ELM Numericl Anlysis Dr Muhrrem Mercimek SOLVING SYSTEMS OF EQUATIONS, ITERATIVE METHODS ELM Numericl Anlysis Some of the contents re dopted from Lurene V. Fusett, Applied Numericl Anlysis using MATLAB. Prentice Hll Inc., 999
ELM Numericl Anlysis Dr Muhrrem Mercimek Tody s lecture Common clssicl itertive techniques for liner eqution systems Jcoi Method Guss- Seidel Method Successive Over Reltion
ELM Numericl Anlysis Dr Muhrrem Mercimek Itertive techniques for liner eqution systems Systems of liner equtions for which numericl solutions re needed re often very lrge. Using generl methods such s Guss elimintion is computtionlly epensive. If the coefficient system is hving specific structure itertive techniques re preferle.
ELM Numericl Anlysis Dr Muhrrem Mercimek 4 Itertive techniques for liner eqution systems For lrge, sprse systems (mny coefficients whose vlue is zero) itertive techniques re preferle.
ELM Numericl Anlysis Dr Muhrrem Mercimek 5 Wht is n eqution System? When you hve A B derive the equivlent system C d nd solve it Generte sequence of pproimtion (), (),..., where ( k) C ( k ) d
ELM Numericl Anlysis Dr Muhrrem Mercimek Jcoi Method Emple : A eqution system A = = C + d A = = y = Strt with y / y y y y Simultneous updting C hs zeros in the digonl New vlues of the vriles re not used until new itertion step is egun y () () y 4 4 4 4
ELM Numericl Anlysis Dr Muhrrem Mercimek 7 Jcoi Method A = = y = / () y 4 4 () y 4 4 () () y 8 8 () () y 8 8 Stopping Criteri: Stop the itertions when The function A (k) less thn tol vlue.(. is -Norm or Eucliden Norm) y The m numer of itertions m _iter hs reched
ELM Numericl Anlysis Dr Muhrrem Mercimek 8 Jcoi Method Emple : Consider the system 0.5 0.5 0.5 0.5 0.5 0.5 0.5.0.5 () () () 0.0 0.5 0.5 Strt with 0.5 0.5 0.0 0.5 0.5 0.0 (0,0,0) 0.5.0.5 () () () 0.0 0.5 0.5 0.5 0.0 0.5 0.5 0.5 0.0 0.5.5 0.5.0.5 The method converges in itertions () = [.00.00 0.9997] T
ELM Numericl Anlysis Dr Muhrrem Mercimek 9 Jcoi Method Emple : A necessry nd sufficient condition for the convergence of the Jcoi method the mgnitude of the lrgest eigenvlue of the itertion mtri C e less thn A necessry condition (not sufficient) for the convergence of the Jcoi method A should e digonlly dominnt. Mgnitude of digonl element should e greter thn sum f mgnitudes of other elements of the row A = = y = y y y y () y 5 y () 5 When you run the other itertions you will see it diverges
ELM Numericl Anlysis Dr Muhrrem Mercimek 0 Guss-Seidel Method Emple 4: A eqution system A = A = = y = Strt with y / y y y y Sequentil updting y () () New vlues of the vriles re used in the sme itertion step y () 4 4 8 8 y () () y () () 5 5
ELM Numericl Anlysis Dr Muhrrem Mercimek Guss-Seidel Method Emple 5: Consider the three-y-three system new ( old) 0.5 0.5 new new 0.5 0.5 new new new 0.5 0.5 Strt with After 0 itertions =[.000.9999 -.000] (0,0,0) ( old) ( old) 0.5.0.5 Stopping Criteri: Stop the itertions when The function A (k) less thn tol vlue.(. is -Norm or Eucliden Norm) The m numer of itertions m _iter hs reched
ELM Numericl Anlysis Dr Muhrrem Mercimek Guss-Seidel Method Discussion The Guss-Seidel method is sensitive to the form of the coefficient mtri A The Guss-Seidel method typiclly converges more rpidly thn the Jcoi method The Guss-Seidel method is more difficult to use for prllel computtion
ELM Numericl Anlysis Dr Muhrrem Mercimek Successive Over Reltion Introduce n dditionl prmeter, ω, tht my ccelerte the convergence of the itertions. A comintion of current updte (From Guss-Seidel) nd previous vlue. A proportion from current updte, s well s proportion from previous vlue is summed up. ω Similry = ω( ) dd ω to oth sides nd use it in nd updtes cn e found multiply ech eqution with ω Similry nd updtes cn e found Guss-Seidel Method updtes new old Similry nd old updtes old ( ) ( ) cn e found new old new old ( ) ( ) new ( ) old ( new new ) A proportion from current updte
ELM Numericl Anlysis Dr Muhrrem Mercimek 4 Successive Over Reltion (SOR) Consider the three-y-three system A 4 0 5 0 5 8 9 4 new old old ( ) ( ) new old new 5 old ( ) ( new old 5 new 4 ( ) ( ) 9 ) 4
ELM Numericl Anlysis Dr Muhrrem Mercimek 5 Successive Over Reltion (SOR) A 4 0 5 0 5 8 9 4 Required numer of itertions for different vlues of the reltion prmeter Strt with Tolernce = 0.0000 ω 0.8 0.9.0..5..4 No. of itertions 44 9 8 5 5
ELM Numericl Anlysis Dr Muhrrem Mercimek Successive Over Reltion (SOR) Discussion The SOR method cn e derived y multiplying the decomposed system otined from the Guss-Seidel method y the reltion prmeter w The itertive prmeter w should lwys e chosen such tht 0 < w <
ELM Numericl Anlysis Dr Muhrrem Mercimek 7 Summry Jcoi method Guss-seidel method k k k SOR method k k k k k k k k k k k k new old old old ( ) ( ) new old new old ( ) ( ) new ( ) old ( new new ) k k k 7