Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution we see point whee the cuve deined by intesects the -is i.e., the line y, with ied point poblem g we see point whee the cuve deined by g intesects the digonl line y. Mny itetive methods o solving nonline equtions use itetion scheme o om g, whee g is unction chosen so tht its ied points e solutions o. Such scheme is clled ied-point itetion o some times unctionl itetion, since unction g is pplied epetedly to initil stting point. Fo given eqution, thee my be mny equivlent ied-pint poblems g with dieent choices o unction g. But not ll ied-point omultions e eqully useul in deiving n itetion scheme o solving given nonline eqution. The esulting itetion schemes my die not only in thei convegence tes but lso in whethe they convege t ll. Convegence o Fied-Point Itetion The behvio o ied-point itetion schemes cn vy widely, om divegence, to slow convegence, to pid convegence. Wht mes the dieence? I g nd g <, then the itetive scheme is loclly convegent, i.e. thee is n intevl contining such tht ied-point itetion with g conveges i stted t point within tht intevl. I g >, on the othe hnd, then ied-point itetion with g diveges o ny stting point othe thn. An itetive method is sid to be o ode o hs the te o convegence, i is the lgest positive el numbe o which thee eits inite constnt C such tht C whee is the eo in the th itetion. C is the symptotic eo constnt usully depends on the deivtives o t. is the tue solution.
Itetive methods Bisection method mes no use o the unction vlue othe thn thei sign, which esults in slow but sue convegence. Using the unction vlues by itetive methods cn deive moe pidly conveging methods. Itetive methods bsed on ist-degee eqution Let is nonline eqution, Thus, i we ppoimte by ist degee eqution in the neighbohood o the oot the we my wite. The solution o this is given by, whee nd e pmetes to be detemined by pescibing two ppopite conditions on nd/o its deivtives. Newton-Rphson Method We detemine nd, using the condition Thus, is: gives Geometic epesenttion nd d i.e. the Newton-Rphson itetion d Newton s method ppoimtes nonline unction ne by tngent line t. O In the limit when, the chod pssing though the points,., becomes the tngent t the point Algoithm, nd Initil guess o,,, 3,. end
Note: It equies two unction evlutions nd pe itetion. Emple: Use Newton s method to ind oot o 4sin 4cos, Thus the Newton s itetion: 4sin,Te 3 4cos Convegence Anlysis Let is the ect solution. Eo t th itetion. Substitute the vlues o nd in the Newton s itetion omul:,weget Epnd by Tylo seies bout the point, we get 3 O on neglecting 3 nd highe powes o,weget C,whee C Hee, the te o convegence. Hence Newton s method hs second ode convegence. Anothe wy o stting this is tht the numbe o coect digits in ppoimte solution is doubled t ech itetion o Newton s method. Rems:. Fo multiple oot, Newton s method is linely convegent, with symptotic constnt C -/m, whee m is the multiplicity o the oot. Fo emple:...5.5.5.5 3.3.5 4.5.65 5..35. Cution: these convegences e locl nd hence i stting point is om solution, method my not convege. e.g. A eltively smll vlue o i.e. nely hoizontl tngent tends to cuse the net itete to lie wy om the coect ppoimtion.
3. One dwbc o Newton s method is, it equies evlution o both unction nd its deivtive t ech itetion. Secnt Method The deivtive my be inconvenient o epnsive to evlute, so we might conside eplcing it by inite dieence ppoimtion using some smll step size h : Thus, the Secnt method is: Algoithm, Initil guesses o,,, 3,. end Geometic epesenttion Appoimting the unction by the secnt line though the pevious two itetes, nd ting the zeo o the esulting line unction to be the net ppoimte solution. Regul-Flsi Method This uses the sme itetive omul s Secnt method nd i the ppoimtions e such tht <, it s clled Regul-Flsi method. Rem:. Since, nd, e nown beoe the stt o the itetion, the Secnt method equies one unction evlution pe step. Convegence Anlysis Let be simple oot o nd substitute in itetive omul, we get Using Tylo seies epnsion bout nd noting tht, we obtin,
L o, O o, C : Eo Eqution, whee C By the deinition o the convegence, whee A nd to be detemined. A / / This lso gives A nd A. Substitution o these vlues, we get / / CA, Comping the powe o on both sides, we get /,whichimplies ± 5, neglecting minus sign gives.68. Thus Secnt method hs supeline te o convegence.