CHAPTER 5d. SIMULTANEOUS LINEAR EQUATIONS
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1 CHAPTE 5. SIUTANEOUS INEA EQUATIONS A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig by Dr. Ibrhim A. Asskkf Sprig ENCE - Compttio thos i Civi Egirig II Dprtmt of Civi Eviromt Egirig Uivrsity of ry, Cog Prk Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig W sw tht th Gssi Eimitio cosists of two stps: Forwr Pss Bck Sbstittio For th foowig st of qtio: [ ]{ } { C } () ij i i ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No.
2 Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Th Forwr Pss c wys rst ito gmt mtrix i th gr form s () Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Th Bck Sbstittio c wys rst ito th foowig gr form:,,,, ()
3 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Gss-Jor Procss A trtiv procss of imitio i which cofficits i com xcpt th pivot mt r imit c so b s to obti sotio. I Gss-Jor imitio, th sotio is obti ircty ftr prformig th forwr pss. Thr is o bck sbstittio. ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 5 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Gss-Jor Procss Giv th foowig st of qtio: C C C C () ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 6
4 Asskkf Si No. 7 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Gss-Jor Procss Or giv th foowig st of qtio i gmt mtrix of th form: C C C C (5) Asskkf Si No. 8 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Gss-Jor Procss Usig Gss-Jor, Eq. 5 c b trsform ito th foowig form: C C C C O (6)
5 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Gss-Jor Procss A i which th sotio is riy obti s C C C C (7) ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 9 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Exmp: Gss-Jor Sov th foowig st of simtos qtios sig th Gss-Jor tho: ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 5
6 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Exmp (cot ): Gss-Jor This systm of qtios c b rprst i mtrix form s ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Exmp (cot ): Gss-Jor Or this systm of qtios c b rprst i gmt mtrix form s ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 6
7 7 Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Exmp (cot ): Gss-Jor Stp : (-) 6 : : : (-) : 9 7 : Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Exmp (cot ): Gss-Jor Stp 7 / / / 5 / / : (-) / : 5 : 7/ : ) ( 5 / 5 : :
8 8 Asskkf Si No. 5 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Exmp (cot ): Gss-Jor Stp ) /(7 / / / 7 / / / : (-) / : / / : : (-) / : / / : Asskkf Si No. 6 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Exmp (cot ): Gss-Jor Th st gmt mtrix givs th sotio for th systm of qtios s foows:
9 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Exmp (cot ): Gss-Jor O ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 7 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig c tht th Gssi imitio procss cosists of two stps: Forwr Pss Bck Sbstittio Th forwr pss trsforms th systm of qtios ito ppr trigr mtrix. ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 8 9
10 Asskkf Si No. 9 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS U Dcompositio Th ppr trigr mtrix c b ot s U it is giv by (8) Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS U Dcompositio c th gmt mtrix tht c rst from th forwr pss of Gssi Eimitio: (9)
11 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Th vs of this mtrix (Eq. 8) c b rt to th vs i Eq. 9 s ij ij for i,,, j,,, Th ppr trigr mtrix U c b rt to th cofficits mtrix A s ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig U A () Whr owr trigr mtrix, which c b viw s to prst th bck sbstittio () ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No.
12 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Thrfor Eq. c b xprss s U A () ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Eq. stts tht th cofficit mtrix A c b compos to two trigr mtrics U. Ths trigr mtrics c b trmi withot th s of Gssi imitio mtho by prformig th mtrix mtipictio U i Eq. ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No.
13 Asskkf Si No. 5 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS U Dcompositio tipictio of U wi rst i () Asskkf Si No. 6 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS U Dcompositio tipictio of U wi rst i ()
14 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig For xmp, th mtipictio of th first row of with th first com of U rsts th foowig v tht is q to. ( ) ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 7 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig I viw of Eq., for xmp ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 8
15 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig From Eq., th vs for U c b compt from: for i,, K, i j ij ji i ij j ji j ik k j jk k jj k k kj ki k for j,, K, (5) for j,, K, i j, j, K, for j,, K, i j, j, K, ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 9 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Oc th cofficit mtrix is compos ito U, ths mtrics c b s to sov systm of qtios with costts C i, i,,,. Th sotio c b obti i two stps, forwr pss bck sbstittio. ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 5
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