convege to the wong oot i multiple oots eist. Newton Raphson s Method Newton s Method NR is o oot inding, wheeas N is o optimization. 55 Eample #1 Let () = sin. What is the oot o () in the poimit o = 4? Step 1 Deive analtical epession o ()/ () sin sin sin tan d sin cos d This means ou update equation is i 1 i i i tan i Step MATLAB code = 4; tol = 1e-6; d = in; Conveges to while abs(d) > tol ate 4 iteations. d = tan(); = - d; end Slide 56 8

29 Secant Method 57 We Stat with Some Function 58 9

30 We Make Two Initial Guesses o the Position o the Root, 1 and 1 59 Evaluate the Function at 1 and

31 Calculate the Equation o the Line Connecting ( 1, 1 ) and (, ) ise slope m un m Calculate Whee the Line Cosses the Ais

32 Evaluate the Function at the New Point Adjust Points to 1 and

33 Calculate Whee the New Line Cosses the Ais Evaluate the Function at the New Point

34 Adjust Points to 1 and And so on

35 Algoithm 1. Detemine two good initial guesses: 1 and.. Evaluate the unction at Iteate until conveged 1. Evaluate unction at.. Calculate Make new ist point the old second point. 4. Calculate new. 1 and I < toleance, Done! 69 Notes on Secant Method Does not equie bounds. Does not equie () o () to be analtical Requies two good initial guesses Full numeical vesion o Newton s method Same weaknesses as Newton Raphson method Algoithm vulneable to instabilit Can convege to the wong oot 7 35

36 Eample #1 Let () = sin. What is the oot o () in the poimit o = 4? 1. Detemine two good initial guesses: 1 =4. and =3.9.. Evaluate the unction at 1. % DASHBOARD 1 1 unc 3. Iteate until conveged 1 = 4.; 1. Evaluate unction at = 3.9;. tol = 1e-6; 1. Calculate. % IMPLEMENT SECANT METHOD 1 = unc(1); d = in; 1 while abs(d) > tol 1 = unc();. Make new ist point the old second point. d = ( - 1)*/( - 1); 1 = ; 1 and 1 1 = ; 3. Calculate new = - d;. end = ; 4. I < toleance, Done! Conveges to ate 5 iteations. Slide 71 36