Chapter Newton-Raphson Method of Solving a Nonlinear Equation
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1 Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson method to solve nonlner equton, nd 4. dscuss the drwbcks of the Newton-Rphson method. Introducton Methods such s the bsecton method nd the flse poston method of fndng roots of nonlner equton f ( requre brcketng of the root by two guesses. Such methods re clled brcketng methods. These methods re lwys convergent snce they re bsed on reducng the ntervl between the two guesses so s to zero n on the root of the equton. In the Newton-Rphson method, the root s not brcketed. In fct, only one ntl guess of the root s needed to get the tertve process strted to fnd the root of n equton. The method hence flls n the ctegory of open methods. Convergence n open methods s not gurnteed but f the method does converge, t does so much fster thn the brcketng methods. Dervton The Newton-Rphson method s bsed on the prncple tht f the ntl guess of the root of f ( s t, then f one drws the tngent to the curve t f (, the pont + where the tngent crosses the -s s n mproved estmte of the root (Fgure. Usng the defnton of the slope of functon, t f ( tn θ f (, + whch gves f ( + ( f (.4.
2 .4. Chpter.4 Equton ( s clled the Newton-Rphson formul for solvng nonlner equtons of the form f (. So strtng wth n ntl guess,, one cn fnd the net guess, +, by usng Equton (. One cn repet ths process untl one fnds the root wthn desrble tolernce. Algorthm The steps of the Newton-Rphson method to fnd the root of n equton (. Evlute f ( symbolclly. Use n ntl guess of the root, f ( + f ( f re, to estmte the new vlue of the root, +, s. Fnd the bsolute reltve ppromte error s Compre the bsolute reltve ppromte error wth the pre-specfed reltve error tolernce, s. If > s, then go to Step, else stop the lgorthm. Also, check f the number of tertons hs eceeded the mmum number of tertons llowed. If so, one needs to termnte the lgorthm nd notfy the user. f ( f ( [, f ( ] f ( + θ + + Fgure Geometrcl llustrton of the Newton-Rphson method.
3 Newton-Rphson Method.4. Emple You hve sphercl storge tnk contnng ol. The tnk hs dmeter of 6 ft. You re sked to clculte the heght h to whch dpstck 8 ft long would be wet wth ol when mmersed n the tnk when t contns 6 ft of ol. Dpstck Sphercl Storge Tnk r h Fgure Sphercl storge tnk problem. The equton tht gves the heght h of the lqud n the sphercl tnk for the gven volume nd rdus s gven by f ( h h 9h Use the Newton-Rphson method of fndng roots of equtons to fnd the heght h to whch the dpstck s wet wth ol. Conduct three tertons to estmte the root of the bove equton. Fnd the bsolute reltve ppromte error t the end of ech terton nd the number of sgnfcnt dgts t lest correct t the end of ech terton. Soluton f ( h h 9h f ( h h 8h Let us tke the ntl guess of the root of ( h Iterton The estmte of the root s f ( h h h f h ( ( 9( + ( 8(.897 f s h.
4 .4.4 Chpter ( The bsolute reltve ppromte error h h h t the end of Iterton s % The number of sgnfcnt dgts t lest correct s, s you need n bsolute reltve ppromte error of 5 % or less for one sgnfcnt dgt to be correct n your result. Iterton The estmte of the root s f ( h h h f h ( (. 7 9(. 7 + (. 7 8( ( The bsolute reltve ppromte error h h h % The number of sgnfcnt dgts t lest correct s. Iterton The estmte of the root s f ( h h h f h.897 ( ( ( ( ( t the end of Iterton s.897
5 Newton-Rphson Method.4.5 ( The bsolute reltve ppromte error h h h t the end of Iterton s % Hence the number of sgnfcnt dgts t lest correct s gven by the lrgest vlue of m for whch m.5 m.78.5 m.46 log (.46 m m log( So m The number of sgnfcnt dgts t lest correct n the estmted root s. Drwbcks of the Newton-Rphson Method. Dvergence t nflecton ponts If the selecton of the ntl guess or n terted vlue of the root turns out to be close to the nflecton pont (see the defnton n the ppend of ths chpter of the functon f ( n the equton f (, Newton-Rphson method my strt dvergng wy from the root. It my then strt convergng bck to the root. For emple, to fnd the root of the equton f ( ( +.5 the Newton-Rphson method reduces to ( ( Strtng wth n ntl guess of 5., Tble shows the terted vlues of the root of the equton. As you cn observe, the root strts to dverge t Iterton 6 becuse the prevous estmte of.9589 s close to the nflecton pont of (the vlue of f ' ( s zero t the nflecton pont. Eventully, fter more tertons the root converges to the ect vlue of..
6 .4.6 Chpter.4 Tble Dvergence ner nflecton pont. Iterton Number Fgure Dvergence t nflecton pont for ( (. Dvson by zero For the equton 6 f the Newton-Rphson method reduces to ( f.
7 Newton-Rphson Method.4.7 For or., dvson by zero occurs (Fgure 4. For n ntl guess close to. such s. 999, one my vod dvson by zero, but then the denomntor n the formul s smll number. For ths cse, s gven n Tble, even fter 9 tertons, the Newton-Rphson method does not converge. Tble Dvson by ner zero n Newton-Rphson method. Iterton f ( Number % E-5 7.5E-6 f( 5.E-6.5E-6.E E E-6-7.5E-6 -.E-5 Fgure 4 Ptfll of dvson by zero or ner zero number.. Osclltons ner locl mmum nd mnmum Results obtned from the Newton-Rphson method my oscllte bout the locl mmum or mnmum wthout convergng on root but convergng on the locl mmum or mnmum. Eventully, t my led to dvson by number close to zero nd my dverge. For emple, for f ( + the equton hs no rel roots (Fgure 5 nd Tble.
8 .4.8 Chpter.4 6 f( Fgure 5 Osclltons round locl mnm for ( + f. Tble Osclltons ner locl mm nd mnm n Newton-Rphson method. Iterton f ( Number % Root jumpng In some cse where the functon f ( s osclltng nd hs number of roots, one my choose n ntl guess close to root. However, the guesses my jump nd converge to some other root. For emple for solvng the equton sn f you choose.4π ( s n ntl guess, t converges to the root of s shown n Tble 4 nd Fgure 6. However, one my hve chosen ths s n ntl guess to converge to π
9 Newton-Rphson Method.4.9 Tble 4 Root jumpng n Newton-Rphson method. Iterton f ( Number % f( Fgure 6 Root jumpng from ntended locton of root for ( sn f. Append A. Wht s n nflecton pont? For functon f (, the pont where the concvty chnges from up-to-down or down-to-up s clled ts nflecton pont. For emple, for the functon f ( (, the concvty chnges t (see Fgure, nd hence (, s n nflecton pont. An nflecton ponts MAY est t pont where f ( nd where f ''( does not est. The reson we sy tht t MAY est s becuse f f (, t only mkes t possble nflecton pont. For emple, for f ( 4 6, f (, but the concvty does not chnge t. Hence the pont (, 6 s not n nflecton pont of f ( 4 6. For f ( (, f ( chnges sgn t ( f ( < for <, nd f ( > for >, nd thus brngs up the Inflecton Pont Theorem for functon f ( tht sttes the followng. If f '( c ests nd f (c chnges sgn t c, then the pont ( c, f ( c s n nflecton pont of the grph of f.
10 .4. Chpter.4 Append B. Dervton of Newton-Rphson method from Tylor seres Newton-Rphson method cn lso be derved from Tylor seres. For generl functon f (, the Tylor seres s f" f ( f ( f + + ( ( + ( + ( + +! As n ppromton, tkng only the frst two terms of the rght hnd sde, f f + f ( + ( ( ( + nd we re seekng pont where f (, f ( +, f ( + f ( ( whch gves + ( ( tht s, f we ssume f + f' Ths s the sme Newton-Rphson method formul seres s derved prevously usng the geometrc method. NONLINEAR EQUATIONS Topc Newton-Rphson Method of Solvng Nonlner Equtons Summry Tet book notes of Newton-Rphson method of fndng roots of nonlner equton, ncludng convergence nd ptflls. Mjor Chemcl Engneerng Authors Autr Kw Dte November 8, Web Ste
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