Chpter 0.04 Newton-Rphson Method o Solvng Nonlner Equton 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 ( 0 requre brcketng o the root by two guesses. Such methods re clled brcketng methods. hese 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. he 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 he Newton-Rphson method s bsed on the prncple tht the ntl guess o the root o ( 0 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 ( tn θ ( 0, + whch gves ( + ( ( 0.04.
0.04. Chpter 0.04 Equton ( s clled the Newton-Rphson ormul or solvng nonlner equtons o the orm ( 0. 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 he steps o the Newton-Rphson method to nd the root o n equton ( 0. Evlute ( symbolclly. Use n ntl guess o the root, ( + ( re, to estmte the new vlue o the root, +, s. Fnd the bsolute reltve ppromte error s + 0 + 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. ( ( [, ( ] ( + θ + + Fgure Geometrcl llustrton o the Newton-Rphson method.
Newton-Rphson Method 0.04. Emple A trunnon hs to be cooled beore t s shrnk tted nto steel hub. Fgure runnon to be sld through the hub ter contrctng. he equton tht gves the temperture to whch the trunnon hs to be cooled to obtn the desred contrcton s gven by 7 0.50598 + 0.89 + 0.746 + 0.888 Use the Newton-Rphson method o ndng roots o equtons to nd the temperture to whch the trunnon hs to be cooled. Conduct three tertons to estmte the root o the bove equton. Fnd the bsolute reltve ppromte error t the end o ech terton nd the number o sgncnt dgts t lest correct t the end o ech terton. Soluton ( 0 7 ( 0.50598 + 0.89 + 0.746 + 0.888 7 (.5794 + 0.76584 + 0.746 Let us ssume the ntl guess o the root o ( 0 Iterton he estmte o the root s,,0 (,0 (,0 0.50598 0.746 0 +.579.890 0 5 6.587 0 8.058 ( s 0., 0 7 ( 0 + 0.89 ( 0 ( 0 + 0.888 7 ( 0 + 0.7658 ( 0 + 0.746
0.04.4 Chpter 0.04 8.06 he bsolute reltve ppromte error,,,0 0 t the end o Iterton s 8.06 ( 0 0 8.06.9% he number o sgncnt dgts t lest correct s 0, s you need n bsolute reltve ppromte error o less thn 5% or one sgncnt dgt to be correct n your result. Iterton he estmte o the root s,,, ( (, 0.50598 + 0.746 8.06.579 + 0.746 5 4.4 8.06 5 6.067 8.06 (0.6965 8.75 he bsolute reltve ppromte error,,, 0 ( 8.06 8.75 0 8.75 0.54076% he number o sgncnt dgts t lest correct s. Iterton he estmte o the root s,, (, (, 0.50598 + 0.746 8.75.579 + 0.746 7 ( 8.06 + 0.89 ( 8.06 ( 8.06 + 0.888 7 ( 8.06 + 0.76584 ( 8.06 t the end o Iterton s 7 ( 8.75 + 0.89 ( 8.75 ( 8.75 + 0.888 7 ( 8.75 + 0.7658 ( 8.75
Newton-Rphson Method 0.04.5 8.800 8.75 5 6.986 8.75 ( 4.575 8.75 he bsolute reltve ppromte error,,, 0 8.75 8.75 ( 8.75 0 t the end o Iterton s.5086 % Hence the number o sgncnt dgts t lest correct s gven by the lrgest vlue o m or whch m 0.5.5086 0.5 7.07 m log 7.07 m ( m log( 7.07 5. 58 m So m 5 he number o sgncnt dgts t lest correct n the estmted root 8.75 s 5. 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 ( 0, 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 ( ( + 0.5 0 the Newton-Rphson method reduces to ( + 0.5 + ( Strtng wth n ntl guess o 5. 0 0, ble 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 0.9589 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 0..
0.04.6 Chpter 0.04 ble Dvergence ner nlecton pont. Iterton Number 0 5.0000.6560.7465.84 4.6000 5 0.9589 6 0.9 7 9.746 8.8 9 8.7 5.498.44.7464 0.8556 4 0.858 5 0.09784 6 0.7475 7 0.994 8 0. Fgure Dvergence t nlecton pont or ( ( 0. Dvson by zero For the equton 6 0. 0 +. 4 the Newton-Rphson method reduces to ( 0 + 0. 0 +. 4 0.06 6.
Newton-Rphson Method 0.04.7 For 0 0 or 0. 0 0, dvson by zero occurs (Fgure 4. For n ntl guess close to 0.0 such s 0. 0 0999, one my vod dvson by zero, but then the denomntor n the ormul s smll number. For ths cse, s gven n ble, even ter 9 tertons, the Newton-Rphson method does not converge. ble Dvson by ner zero n Newton-Rphson method. Iterton ( Number % -6 0 0.09990.60000.6480 8.778 0.75.760 5.568 50.8.74.6485 50.4 4 0.77765 0.4884 50.6 5 0.558 0.4470 50.946 6 0.405 0.0486 5.4 7 0.69 0.069 5.7 8 0.4608 0.00755 5.7 9 0.094490 0.009 54.60.00E-05 7.50E-06 ( 5.00E-06.50E-06 0.00E+00-0.0-0.0-0.0 0 0.0 0.0 0.0 0.04 -.50E-06 0.0-5.00E-06-7.50E-06 -.00E-05 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 ( + 0 the equton hs no rel roots (Fgure 5 nd ble.
0.04.8 Chpter 0.04 6 ( 5 4 4 0 - - 0 -.75-0.040 0.5.4 - Fgure 5 Osclltons round locl mnm or ( +. ble Osclltons ner locl mm nd mnm n Newton-Rphson method. Iterton ( Number % 0 4 5 6 7 8 9.0000 0.5.75 0.057.4.59 0.766 5.795.6955 0.97678.00.5 5.06.09.874.570.09 4.94 9.66.954 00.00 8.57 476.47 9.66 50.80 89.88.99.9 75.96 4. 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 0 you choose 0.4π ( 7.598 s n ntl guess, t converges to the root o 0 s shown n ble 4 nd Fgure 6. However, one my hve chosen ths s n ntl guess to converge to π 6. 885.
Newton-Rphson Method 0.04.9 ble 4 Root jumpng n Newton-Rphson method. Iterton ( Number % 0 4 5 7.598 4.46 0.5499 0.0607 8.76.9586 0.95 0.969 0.56 0.060 5 8.75.9586 68.97 7.44 97.9 7.54 4.8 4 (.5 0.5 0-0 4 6 8-0.0607 0.5499 4.46 7.598-0.5 - -.5 Fgure 6 Root jumpng rom ntended locton o root or ( sn 0. 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 ( (, the concvty chnges t (see Fgure, nd hence (,0 s n nlecton pont. An nlecton ponts MAY est t pont where ( 0 nd where ''( does not est. he reson we sy tht t MAY est s becuse ( 0, t only mkes t possble nlecton pont. For emple, or ( 4 6, ( 0 0, but the concvty does not chnge t 0. Hence the pont (0, 6 s not n nlecton pont o ( 4 6. For ( (, ( chnges sgn t ( ( < 0 or <, nd ( > 0 or >, nd thus brngs up the Inlecton Pont heorem 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.
0.04. Chpter 0.04 Append B. Dervton o Newton-Rphson method rom ylor seres Newton-Rphson method cn lso be derved rom ylor seres. For generl uncton (, the ylor seres s " ( ( + + ( ( + ( + ( + +! As n ppromton, tkng only the rst two terms o the rght hnd sde, + ( + ( ( ( + nd we re seekng pont where ( 0, ( + 0, 0 ( + ( ( whch gves + ( ( tht s, we ssume + ' hs s the sme Newton-Rphson method ormul seres s derved prevously usng the geometrc method. NONLINEAR EQUAIONS opc Newton-Rphson Method Summry etbook emples o the Newton-Rphson Method Mjor Mechncl Engneerng Authors Autr Kw Dte November 7, 0 Web Ste http://numerclmethods.eng.us.edu