SYSTEMS OF NON-LINEAR EQUATIONS. Introduction Graphical Methods Close Methods Open Methods Polynomial Roots System of Multivariable Equations
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1 SYSTEMS OF NON-LINEAR EQUATIONS Itoduto Gaphal Method Cloe Method Ope Method Polomal Root Stem o Multvaale Equato
2 Chapte Stem o No-Lea Equato /. Itoduto Polem volvg o-lea equato egeeg lude optmato olvg deetal equato ad ege value. Uuall a tpal polem to d oot o a olea equato e.g. whee t aaltal oluto ae ( a (. ± 4a (. a Howeve o othe o-lea equato dg a oot ma ot e a mple ta ad ma a ol e detemed va umeal appoah ol. Code the ollowg uto: ( ( 4 FIGURE. Root o ( The uto ( ma tae the om o oe o the ollowg ategoe:. Lea uto.e. ( a whh mple to olve. Polomal o algea uto.e. the ollowg om: L o ( a a L a. Taedet o o-algea uto.e. a e epaded to te ee the om: e.g. ( e L ( a!!
3 Chapte Stem o No-Lea Equato /. Gaphal Method The mplet wa to etmate the oot o oluto o ( om t gaph.e. om the teeto o the gaph wth the - a. Code the ollowg uto:. 5 (.5( e. 5. (.4 ( FIGURE. A gaphal method to d the oot o a uto Th method ot auate ut a e ued to ota a appomate value.
4 Chapte Stem o No-Lea Equato / 4. Cloe Method The loe o aetg method d oot aed o a age peed two value whh aumed to ota a oot. Fo the eto method the oot whh le the age o ( detemed loog at the g o the uto at oth ed: ( ( < ( 5 4 FIGURE. The eto method o eahg oot I the podut o ( ( egatve the the et appomate oot (. Fo the et teato ( ( < othewe ( ( < the the oot le ( the the oot le (. I Fg..: utl ( a deed a ovegee o temato teo. Fo the ale poto method a lea tepolato ued to gve a ette appomato (ee Fg..4 thu leadg to ate ovegee tha the eto method: ST PT RQ PQ ( ( (
5 Chapte Stem o No-Lea Equato / 5 ( P T S R Q FIGURE.4 The ale poto method o eahg oot Upo eaagemet: ( ( ( ( Hee the omula o eahg oot a e geealed a ollowed: whee 4 ( ( ed ed ed (.4 ( ed ( Eample. Ue the eto method to d the oot o Compae wth the eal value o Soluto ( ( e e (.6 the age o [ ]. Lelaa ( ε a % ε t % M M M M M M M E6.5E4.95E4
6 Chapte Stem o No-Lea Equato / 6 Eample.4 Ue the ale poto method to d the oot o [ ]. Compae wth the eal value o Soluto ( e ( (.6 Fo 4 Eq. (.4 a e wtte a ad.67 ( ( ( ( (.6 (.78 e the age o Lelaa ( ε a % ε t % E E5
7 Chapte Stem o No-Lea Equato / 7.4 Ope Method The ope method d oot though teato ug oe o two pot a tal pot. The mplet method the ed pot teato method whee the ogal equato ( moded to e: ( g (.5 ad the teato tat wth a tal value. Th method lead to a ve low ovegee ad ma e dvege. Eample.6 Ue the ed pot teato method to ota the oot o to thee demal plae. Tae the tal value o. Peeleaa ( e e e auate ε t % ε t %
8 Chapte Stem o No-Lea Equato / 8 The mot popula method o eahg oot the Newto-Rapho method whh uuall lead to a at ovegee (ee Fg.4. ( ( ( FIGURE.5 The Newto-Rapho method o eahg oot Th a gadet-aed method whee the omula a e deved om the t ode devatve: ( ( o ( ( The geealed om o the Newto-Rapho omula ( ( (.6 whee. The eo o th method a e deved om the Talo ee epao aumg the eal oot a ε the ( ( ε ( ε ( ( L B egletg ε ad othe hghe ode tem the elatve eo a e otaed: ε ε! ( ε ( ( ( ( (
9 Chapte Stem o No-Lea Equato / 9 Howeve the Newto-Rapho method ha lmtato the ollowg ae: o o Some uto ma have t devatve dult to deve ad ma eque legth tep Some mple uto ma have a mall taget gadet ad thu eed ma teato to ovege e.g. ( wth.5 o Some ae ma lead to dvegee (ee Fg..6 ( FIGURE.6 The Newto-Rapho teato poe whh dveged o Ollato ma happe ad th wll ot temate (ee Fg..7 ( 4 FIGURE.7 Ollato the Newto-Rapho method o The method ma e ot auate the ae o multple oot (ee Fg..8. ( FIGURE.8 A ae o multple oot ug the Newto-Rapho method
10 Chapte Stem o No-Lea Equato / Eample.7 Ue the Newto-Rapho method to deteme the oot o the tal value o. Soluto The Newto-Rapho omula: ( e ( e ( ( e. Tae e e ( ( ( ( ε % E E7.5E E E5 7.5E4 t
11 Chapte Stem o No-Lea Equato / The moded Newto-Rapho method mple the omula uh that the devatve ha to e evaluated oe ol ut lead to moe teato: ( ( (.7 Moeove ( ma eve evaluated though a mall hage o : ( ( Eample.8 Repeat Eample.7 ug the moded Newto-Rapho method.7. Soluto Fom Eample.7: ( ( Hee the omula o the moded Newto-Rapho e e ( ( ( ε % E5.58E-5 5.7E E5 5.4E-6.49E E6.5E-6.486E E7.9E-7 5.8E E7 5.74E-8.64E E8.E-8.5E E9.44E E7 t
12 Chapte Stem o No-Lea Equato / The eat method a avod ug a the devatve ad the gadet tae om the omula o te dvded deee: ( ( ( Hee the omplete omula o the eat method ( ( ( ( (.8 ( ( ( FIGURE.9 The eat method o eahg oot Howeve t eed two tal odto.e. o ad. Eample.9 Ue the eat method to deteme the oot o value o ad.. Soluto e. Tae the tal ( ε % E E8.98E6 t
13 Chapte Stem o No-Lea Equato / The ollowg gaph gve the ompao o the poee: Fed pot teato Beto method Relatve eo εt (% Newto-Rapho method Seat method Moded Newto- Rapho method Fale poto method Nume o teato FIGURE. Compao o oot eahg method o ( e
14 Chapte Stem o No-Lea Equato / 4.5 Polomal Root The tpal om o -th ode polomal uto o havg oot: ( a a a a L (.9 The Mülle method ue the mla appoah a the eat method ut eque thee tal pot (ee Fg... ( FIGURE. Paaol pojeto the Mülle method I th method the quadat polomal ha ee ued: ( ( ( a (. Wth the thee tal pot [ ] ( [ ] ( ad [ ] ( : ( ( ( ( ( ( ( ( ( a a a B olvg thee multaeou equato: ( ( ( ( ( ( ( ( a a (.
15 Chapte Stem o No-Lea Equato / 5 B ug the value o a ad Eq. (. a e eued to ota : ( a( ( ± 4a (. Eample. Ue the Mülle method to get the oot o the ollowg u polomal: (.5.5 Tae.5.4 da.6 a the tal value ad peom teato utl the elatve eo le tha.5%. Soluto I the t teato: The value o ( o the thee tal value ae Fom Eq. (.: Hee ug Eq. (.: ( (.5.5(.5.5(.5.75 ( (.4.5(.4.5(.4.4 ( (.6.5(.6.5( a ( (.576 ( ( 5(
16 Chapte Stem o No-Lea Equato / 6 The et o the poee ae a ollow: ( a ε a (% The Batow method ome oth the Mülle ad Newto-Rapho method ad eale the detemato o all oot ethe eal o omple. Dvdg Eq. (.9 wth a quadat ato podue ( L (. eultg the edual tem a ollowed Hee Eq. (.9 a e ewtte a ( R (.4 a a a ( ( ( R ( L ( ( L a ( Hee a a a o L (.5 I da a appomato o da the da ae the epetve hage.e.
17 Chapte Stem o No-Lea Equato / 7 Se da ae uto o da the ollowg equato a e etalhed ( ( (.6 The dvdg Eq. (. wth the ame quadat ato eld: o L (.7 Th podue whh a e olved a ollowed (.8 Thee equato ae teated utl da ae wth the ovegee tea ad the the oot a e etmated a ollowed: 4 ± (.9
18 Chapte Stem o No-Lea Equato / 8 Eample. Ue the Batow method to ota all oot o the ollowg polomal: (.5.5 Tae a the tal value ad peom teato o the ovegee tea o.5%. Soluto I the t teato: Ue Eq. (.5 ad Eq. (.7: a a a a The ue Eq. (.8: (.5 ( (.5 (.5 ( (.5 ( (.5 ( ( (.5 ( ( (.5 ( (.5 (.5 ( (.5 ( ( The et o the poee ae a ollow: ε a (% ε a (%
19 Chapte Stem o No-Lea Equato / 9 Fom th tale the quadat ato Fom Eq. (.9:.5 ±.5 ± (.5 4( Oe moe oot a e otaed a ollowed (ue the value o da :.5 Hee the lat oot.5 (.5.5 ( (.5.5 (
20 Chapte Stem o No-Lea Equato /.6 Stem o Multvaale Equato Code the ollowg tem o o-lea equato: ( ( ( M (. Th tem a e olved ug the Newto-Rapho method a ollowed (va the t ode Talo ee: ( ( ( ( ( L ( ( ( ( ( L M ( ( ( ( ( L whee M
21 Chapte Stem o No-Lea Equato / Fo ( ( ( L the aove elato a e eaaged to a mat equato: ( ( ( M M L M O M M L L (. o a moe ompat om ( J whee the let-had de mat J ow a the Jaoa mat. Eq. (. a e teated utl oveged. Eample. Ue the Newto-Rapho method to olve the ollowg tem: Tae da a the tal value ad peom teato o the utl the elatve eo om le tha.5%. Soluto Fo the gve tem: ( ( The the Jaoa mat J a e omed a ollowed: 4 J ad the teato omulato a ollowed: ( ( 4
22 Chapte Stem o No-Lea Equato / B ug the tal value o da : ( ( ε a e (%
23 Chapte Stem o No-Lea Equato / Eee. Deteme the teeto o the two ollowg equato: g( 6 h( ug the ale poto method ad the the Newto-Rapho method. B ug the age o [.5] peom alulato utl t ovege. Alo at eah teato alulate the appomate ad atual eo the atual oluto.47.. The elatohp o to ato o a low a dampg elemet wth the Reold ume R e gve : l 5.6 ( R 4 e whee a otat o teal wall oughe o the dampg elemet ad equal to.8. Calulate the value o R e 75.. Solve the ollowg tem o o-lea equato: z e z e.4.9 z.4 Ue the tal value o ( z ( ad the temato atea o.% o the appomate eo om.
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