Root Finding COS 323

Transcription

1 Root Fidig COS 323

2 Remider Sig up for Piazza Assigmet 0 is posted, due Tue 9/25

3 Last time.. Floatig poit umbers ad precisio Machie epsilo Sources of error Sesitivity ad coditioig Stability ad accuracy Asymptotic aalysis ad covergece order

4 Today Root fidig defiitio & motivatio Stadard techiques for root fidig Algorithms, covergece, tradeoffs Eample applicatios of Newto s Method Root fidig i > 1 dimesio

5 1-D Root Fidig Give some fuctio, fid locatio where f0

6 Why Root Fidig? Solve for i ay equatio: f b where? fid root of g f b 0 Might ot be able to solve for directly e.g., f e -0.2 si3-0.5 Evaluatig f might itself require solvig a differetial equatio, ruig a simulatio, etc.

7 Why Root Fidig? Egieerig applicatios: Predict depedet variable e.g., temperature, force, voltage give idepedet variables e.g., time, positio Focus o fidig real roots

8 Bracket-Based Methods Give: Poits that bracket the root A well-behaved fuctio Ca always fid some root f + > 0 f < 0

9 What Goes Wrog? Taget poit: very difficult to fid Sigularity: brackets do t surroud root Pathological case: ifiite umber of roots e.g. si1/

10 Bisectio Method Give poits + ad that bracket a root, fid half ½ + + ad evaluate f half If positive, + half else half Stop whe + ad close eough If fuctio is cotiuous, this will succeed i fidig some root

11 Error Covergece of Iterative Methods Absolute error boud ε at step : ε bouds estimated at step true Covergece: describes how ε +1 relates to ε Liear covergece: ε +1 c ε for some c 0,1 Superliear covergece: ε +1 c ε q for some c 0,1, q > 1

12 Liear: Superliear: Subliear:

13 Bisectio Error Covergece Very robust method: guarateed to fid root! Covergece rate: Error bouded by size of [ + ] iterval Iterval shriks i half at each iteratio So, error boud cut i half at each iteratio: ε +1 ½ ε Liear covergece! Oe etra bit of accuracy i at each iteratio

14 Faster Root-Fidig Facier methods get super-liear covergece Typical approach: model fuctio locally by somethig whose root you ca fid eactly Model did t match fuctio eactly, so iterate I may cases, these are less safe tha bisectio

15 Secat Method Iterpolate or etrapolate through two most recet poits

16 Secat Method Covergece Faster tha bisectio: ε +1 c ε 1.6 Faster tha liear: umber of correct bits multiplied by 1.6 Drawback: oly true if sufficietly close to a root of a sufficietly smooth fuctio Does ot guaratee that root remais bracketed

17 False Positio Method Similar to secat, but guaratees bracketig Stable, but liear i bad cases

18 False Positio Failure 2 1 3

19 Other Iterpolatio Strategies Ridders method: fit epoetial to f +, f, ad f half Va Wijgaarde-Dekker-Bret method: iverse quadratic fit to 3 most recet poits if withi bracket, else bisectio Both of these safe if fuctio is asty, but fast super-liear if fuctio is ice

20 Demo

21 Newto-Raphso Best-kow algorithm for gettig quadratic covergece whe derivative is easy to evaluate Quadratic: # correct bits doubles each iteratio! ε +1 c ε 2 Aother variat o the etrapolatio theme Slope derivative at f f

22 Newto-Raphso covergece Begi with Taylor series Divide by derivative ca t be zero! f +δ f +δ f +δ 2 f wat ~ Newto Newto f f f f f f f f ε ε δ δ δ δ δ δ δ δ

23 Newto-Raphso Method fragile: ca easily get cofused Good startig poit critical Newto popular for polishig off a root foud approimately usig a more robust method Quadratic oly for simple root

24 Newto-Raphso Covergece Ca talk about basi of covergece : rage of 0 for which method fids a root Ca be etremely comple: here s a eample i 2-D with 4 roots

25 Commo Eample of Newto: Square Root Let f 2 a: zero of this is square root of a f 2, so Newto iteratio is Divide ad average method ~2000 B.C. a a f f +

26 Reciprocal via Newto Divisio is slowest of basic operatios O some computers, hardware divide ot available!: simulate i software Need oly subtract ad multiply b b a b b f b f a *

27 Rootfidig i >1D Behavior ca be comple: e.g. i 2D wat f, y wat g, y 0 0

28 Rootfidig i >1D Ca t bracket ad bisect Result: few geeral methods

29 Newto i Higher Dimesios Start with Write as vector-valued fuctio 0, 0, wat wat y g y f,, y g y f f

30 Newto i Higher Dimesios Epad i terms of Taylor series wat f + δ f + f δ f is a Jacobia f f J y f

31 Newto i Higher Dimesios 1-dimesioal case: δ f / f N-dimesioal: Solve for δ δ J 1 f Requires matri iversio we ll see this later Ofte fragile must be careful Keep track of whether error decreases If ot, try a smaller step i directio δ

32 Recap: Tradeoffs Bracketig methods Bisectio, False-positio Stable, slow Ope methods Secat, Newto Possibly diverget, fast Newto requires derivative Hybrid methods Bret Combie bracketig & ope methods i a pricipled way

33 Practical otes Root-fidig i Matlab: fzero: For fidig root of a sigle fuctio Combies safe ad fast methods roots: For fidig polyomial roots Ecel: Goal Seek: Drive a equatio to 0 by adjustig 1 parameter Solver: Ca vary multiple parameters simultaeously, also miimize & maimize Tip: Plot your fuctio first!!!