. Determine these to one correct decimal accuracy using the bisection method: (a) 2. The following equations all have a root in the interval ( 0,1.

Transcription

1 PROBLEMS Us grhic rrsttio to dtrmi th zros of th followig fuctios to o corrct dciml : ( 4 4si ; (b ; (c ( ; (d 4 8 ; ( ; (f ; (g t I ordr to obti grhicl solutio of f ( o th itrvl [,b], ty th followig sttmts ito th MATLAB commd widow: Hit : Ply with th brct [,b] zlot( f(,,b hold o, zlot(,,b Th followig qutios ll hv root i th itrvl (,6 Dtrmi ths to o corrct dciml ccurcy usig th bisctio mthod: ( (c cos l ; (b ; Us hd comuttio d th MATLAB cod bisctiom Hit : You my us zlot to dtrmi suitbl brct withi th giv itrvl Rcll tht dciml ccurcy cotrol is chivd by th us of bsolut rror ty rssio with suitbl form of TOLrc Us th bisctio mthod to fid solutios ccurt to two dcimls for o ( [, ] ; (b [,] ; (c [,] ; (d [,] Us hd comuttio d th MATLAB cod bisctiom Not : I rogrmmig this olyomil fuctio, us std multilictio form 4 Rcll tht i th th st of th bisctio mthod th root r c b stimtd s (b (b r c (b whr c ( b is th midoit If rror tolrc hs b rscribd i dvc, it is ossibl to dtrmi th umbr of sts rquird i th bisctio mthod Suos tht w wt ( r c : b By tig logrithms, o obtis, th it is cssry to solv th followig iqulity for l(b l( l If d b, how my sts of th bisctio mthod r dd to dtrmi th root r to ight dcimls ccurcy? Hit : This lysis should giv hit for ossibl bsolut rror ty rssio for dciml cotrol 5 Try to dvis stoig critrio for th bisctio mthod to gurt tht th root is dtrmid with rltiv rror t most Qutify for t digit ccurcy Hit : At y giv stg of bisctio rocss, th brct [,b] d th midoit c r ow A vilbl stimt of th uow ct qutity (th root my b usd to ormliz bsolut rror to gt th corrsodig rltiv rror Th bcoms suitbl form of TOLrc

2 6 Us th bisctio mthod to dtrmi th oit of itrsctio of th curvs giv by y d MATLAB cod bisctiom y to two sigiifict digits Us hd comuttio d th Hit : Covrt ito root-fidig roblm d imos digit ccurcy cotrol s chivd by th us of rltiv rror ty rssio d suitbl form of TOLrc 7 Fid roimtio to 5 corrct to 4 sigiifict digits usig th MATLAB cod bisctiom (cosidr f ( 5 8 Rviw qustios : ( How my biry digits of rcisio r gid i ch st of th bisctio mthod d how my sts r rquird for ch dciml digit of rcisio? (b Wht do w m wh w sy tht itrtio mthod is qudrticlly covrgt; (c Udr wht ssumtios is Nwto s mthod qudrticlly covrgt? Hit : Dig ito th lctur ots 9 Us Nwto s mthod to dtrmi th ozro roots of ( 4 ; (b ; (c l to two sigiifict digits Us hd comuttio d th MATLAB cod wtom Hit : Nwto grts squc of imrovd stimts of th root So, othig is mor turl th chcig th cosqutiv lmts of th squc for digits grmt (rl rr with scil TOL Wht lir fuctio y b roimts f ( si bst i th viciity of 4? How dos this roblm rlt to Nwto s mthod? Hit : Dos tgt-li rig bll? Vrify tht wh Nwto s mthod is usd to comut R (by solvig th qutio R d show tht, th squc of itrts is dfid by R R R Itrrt this qutio i trms of qudrtic covrgc Hit : C you fid R d R i th lst rssio? How r thy rltd? Us th sct mthod to dtrmi th roots of th followig qutios to two corrct dcimls : ( cod sctm ; (b t cosh Us hd comuttio d th MATLAB Hit : zlot to th rsqu to gt brct or suitbl iitil gusss How to imos dciml cotrol?

3 O wts to solv th qutio l, whos root is r 5, by itrtio, do chooss mog th followig itrtio formuls: (i ( l ; (ii ; (iii ( Which of th formuls c b usd? (b Which formul should b usd? (c Giv v bttr formul Hit : For (, umricl rimts with 5 or th coditio ( r 5 K my hl For (b, thi of th rltio btw K d th sd of covrgc For (c, dos Nwto rig bll? 4 Th qutio (for th squr root r c b writt quivltly i th form ( i my diffrt wys: (i ( ; (ii ( ( ( ; (iii Discuss th covrgc (or ocovrgc bhvior of th itrtio (,,,,, for ch of ths thr itrtio fuctios I cs of covrgc, which o is th fstst covrgt? Hit : Agi, th coditio ( r K my hl 5 O wts to us th itrtio formul to solv th qutio Ivstigt if d to wht th itrtio squc covrgs for vrious choics of Hit : This clls for umricl rimts d grhicl tchiqus (s lctur ots 6 Dtrmi rmtrs, b, d c so tht th ordr of th itrtiv mthod for i s b s c s bcoms s high s ossibl For this choic of, b, d c, idict how th rror ( r dds o th rror i ( 5 r Hit : Wll, thr uows, b d c rquir thr qutios for uiqu dtrmitio O qutio, of cours, coms from th fid oit chc ( s s to do with th sd (ordr of covrgc Wht bout th othr two? Thy my b somthig 7 (i Show tht ch of th followig fuctios hs fid oit t r rcisly wh f (r, 4 4 whr f ( : ( ; (b ( ( ; (c ( ( ( 4 ( ( 4 ; (d ( ( (4 4 (ii Prform 4 four itrtios, by lttig d (,,,, o ch of th fuctios i (iii Which fuctio do you thi givs th bst roimtio to th solutio? Hit : Elor th wys th qutio f ( c b ut ito th forms ( 8 Us fid-oit itrtio mthod to dtrmi solutio ccurt to two dcimls for si o [, ] Us

4 Hit : Costruct suitbl itrtio fuctio ( to gt to th root i [, ] usig Aly th dciml cotrols to th rsultig squc 9 Solv for th root i [, ], usig fid-oit itrtio mthod Obti roimtio to th root ccurt to two dcimls Hit : Costruct suitbl itrtio fuctio ( to gt to th root i [, ] Aly th dciml cotrols to th rsultig squc For ch th followig qutios, dtrmi fuctio d itrvl [, b ] o which fid-oit itrtio will covrg to ositiv solutio of th qutio: ( ; (b cos Fid th solutios to four sigifict digits Hit : This clls for grhicl d umricl rimttio Do ot forgt to ly th digit cotrols to th rsultig squc Cosidr th siml olir qutio f ( Kowig tht this qutio hs two roots or 8, ivstigt th rcticbility of th fid-oit itrtio First cosidr th followig itrtiv formul : ( ( Notig tht th first drivtiv of this itrtio fuctio is (, dtrmi which solutio ttrcts this itrtio I dditio, ru th MATLAB routi fidoitm to rform th itrtio with iitil oits,, d Wht dos th routi yild for ch iitil oit? b Now, cosidr th followig itrtiv formul: ( b Notig tht th first drivtiv of this itrtio fuctio is ( dtrmi which solutio ttrcts I dditio, ru th MATLAB routi fidoitm to rform th itrtio with iitil oits,, d Wht dos th routi yild for ch iitil oit? Not : This rcis illustrts tht th outcom of lgorithm my dd o th strtig oit Cosidr th olir qutio f ( t( Obti grhicl solutio of this qutio by tyig th followig sttmts ito th MATLAB commd widow: zlot( t(i-,-i/,*i/ hold o, zlot(,-i/,*i/ to dmostrt tht th qutio hs ifiitly my roots Us th bisctio mthod for fidig th solutio btw 5 d to 4-dciml ccurcy Could you gt th right solutio? If ot, li why you fild d suggst how to m it

5 b Show tht th sig of f ( chgs btw 5 d d lso btw d 5 Notig this, try usig th bisctio mthod to fid solutio btw 5 d Chc th vlidity of th solutio, tht is, chc if f ( or ot If th solutio is ot good, li th rso c I ordr to fid th solutio roud by usig th fid-oit itrtio with th iitil oit, us th itrtio fuctio ( t( Could you gt th solutio r? Will it b bttr if you strt th routi with y diffrt iitil oit? Wht is wrog? d Now, try with othr itrtio fuctio ( t Wht could you gt? Is it th right solutio? Dos this itrtio fuctio wor with diffrt iitil vlu, li or 6, which r fr from th solutio w wt to fid? Rcll tht th sct mthod ws dvisd to rmov th cssity of th drivtiv d imrov th covrgc But, it somtims turs out to b wors tht th Nwto mthod Aly th routis wtom d sctm to solv f ( strtig with th iitil oit o tim d for othr tim 4 ( For wht strtig vlus will Nwto s mthod covrg for th fuctio f ( (? (b Wht hs if th Nwto itrtio is lid to f ( rct with? For wht strtig vlus will Nwto s mthod covrg? (c Strtig t,, or, lyz wht hs wh Nwto s mthod is lid to th fuctio f ( 9 5 ; (d Rt for f (, strtig with or Hit : This clls for umricl rimts d grhicl tchiqus (s lctur ots 5 Usig clcultor, obsrv th sd with which Nwto s mthod covrgs i th cs of f ( m ( with m d 8 Us 6 I ordr to cclrt Nwto mthod for multil roots, Nwto mthod is modifid s ( f ( M d (b f ( u( whr u( f ( u( f ( with M : th ordr of multilicity of th root (if ow w wt to fid Bsd o ths two ids, modify th routi wtom to solv f ( ( Us Nwto s mthod to solv th qutio strtig with d itrt util fiv-dciml ccurcy is obtid Do th rsults sm uusul** for Nwto s mthod? Us suitbly modifid Nwto s mthod to rdo th clcultios **Not: O my us th followig stimtor for th ordr (sd of covrgc :

6 c c l l whr Hit : Th root my hv (uow multilicity > 8 Aly Nwto mthod to fid th zro of th fuctio f ( sig( for which f ( f ( ( Could you gt th right solutio? If ot, li why you fild A lot of f ( my b obtid by tyig th followig sttmts ito th MATLAB commd widow: zlot( sig(-*sqrt(bs(-,,4 Similrly, ivstigt th bhvior of th sct mthod o this fuctio 9 Hr is cubic olyomil with thr closly scd rl roots ( ( Plot ( for 4 7 Show th loctio of th thr roots (b Strtig with 5, wht dos Nwto's mthod do? (c Strtig with d, wht dos th sct mthod do? (d Strtig with th itrvl [,], wht dos bisctio do? Not : I rogrmmig this olyomil fuctio, us std multilictio form Th formul for th sct mthod c lso b writt s f ( f ( f ( f ( f ( f ( f ( Estblish this, d li why th lttr form is ifrior i fiit rcisio viromt Hit : I th first cs, is imrovd uo s much s th right-most rssio rmits I th scod cs, is comutd usig covolutd rssio tht my cus cclltio rrors Cosidr Nwto mthod with Armijo rul whr th Nwto itrtio for fidig th root r, f( r, is modifid to rd f( f( with m, m itgr It is usful i th globliztio of th Nwto mthod i thos css whr th w guss f ( h f ( d thus h lds to icrs i th mgitud of th fuctio o should try smllr st siz h If th mgitud of f still icrss, o

7 should ly th rductio, h, h, rtdly, util it lds to locl 4 8, dcrs i f( Modify Nwtom to imlmt th Armijo rul log th suggstd lis (or writ your ow cod: F = F(P; FP = FP(P; D = -F/FP; PN = P + D; FN = F(PN; d if bs(fn < bs(f, P = PN; ls D = D/; Tst th cod o th roblm: Solv f ( rct( for r by strtig th Nwto itrtio t