6 Roots of Equations: Open Methods

Transcription

1 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 6 Roots of Equtons: Open Methods Smple Fed-Pont Iterton Newton-Rphson Secnt Methods MATLAB Functon: fzero Polynomls Cse Study: Ppe Frcton Brcketng vs. open methods Grphcl depcton of the fundmentl dfference etween the rcketng nd nd c open methods for root locton. In, whch s secton, the root s constrned wthn the ntervl prescred y l nd u. In contrst, for the open method depcted n nd c, whch s Newton-Rphson, formul s used to project from to n n tertve fshon. Thus the method cn ether dverge or c converge rpdly, dependng on the shpe of the functon nd the vlue of the ntl guess. DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t

2 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Smple Fed-Pont Iterton lso clled, one-pont terton or successve susttuton rerrngng f = so tht s on the LHS; = g y lgerc mnpulton or y smply ddng to oth sdes - formul predctng new vlue of s functon of n old vlue of ; - ppromte error; ε = % Emple Use smple fed-pont terton to locte the root of f = e. Sol Notce tht the true % reltve error s roughly proportonl to the error from the prevous terton y fctor of out.5 ~.6 lner convergence; chrcterstc of fed-pont terton; E t, Et, DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t

3 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Posslty of convergence convergence dvergence Grphcl depcton of nd convergence nd c nd d dvergence of smple fed-pont terton. Grphs nd c re clled monotone ptterns wheres nd d re clled osclltng or sprl ptterns. Note tht convergence occurs when g' <. Fed-pont terton converges f g < or f the mgntude of the slope of g s less thn the slope of the lne f =. Recll, = g. Suppose tht the true soluton s = g. r r Sutrctng, then we hve = g g r r From the dervtve men-vlue theorem, nd n whch f lettng = nd = r, or g ξ g g = g g = g ξ r r r r whch yelds r = r g ξ If the true error for terton s defned s = g ξ E E t, t, E t, = r, then we hve Consequently, f g < the errors ; f g > the errors f f g > g < the errors wll e postve monotonc the errors wll oscllte DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t 3

4 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Coffee Brek Dervtve men-vlue theorem sttng tht f functon g nd ts frst dervtve re contnuous over n ntervl, then there ests t lest one vlue of = ξ wthn the ntervl such tht g ξ = g g sttng tht there s t lest one pont etween nd tht hs slope, desgnted g'ξ, whch s prllel to the lne jonng g nd g DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t 4

5 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Newton-Rphson most wdely used root-loctng formul Grphcl depcton of the Newton- Rphson method. A tngent to the functon of [tht s, f'] s etrpolted down to the s to provde n estmte of the root t. ntl guess t the root = etendng tngent from [, f ] n mproved estmted root s the pont where the tngent crosses the s f f = Newton-Rphson formul Emple Use the Newton-Rphson method to estmte the root of f = e employng n ntl guess of =. Sol DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t 5

6 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 the error s roughly proportonl to the squre of the prevous error; qudrtc convergence; poor performnce for multple roots E E t, t, Recll the Tylor seres epnson, f f ξ! = f f T runctng fter the frst dervtve term, f f f At the ntersecton wth the s, f f Or = f = Newton-R t f phson formul derved from he Tylor seres epnson ettng nd susttutng long wth L = r r = f f f ξ! = r r f, S utrctng = f f from the ove equton; And relzng th e error, E t, = r The n we hve, f ξ! = f Et, Et, If ssumng convergence, oth nd ξ should eventully e ppromted y the root r, E t, f r = E f r t, DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t 6

7 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Emple Determne the postve root of f = usng the Newton-Rphson method nd n ntl guess of =.5. Sol DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t 7

8 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Four cses where the Newton-Rphson method ehts poor convergence zero slope [f' = ] cuses dvson y zero n N-R formul convergence of N-R method ~ nture of the functon ccurcy of the ntl guess <An M-fle to mplement the Newton-Rphson method> DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t 8

9 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Secnt Methods N-R method wthout hvng to compute dervtves Susttutng ckwrd fnte dvded dfference of the dervtve to the N-R formul, or = f f then we hve secnt method requrng two ntl estmtes of ; - nd rcketng method? ut not requred to chnge sgns of f etween the estmtes open method Alterntvely, tkng frctonl perturton whch requres one ntl guess; f f δ f δ then we hve modfed secnt method Wht s the dfference etween the Newton-Rphson nd secnt methods? DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t 9

10 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Emple Use the modfed secnt method to determne the mss of the ungee jumper wth drg coeffcent of.5 kg/m to hve velocty of 36 m/s fter 4 s of free fll. Note: The ccelerton of grvty s 9.8 m/s. Use n ntl guess of 5 kg nd vlue of -6 for the perturton frcton. Sol DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t

11 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t Müller's Method nsted of projectng strght lne to the s through two functon vlues secnt method, projectng prol through three ponts use of the qudrtc formul whch mens tht oth rel nd comple roots cn e locted mjor eneft of Müller's method Wrtng the prolc equton n convenent form; c f = Evlutng y susttutng ech of three ponts [, f ], [, f ], nd [, f ]; c f = c f = c c f = = Note tht,, nd re ntl guesses. Reducng; f f = f f =

12 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Lettng; Hence we hve; h = h δ h δ h h h h h = hδ Solvng for nd ; Applyng the qudrtc formul whch s roust to the round-off error, ± = 4c It s noted tht f >> 4c ± zero vlue s possle sutrctve cncellton Then we hve the reltonshp etween the present 3 nd the prevous root estmte ; f = c c = or ± 4c 3 = ± c 4c The sgn s chosen to gree wth the sgn of, whch wll result n the lrgest denomntor, nd gve the root estmte tht s close to. And the error cn e clculted s; ε = 3 % 3 DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t

13 HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 Emple Use Müller's method wth guesses of, nd = 4.5, 5.5, nd 5, respectvely, to determne root of the equton 3 f = 3. Note tht the roots of ths equton re -3, -, nd 4. Sol DM869/Computtonl Numercl Anlyss/6_cn.doc Avlle t 3