Root Finding COS 323

Transcription

1 Root Fidig COS 33

2 Why Root Fidig? Solve or i ay equatio: b where? id root o g b 0 Might ot be able to solve or directly e.g., e -0. si3-0.5 Evaluatig might itsel require solvig a dieretial equatio, ruig a simulatio, etc.

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

4 1-D Root Fidig Give some uctio, id locatio where 0 Need: Startig positio 0, hopeully close to solutio Ideally, poits that bracket the root + > 0 < 0

5 1-D Root Fidig Give some uctio, id locatio where 0 Need: Startig positio 0, hopeully close to solutio Ideally, poits that bracket the root Well-behaved uctio

6 What Goes Wrog? Taget poit: very diicult to id Sigularity: brackets do t surroud root Pathological case: iiite umber o roots e.g. si1/

7 Bisectio Method Give poits + ad that bracket a root, id hal ½ + + ad evaluate hal I positive, + hal else hal Stop whe + ad close eough I uctio is cotiuous, this will succeed i idig some root

8 Bisectio Very robust method Covergece rate: Error bouded by size o [ + ] iterval Iterval shriks i hal at each iteratio Thereore, error cut i hal at each iteratio: ε +1 ½ ε This is called liear covergece Oe etra bit o accuracy i at each iteratio

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

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

11 Secat Method Simple etesio to bisectio: iterpolate or etrapolate through two most recet poits 3 1 4

12 Secat Method Faster tha bisectio: ε +1 cost. ε 1.6 Faster tha liear: umber o correct bits multiplied by 1.6 Drawback: the above oly true i suicietly close to a root o a suicietly smooth uctio Does ot guaratee that root remais bracketed

13 False Positio Method Similar to secat, but guaratee bracketig Stable, but liear i bad cases

14 Other Iterpolatio Strategies Ridders s method: it epoetial to +,, ad hal Va Wijgaarde-Dekker-Bret method: iverse quadratic it to 3 most recet poits i withi bracket, else bisectio Both o these sae i uctio is asty, but ast super-liear i uctio is ice

15 Newto-Raphso Best-kow algorithm or gettig quadratic covergece whe derivative is easy to evaluate Aother variat o the etrapolatio theme Slope derivative at 1 + 1

16 Newto-Raphso Begi with Taylor series Divide by derivative ca t be zero! 0... wat δ δ δ 1 ~ 0 0 Newto Newto ε ε δ δ δ δ δ δ δ δ

17 Newto-Raphso Method ragile: ca easily get coused Good startig poit critical Newto popular or polishig o a root oud approimately usig a more robust method

18 Newto-Raphso Covergece Ca talk about basi o covergece : rage o 0 or which method ids a root Ca be etremely comple: here s a eample i -D with 4 roots

19 Popular Eample o Newto: Square Root Let a: zero o this is square root o a, so Newto iteratio is a Divide ad average method a +

20 Reciprocal via Newto Divisio is slowest o basic operatios O some computers, hardware divide ot available!: simulate i sotware Need oly subtract ad multiply b b a b b b a + 0 *

21 Rootidig i >1D Behavior ca be comple: e.g. i D wat, y wat g, y 0 0

22 Rootidig i >1D Ca t bracket ad bisect Result: ew geeral methods

23 Newto i Higher Dimesios Start with Write as vector-valued uctio 0, 0, wat wat y g y,, y g y

24 Newto i Higher Dimesios Epad i terms o Taylor series wat + δ + δ is a Jacobia J y

25 Newto i Higher Dimesios Solve or δ δ J 1 Requires matri iversio we ll see this later Ote ragile, must be careul Keep track o whether error decreases I ot, try a smaller step i directio δ