SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS
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1 ELM Numecl Alyss D Muhem Mecmek SOLVING SYSTEMS OF EQUATIONS DIRECT METHODS ELM Numecl Alyss Some of the cotets e dopted fom Luee V. Fusett Appled Numecl Alyss usg MATLAB. Petce Hll Ic. 999
2 ELM Numecl Alyss D Muhem Mecmek Tody s lectue Commo dect techques fo le equto systems Guss Elmto Guss Elmto wth ow pvotg Guss Elmto fo tdgol systems
3 ELM Numecl Alyss D Muhem Mecmek Defto of the Polem We wt to solve: A = whee s the vecto of the ukows whle A d e gve. Codtos:. The ume of equtos s equl to the ume of ukows (tht s A s sque mt). The coeffcets of A d e el. The soluto of the system ests d t s uque
4 ELM Numecl Alyss D Muhem Mecmek Defto of the Polem The soluto of the system ests d t s uque A ests A s ot sgul A's colums e lely depedet A's les e lely depedet det(a) s ozeo k(a) s equl to A = oly f s ull vecto
5 ELM Numecl Alyss D Muhem Mecmek 5 Defto of the Polem Cme s ule The soluto of system of equtos: det det A A A eplces the th colum
6 ELM Numecl Alyss D Muhem Mecmek 6 Clculto of the detemt How to compute the detemt of sque mt whee C j s the cofcto of elemet j. det( A) N C j The cofcto C j s the detemt of the sumt oted y emovg the th ow d the j th colum of the mt multpled y () +j : j det M C j j No th ow M j j j j j j j j j No j th colum
7 ELM Numecl Alyss D Muhem Mecmek 7 Guss Elmto Emple : Solvg thee equtos thee ukows y 6y y z z 8z 8 y z y z y z y z 8y 9z z z y ( ) () ( ) z y
8 8 Usg Mt Notto: mtvecto fom : A = Fom ugmeted mt Guss elmto pocedue j j m Guss Elmto A 8 ELM Numecl Alyss D Muhem Mecmek
9 Pvot At the kth stge of Guss elmto pocedue the ppopte multple of the kth ow s used to educe ech of the etes the kth colum elow the kth ow to zeo elemet kk : pvot elemet Guss Elmto 9 ELM Numecl Alyss D Muhem Mecmek
10 ELM Numecl Alyss D Muhem Mecmek Guss Elmto Emple : V left loop R Ω R 5Ω ( ) ( ) V R Ω Ω uppe 5 ght loop ( ) ( ) R Ω R 5 Ω lowe ght loop ( ) ( ) V
11 ELM Numecl Alyss D Muhem Mecmek Guss Elmto Step 55 5 The pvot s = A 5 Multply the fst ow y / d dd t to the secod ow Multply the fst ow y / d dd t to the thd ow 55 A 5/ 5 / 5 / /
12 ELM Numecl Alyss D Muhem Mecmek Guss Elmto Step The pvot s = 5/ Multply the secod ow y /5 d dd t to the thd ow to get A 5/ By ck susttuto 5 /. / / 5 ( 5 / )(5) / 5/ ( )() ( )(5) /
13 ) ( ) ( ) )( ( k k k k k k k k k k k k Mesug computtol effot Mesue the ume of multplcto d dvsos The totl ume of multplcto d dvsos Guss Elmto Dscusso ELM Numecl Alyss D Muhem Mecmek
14 Emple : A IllCodtoed mt 6 5 Guss Elmto Dscusso ELM Numecl Alyss D Muhem Mecmek
15 ELM Numecl Alyss D Muhem Mecmek 5 Guss Elmto wth Row Pvotg I cet cses Reducg the ccuces Moe ccute th Guss Elmto Avodg (f possle) the flue Dvde y zeo Pvotg s eeded
16 ELM Numecl Alyss D Muhem Mecmek 6 Guss Elmto wth Row Pvotg Emple : Roudg to two sgfct dgts Wthout ow pvotg Wth ow Pvotg
17 8 8 6 z z z y y y ) ( 6() 8 ) ( 5 Guss Elmto wth Row Pvotg Emple 5: 7 ELM Numecl Alyss D Muhem Mecmek
18 ELM Numecl Alyss D Muhem Mecmek 8 Guss Elmto fo Tdgol System Specl Le System Asg Applcto A geel tdgol mt s mt whose ozeo elemets e foud oly o the dgol sudgol d supedgol of the mt.
19 ELM Numecl Alyss D Muhem Mecmek 9 Guss Elmto fo Tdgol System * Emple of Tdgol Mt = + = + = + =. We c tke dvtge of the zeo elemets tht e ledy peset the coeffcet mt d vods uecessy thmetc opetos. Thus we eed to stoe oly the ew vectos d.
20 .... )() ( ; )() ( ; )() ( ; Guss Elmto fo Tdgol System... ELM Numecl Alyss D Muhem Mecmek Emple 6:
21 d d d d d... Step : Fo the fst equto Step : Fo ech of the equto Step : Fo the lst equto Step : y ck susttuto Thoms Method ELM Numecl Alyss D Muhem Mecmek
22 ELM Numecl Alyss D Muhem Mecmek Thoms Method Emple 7: (6) d (); ( );. ( ); ().
23 . d d. ) / )( ( ) )(/ ( ) / )( ( d d. )() / ( / )() / ( / ) / ( / ELM Numecl Alyss D Muhem Mecmek Thoms Method
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