NewtonRaphson Method of Solving a Nonlinear Equation Autar Kaw


 Gwendoline Moore
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1 NewtonRphson Method o Solvng Nonlner Equton Autr Kw Ater redng ths chpter, you should be ble to:. derve the NewtonRphson method ormul,. develop the lgorthm o the NewtonRphson method,. use the NewtonRphson method to solve nonlner equton, nd 4. dscuss the drwbcks o the NewtonRphson method. Introducton Methods such s the bsecton method nd the lse poston method o ndng roots o nonlner equton ( ) requre brcketng o 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 o the equton. In the NewtonRphson method, the root s not brcketed. In ct, only one ntl guess o the root s needed to get the tertve process strted to nd the root o n equton. The method hence lls n the ctegory o open methods. Convergence n open methods s not gurnteed but the method does converge, t does so much ster thn the brcketng methods. Dervton The NewtonRphson method s bsed on the prncple tht the ntl guess o the root o ( ) s t, then one drws the tngent to the curve t ( ), the pont + where the tngent crosses the s s n mproved estmte o the root (Fgure ). Usng the denton o the slope o uncton, t ( ) θ tn ( ), + Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 5
2 whch gves ( ) ( ) + () Equton () s clled the NewtonRphson ormul or solvng nonlner equtons o the orm ( ). So strtng wth n ntl guess,, one cn nd the net guess, +, by usng Equton (). One cn repet ths process untl one nds the root wthn desrble tolernce. Algorthm The steps o the NewtonRphson method to nd the root o n equton ( ). Evlute ( ) symbolclly. Use n ntl guess o the root, s + ( ) ( ). Fnd the bsolute reltve ppromte error s + + re, to estmte the new vlue o the root, +, 4. Compre the bsolute reltve ppromte error wth the prespeced reltve error tolernce, s. I > s, then go to Step, else stop the lgorthm. Also, check the number o tertons hs eceeded the mmum number o tertons llowed. I so, one needs to termnte the lgorthm nd noty the user. Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 5
3 () ( ) [, ( )] ( + ) θ + + Fgure Geometrcl llustrton o the NewtonRphson method. Emple You re workng or DOWN THE TOILET COMPANY tht mkes lots or ABC commodes. The lotng bll hs specc grvty o.6 nd hs rdus o 5.5 cm. You re sked to nd the depth to whch the bll s submerged when lotng n wter. Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 5
4 Fgure Flotng bll problem. The equton tht gves the depth n meters to whch the bll s submerged under wter s gven by Use the NewtonRphson method o ndng roots o equtons to nd ) the depth to whch the bll s submerged under wter. Conduct three tertons to estmte the root o the bove equton. b) the bsolute reltve ppromte error t the end o ech terton, nd c) the number o sgncnt dgts t lest correct t the end o ech terton. Soluton 4 ( ) ( ). Let us ssume the ntl guess o the root o ( ) Source URL: Sylor URL: s. 5 m. Ths s resonble guess (dscuss why nd.m re not good choces) s the etreme vlues o the depth would be nd the dmeter (. m) o the bll. Iterton The estmte o the root s ( ) ( ) (. 5) 65. (. 5) + (. 5). (. 5) (.4) The bsolute reltve ppromte error.99 4 t the end o Iterton s Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 4 o 5
5 % The number o sgncnt dgts t lest correct s, s you need n bsolute reltve ppromte error o 5% or less or t lest one sgncnt dgt to be correct n your result. Iterton The estmte o the root s ( ) ( ) (. 64) 65. (. 64) + (. 64). (. 64) ( ) The bsolute reltve ppromte error.99 4 t the end o Iterton s % Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 5 o 5
6 The mmum vlue o m or whch sgncnt dgts t lest correct n the nswer s. Iterton The estmte o the root s ( ) ( ) (. 68) 65. (. 68) + (. 68). (. 68) ( 4.98 ) The bsolute reltve ppromte error m.5 s.844. Hence, the number o.99 4 t the end o Iterton s The number o sgncnt dgts t lest correct s 4, s only 4 sgncnt dgts re crred through n ll the clcultons. Drwbcks o the NewtonRphson Method. Dvergence t nlecton ponts I the selecton o the ntl guess or n terted vlue o the root turns out to be close to the nlecton pont (see the denton n the ppend o ths chpter) o the uncton n the equton ( ), NewtonRphson method my strt dvergng wy rom ( ) the root. It my then strt convergng bck to the root. For emple, to nd the root o the equton Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 6 o 5
7 ( ) ( ) +.5 the NewtonRphson method reduces to + ( ) +.5 ( ) Strtng wth n ntl guess o 5., Tble shows the terted vlues o the root o the equton. As you cn observe, the root strts to dverge t Iterton 6 becuse the prevous estmte o.9589 s close to the nlecton pont o ' (the vlue o ( ) s zero t the nlecton pont). Eventully, ter more tertons the root converges to the ect vlue o.. Tble Dvergence ner nlecton pont. Iterton Number Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 7 o 5
8 Fgure Dvergence t nlecton pont or ( ) ( ).. Dvson by zero For the equton 6 ( ) the NewtonRphson method reduces to Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 8 o 5
9 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 ormul s smll number. For ths cse, s gven n Tble, even ter 9 tertons, the NewtonRphson method does not converge. Tble Dvson by ner zero n NewtonRphson method. Iterton Number ) % ( Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 9 o 5
10 .E5 7.5E6 () 5.E6.5E6.E E E67.5E6 .E5 Fgure 4 Ptll o dvson by zero or ner zero number.. Osclltons ner locl mmum nd mnmum Results obtned rom the NewtonRphson 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, or ( ) + the equton hs no rel roots (Fgure 5 nd Tble ). Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 5
11 6 () Fgure 5 Osclltons round locl mnm or ( ) +. Tble Osclltons ner locl mm nd mnm n NewtonRphson method. Iterton Number ) % ( Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 5
12 Root jumpng In some cse where the uncton () s osclltng nd hs number o roots, one my choose n ntl guess close to root. However, the guesses my jump nd converge to some other root. For emple or solvng the equton sn you choose.4π s n ntl guess, t converges to the root o s shown n ( ) Tble 4 nd Fgure 6. However, one my hve chosen ths s n ntl guess to converge to π Tble 4 Root jumpng n NewtonRphson method. Iterton Number ) % ( Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 5
13 () Fgure 6 Root jumpng rom ntended locton o root or ( ) sn Append A. Wht s n nlecton pont? For uncton ( ), the pont where the concvty chnges rom uptodown or downtoup s clled ts nlecton pont. For emple, or the uncton ( ) ( ) concvty chnges t (see Fgure ), nd hence (,) s n nlecton pont., the Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 5
14 An nlecton ponts MAY est t pont where ( ) nd where ''( ) does not est. The reson we sy tht t MAY est s becuse ( ), t only mkes t possble nlecton pont. For emple, or ( ) 4 6, ( ), but the concvty does not chnge t. Hence the pont (, 6) s not n nlecton pont o ( ) 4 6., ( ) chnges sgn t ( ( ) < or <, nd ( ) > For ( ) ( ) or > ), nd thus brngs up the Inlecton Pont Theorem or uncton () tht sttes the ollowng. I '( c) ests nd (c ) chnges sgn t c, then the pont ( c, ( c)) s n nlecton pont o the grph o. Append B. Dervton o NewtonRphson method rom Tylor seres NewtonRphson method cn lso be derved rom Tylor seres. For generl, the Tylor seres s uncton ( ) ( ) ( ) + ( )( ) ( ) ( ) + L "! + As n ppromton, tkng only the rst two terms o the rght hnd sde, ( ) ( ) + ( )( ) + + nd we re seekng pont where ( ), ( ), + ( ) + ( )( ) + tht s, we ssume whch gves + ' ( ) ( ) Ths s the sme NewtonRphson method ormul seres s derved prevously usng the geometrc method. Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 4 o 5
15 Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 5 o 5
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