Chapter Newton-Raphson Method of Solving Simultaneous Nonlinear Equations

Transcription

1 Chapter 7 Newto-Rapho Method o Solg Smltaeo Nolear Eqato Ater readg th chapter o hold be able to: dere the Newto-Rapho method ormla or mltaeo olear eqato deelop the algorthm o the Newto-Rapho method or olg mltaeo olear eqato e the Newto-Rapho method to ole a et o mltaeo olear eqato 4 model a real-le problem that relt a et o mltaeo olear eqato Itrodcto Seeral phcal tem relt a mathematcal model term o mltaeo olear eqato A et o ch eqato ca be wrtte a The olto to thee mltaeo olear eqato are ale o whch at all the aboe eqato The mber o et o olto to thee eqato cold be oe qe more tha oe bt te or te I th chapter we e the Newto- Rapho method to ole thee eqato The Newto-Rapho method o olg a gle olear eqato ca be dered g rt-order Talor ere rt two term o Talor ere ad are ge b where preo etmate o root preet etmate o root Sce we are lookg or where become zero Eqato ca be re-wrtte a 7

2 7 Chapter 7 ad the a a recre ormla a Derato Now how do we eted the ame to mltaeo olear eqato? For ake o mplct let lmt the mber o olear eqato to two a 4a 4b The rt order Talor-ere or olear eqato 4a & 4b are 5a 5b We are lookg or where ad are zero Hece 6a 6b Wrtg 7a 7b we get 8a 8b Rewrtg Eqato 8a ad 8b 9a 9b ad the the matr orm

3 Newto-Rapho Method o Solg Smltaeo Nolear Eqato 7 Solg Eqato wold ge ad Sce the preo etmate o the root oe ca d rom Eqato 7 a b Th proce repeated tll oe obta the root o the eqato wth a prepeced tolerace ε ch that ε a < ε ε a < ε Eample Fd the root o the mltaeo olear eqato 5 Ue a tal ge o 4 Codct two terato Solto Frt pt the eqato the orm to ge 5 Now Hece rom Eqato

4 74 Chapter 7 5 Iterato The tal ge 4 Hece Solg the eqato b a method o or choce we get 5 5 Sce 5 5 we get The abolte relate appromate error at the ed o the rt terato are % 5 5 ε a

5 Newto-Rapho Method o Solg Smltaeo Nolear Eqato 75 ε a 4667% Iterato The etmate o the root at the ed o terato# 5 75 Hece Solg the aboe eqato b a method o or choce ge 5 Sce gg The abolte relate appromate error at the ed o the ecod terato a %

6 76 Chapter 7 a % Althogh ot aked the eample problem tatemet the etmated ale o the root ad the abolte relate appromate error are ge below Table Table : Etmate o the root ad abolte relate appromate error Iterato a % a % mber The eact olto to whch the aboe cheme coergg to 5 5 Eample A 5m log gtter made rom a lat heet o almm whch 5m m The hape o the gtter cro-ecto how Fgre ad made b bedg the heet at two locato at a agle θ Fgre What are the ale o ad θ that wll mamze the olme capact o the gtter o that t dra water qckl drg a hea raall? Fgre Sheet metal bet to orm a gtter

7 Newto-Rapho Method o Solg Smltaeo Nolear Eqato 77 θ θ L Fgre : Parameter o the gtter D C θ θ E coθ A L B coθ F Solto Fgre : Labelg the parameter ad pot o the gtter Oe eed to mamze the cro-ectoal area o the gtter The cro-ectoal area G o the gtter ge b

8 78 Chapter 7 where G Area o trapezod ABCD AB CD FC E AB L E CD AB EA BF L co θ co θ L co θ E FC BC θ θ E4 hece G AB CD DE G θ [ L L co θ ] θ [ L co θ ] θ E5 For eample or θ the cro-ectoal area o the gtter become G a t repreet a lat heet or θ 8 the cro-ectoal area o the gtter become G a t repreet a oerlapped heet ad or θ 9 t repreet a rectaglar croectoal area wth G L For the ge L m G θ coθ θ The G θ coθ coθ G coθ θ coθ θ θ B olg the eqato θ θ coθ coθ θ coθ θ coθ θ we ca d the local mma ad mama o G θ Oe o the local mama ma alo be the abolte mamm The ale o ad θ that correpod to the mamm o G θ are what we are lookg or Ca o ole the eqato to d the correpodg ale o ad θ or mamm ale o G? Ittel what do o thk wold be thee ale o ad θ?

9 Newto-Rapho Method o Solg Smltaeo Nolear Eqato 79 Apped A: Geeral matr orm o olg mltaeo olear eqato The geeral tem o eqato ge b A We ca rewrte a orm a ˆ F A where F A A4 T A5

10 7 Chapter 7 The Jacoba o the tem o eqato b g Newto-Rapho method the J A6 Ug the Jacoba or the Newto-Rapho method ge where [ ] F J A7 ew old A8 Hece ew [ J ] F old [ J ] F [ J ] F ew old A Ealatg the ere o [ J ] Eqato A comptatoall more tee tha olg Eqato A7 or ad calclatg ew a ew old A