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.
.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 = 4. 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.
Newton-Rphson Method.4. Emple You re workng for DOWN THE TOILET COMPANY tht mkes flots for ABC commodes. The flotng bll hs specfc grvty of.6 nd hs rdus of 5.5 cm. You re sked to fnd the depth to whch the bll s submerged when flotng n wter. Fgure Flotng bll problem. The equton tht gves the depth n meters to whch the bll s submerged under wter s gven by 4.65.99 Use the Newton-Rphson method of fndng roots of equtons to fnd ) the depth to whch the bll s submerged under wter. Conduct three tertons to estmte the root of the bove equton. b) the bsolute reltve ppromte error t the end of ech terton, nd c) the number of sgnfcnt dgts t lest correct t the end of ech terton. Soluton 4 f 65..99 f. Let us ssume the ntl guess of the root of f s. 5 m. Ths s resonble guess (dscuss why nd.m re not good choces) s the etreme vlues of the depth would be nd the dmeter (. m) of the bll. Iterton The estmte of the root s f f. 5 65.. 5. 5.. 5. 5.8. 9. 5.4.64 4 5.99 4
.4.4 Chpter.4 The bsolute reltve ppromte error 9.9%. 64. 5. 64 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 t lest one sgnfcnt dgt to be correct n your result. Iterton The estmte of the root s f f. 64 65.. 64. 64.. 64. 64 7 9778.. 64 8.997 5. 64 4.4646.68 The bsolute reltve ppromte error. 68. 64. 68.76% The mmum vlue of m for whch sgnfcnt dgts t lest correct n the nswer s. Iterton The estmte of the root s f f.99 4 t the end of Iterton s. 68 65.. 68. 68.. 68. 68 4.44. 68 8.97 9. 68 4.98.68 The bsolute reltve ppromte error m.5 s.844. Hence, the number of.99 4 t the end of Iterton s
Newton-Rphson Method.4.5. 68. 68. 68 The number of sgnfcnt dgts t lest correct s 4, s only 4 sgnfcnt dgts re crred through n ll the clcultons. 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 ( ).5 = ( ) 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.. Tble Dvergence ner nflecton pont. Iterton Number 5..656.7465.84 4.6 5.9589 6.9 7 9.746 8.8 9 8.7 5.498.44.7464.8556 4.858 5.9784 6.7475 7.994 8.
.4.6 Chpter.4 Fgure Dvergence t nflecton pont for. Dvson by zero For the equton 6 f.. 4 the Newton-Rphson method reduces to.. 4 =.6 6 f. 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 % -6.999.6.648 8.778.75.76 5.568 5.8.74.6485 5.4 4.77765.4884 5.6 5.558.447 5.946 6.45.486 5.4 7.69.69 5.7 8.468.755 5.7 9.9449.9 54.6
Newton-Rphson Method.4.7. E- 5 7.5E-6 f() 5.E-6.5E-6.E+ -. -. -.....4 -.5E-6. -5.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 ). 6 f() 5 4 4 - - -.75 -.4.5.4 - Fgure 5 Osclltons round locl mnm for f.
.4.8 Chpter.4 Tble Osclltons ner locl mm nd mnm n Newton-Rphson method. Iterton f ( ) Number % 4 5 6 7 8 9..5.75.57.4.59.766 5.795.6955.97678..5 5.6.9.874.57.9 4.94 9.66.954. 8.57 476.47 9.66 5.8 89.88.99.9 75.96 4. 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.4 7.598 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 6. 885. Tble 4 Root jumpng n Newton-Rphson method. Iterton f ( ) Number % 4 5 7.598 4.46.5499.67 4 8.76.9586.95.969.56.6 5 8.75.9586 68.97 7.44 97.9 7.54 4.8 4
Newton-Rphson Method.4.9 f().5.5-4 6 8 -.67.5499 4.46 7.598 -.5 - -.5 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. 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, tht s, f we ssume f,
.4. Chpter.4 f whch gves f 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 Generl Engneerng Authors Autr Kw Dte December, 9 Web Ste http://numerclmethods.eng.usf.edu