Roots: Open Methods. Berlin Chen Department of Computer Science & Information Engineering National Taiwan Normal University

Transcription

1 Roots: Open Methods Berln Chen Department of Computer Scence & Informaton Engneerng Natonal Tawan Normal Unverst Reference:. Appled Numercal Methods wth MATLAB for Engneers, Chapter 6 & Teachng materal

2 Chapter Objectves / Recognzng the dfference between bracketng and open methods for root locaton Understandng the fed-pont teraton method and how ou can evaluate ts convergence characterstcs Knowng how to solve a roots problem wth the Newton- Raphson method and apprecatng the concept of quadratc convergence a b c 0 b b a 4ac a 5 b 4 c 3 d e f 0? sn 0? NM Berln Chen

3 Chapter Objectves / Knowng how to mplement both the secant and the modfed secant methods Knowng how to use MATLAB s fzero functon to estmate roots Learnng how to manpulate and determne the roots of polnomals wth MATLAB NM Berln Chen 3

4 Recall: Taonom of Root-fndng Methods Nonlnear Equaton Solvers Bracketng Graphcal Open Methods Chapter 5 Incremental Search Bsecton False Poston Smple Fed-Pont Iteraton Newton Raphson Secant Chapter 5 Chapter 6 We can also emplo a hbrd approach Bracketng + Open Methods NM Berln Chen 4

5 Open Methods Open methods dffer from bracketng methods, n that open methods requre onl a sngle startng value or two startng values that do not necessarl bracket a root Open methods ma dverge as the computaton progresses, but when the do converge, the usuall do so much faster than bracketng methods NM Berln Chen 5

6 Graphcal Comparson of Root-fndng Methods NM Berln Chen 6

7 Smple Fed-Pont Iteraton Rearrange the functon f=0 so that s on the lefthand sde of the equaton: =g Use the new functon g to predct a new value of -that s, + =g The appromate error s gven b: a 00% NM Berln Chen 7

8 Smple Fed-Pont Iteraton: An Eample / Solve f=e - - Re-wrte as =g b solatng eample: =e - Start wth an ntal guess here, 0 a % t % t / t The true percent relatve error s roughl proportonal a factor of about 0.5 to 0.6 to the error from the prevous teraton. Contnue untl some tolerance s reached NM Berln Chen 8

9 Smple Fed-Pont Iteraton: An Eample / 0 3 NM Berln Chen 9

10 Convergence Convergence of the smple fedpont teraton method requres that the dervatve of g near the root has a magntude less than Convergent, 0 g < Convergent, -<g 0 3 Dvergent, g > 4 Dvergent, g <- Chapra and Canale 00 have shown that the error for an teraton s lnearl proportonal to the error from the prevous teraton multpled b the absolute value of the slope dervatve of g: E g E NM Berln Chen 0

11 Newton-Raphson Method Based on formng the tangent lne to the f curve at some guess, then followng the tangent lne to a pont where t crosses the -as Such a pont usuall represents an mproved estmate of the root NM Berln Chen 0 f f f f

12 Newton-Raphson Method: Pros and Cons Pro: The error of the + th teraton s roughl proportonal to the square of the error of the th teraton - ths s called quadratc convergence Con: Some functons show slow or poor convergence Chapra and Canale 00 have shown that the error s roughl proportonal to the square of the prevous error: E t, f f E t, NM Berln Chen

13 Secant Methods / A potental problem n mplementng the Newton- Raphson method s the evaluaton of the dervatve - there are certan functons whose dervatves ma be dffcult or nconvenent to evaluate For these cases, the dervatve can be appromated b a backward fnte dvded dfference: f ' f f NM Berln Chen 3

14 Secant Methods /3 Substtuton of ths appromaton for the dervatve to the Newton-Raphson method equaton gves: f f f Note - ths method requres two ntal estmates of but does not requre an analtcal epresson of the dervatve NM Berln Chen 4

15 Secant Methods 3/3 Modfed Secant Method Rather than usng two arbtrar values to estmate the dervate, an alternatve approach nvolves a fractonal perturbaton of the ndependent varable to estmate f NM Berln Chen 5 f f f f f f

16 Brent s Root-locaton Method A hbrd approach that combnes the relablt of bracketng wth the speed of open methods Tr to appl a speed open method whenever possble, but revert to a relable bracketng method f necessar That s, n the event that the open method generate an unacceptable result.e., an estmate fallng outsde the bracket, the algorthm reverts to the more conservatve bsecton method Developed b Rchard Brent 973 Here the bracketng technque beng used s the bsecton method, whereas two open methods, namel, the secant method and nverse quadratc nterpolaton, are emploed Bsecton tpcall domnates at frst but as root s approached, the technque shfts to the fast open methods NM Berln Chen 6

17 Inverse Quadratc Interpolaton /4 Inverse quadratc nterpolaton s smlar n sprt to the secant method The secant method: compute a straght lne that goes through two guesses and take the ntersecton of the straght lne wth the as as the new root estmate Inverse quadratc nterpolaton: compute parabola quadratc curve, a functon of, that goes through three ponts and take the ntersecton of the parabola wth the as as the new root estmate However, t s possble that the parabola mght not ntersect the as Inverse quadratc nterpolaton rectfes the dffcult b fttng the ponts wth a parabola n a functon of NM Berln Chen 7 g Ths form s also called a Lagrange polnomal.

18 Inverse Quadratc Interpolaton /4 NM Berln Chen 8

19 Inverse Quadratc Interpolaton 3/4 The nverse quadratc nterpolaton =f alwas ntersect the as. NM Berln Chen 9

20 Inverse Quadratc Interpolaton 4/4 The new root estmate, +, therefore corresponds to =0 -Substtuted nto the equaton shown above, we can have NM Berln Chen 0

21 An Eample Functon for the Brent s Method NM Berln Chen

22 MATLAB s fzero Functon MATLAB s fzero provdes the best qualtes of both bracketng methods and open methods. Usng an ntal guess: = fzerofuncton, 0 [, f] = fzerofuncton, 0 functon s a functon handle to the functon beng evaluated 0 s the ntal guess s the locaton of the root f s the functon evaluated at that root Usng an ntal bracket: = fzerofuncton, [0 ] [, f] = fzerofuncton, [0 ] As above, ecept 0 and are guesses that must bracket a sgn change NM Berln Chen

23 fzero Optons Optons ma be passed to fzero as a thrd nput argument - the optons are a data structure created b the optmset command optons = optmset par, val, par, val, par n s the name of the parameter to be set val n s the value to whch to set that parameter The parameters commonl used wth fzero are: dspla: when set to ter dsplas a detaled record of all the teratons tol: A postve scalar that sets a termnaton tolerance on NM Berln Chen 3

24 fzero Eample optons = optmset dspla, ter ; Sets optons to dspla each teraton of root fndng process [, f] = fzero@ ^0-, 0.5, optons Uses fzero to fnd roots of f= 0 - startng wth an ntal guess of =0.5 MATLAB reports =, f=0 after 35 functon counts NM Berln Chen 4

25 Polnomals / MATLAB has a bult n program called roots to determne all the roots of a polnomal - ncludng magnar and comple ones. = rootsc s a column vector contanng the roots cs a row vector contanng the polnomal coeffcents Eample: Fnd the roots of f= = roots[ ] NM Berln Chen 5

26 Polnomals / MATLAB s pol functon can be used to determne polnomal coeffcents f roots are gven: b = pol[0.5 -] Fnds f where f =0 for =0.5 and =- MATLAB reports b = [ ] Ths corresponds to f= MATLAB s polval functon can evaluate a polnomal at one or more ponts: a = [ ]; If used as coeffcents of a polnomal, ths corresponds to f= polvala, Ths calculates f, whch MATLAB reports as NM Berln Chen 6