Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw

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1 Newton-Rphson Method o Solvng Nonlner Equton Autr Kw Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson method to solve nonlner equton, nd 4. dscuss the drwbcks o the Newton-Rphson 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 Newton-Rphson 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 Newton-Rphson 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 Newton-Rphson 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 Newton-Rphson 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 pre-speced 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 Newton-Rphson 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 Newton-Rphson 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 Newton-Rphson 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 ( ), Newton-Rphson 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 Newton-Rphson 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 Newton-Rphson 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 Newton-Rphson method does not converge. Tble Dvson by ner zero n Newton-Rphson method. Iterton Number ) % ( Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 9 o 5

10 .E-5 7.5E-6 () 5.E-6.5E-6.E E E-6-7.5E-6 -.E-5 Fgure 4 Ptll o dvson by zero or ner zero number.. Osclltons ner locl mmum nd mnmum Results obtned rom 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, 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 Newton-Rphson 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 Newton-Rphson 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 up-to-down or down-to-up 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 Newton-Rphson method rom Tylor seres Newton-Rphson 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 Newton-Rphson 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