GAUSS ELIMINATION. Consider the following system of algebraic linear equations

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1 Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION Consder the followng system of lgebrc lner equtons To solve the bove system usng clsscl methods, equton () s subtrcted from equton () multpled by. Ths wll elmnte the vrble from equton (). Also, equton () cn be multpled by nd subtrcted from equton () multpled by. Ths wll elmnte the vrble from equton (). The resultng system of equtons becomes Now, to elmnte the vrble from equton (), equton () s multpled by nd dded to equton () multpled by 7. The resultng system of equtons becomes 7 7 Obvously, from equton (),. Substtutng n equton () results s Now, substtutng the vlues of both nd n equton () leds to.. Now, we wll repet the bove process but workng wth mtrces. The system cn be presented s mtr form s stes.google.com/ste/zydmsoud/numercl 88

2 Numercl Anlyss for Engneers Germn Jordnn Unversty The ugmented system mtr s A b ( ) ( ) Obvously, from equton () (row ),. Substtutng n equton () (row ) results s. Then, substtutng the vlues of both nd n equton () (row )leds to. The bove nlyss llustrtes three rules:. Any equton cn be multpled by constnt.. Any equton cn be replced by ts sum wth nother equton.. The order of the equtons cn be chnged. Now, let us repet the prevous clcultons n computerzed fshon, whle mntnng sgnfcnt fgures clcultons wth roundng. stes.google.com/ste/zydmsoud/numercl 89

3 Numercl Anlyss for Engneers Germn Jordnn Unversty A b ( ) ( ) (.) (.6666) (.)( ) (.)() (.6667)( ) (.6667)() (.)() (.6667)() ( ) (.57). (.57)(.) 6 (.57)(7)... 7 From equton () (row ),. Substtutng n equton () (row ) results s. Then, substtutng the vlues of both nd n equton () (row )leds to. The bove computerzed steps descrbe the Guss Elmnton method. The lgorthm of the Guss Elmnton method s stes.google.com/ste/zydmsoud/numercl 9

4 Numercl Anlyss for Engneers Germn Jordnn Unversty where k,,, n nd k,,, n. For emple, for nd nd, for nd k, nd, for nd k, k, k ( k) PIVOTING Snce the lgorthm of the Guss Elmnton method s k k ( k) problems rse when s zero or too smll. zeros my be creted on the dgonl even f they were not present n the orgnl mtr. To vod these zeros on the dgonl, equtons (rows) cn be rerrnged so tht the lrgest coeffcents re on the dgonl. Ths process s clled Pvotng. stes.google.com/ste/zydmsoud/numercl 9

5 Numercl Anlyss for Engneers Germn Jordnn Unversty Emple Consder the followng ugmented system mtr: A b Perform Guss Elmnton steps s follows: ( ( ) ) STOP To get rd of the zero on the dgonl of row, rows nd cn be nterchnged s follows: Then stes.google.com/ste/zydmsoud/numercl 9

6 Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION STEPS. Augment the n n coeffcent mtr wth the vector of the rght-hnd-sdes to form n ugmented n ( n ) mtr.. Interchnge rows so tht the lrges element n the frst column s on the dgonl, ( ).. Crete zeros n the frst column by pplyng the Guss Elmnton lgorthm k k, where k,,, n k th. epet steps nd for the second column to the ( n ) column, puttng the lrgest coeffcent on the dgonl of the workng column. th 5. Solve for n from the n equton (row). 6. Bck substtute nd solve for the rest of the vlues of the vector. DETEMINANT EVALUATION USING GAUSS ELIMINATION Usng Guss Elmnton, the mtr cn be reduced to A A stes.google.com/ste/zydmsoud/numercl 9

7 Numercl Anlyss for Engneers Germn Jordnn Unversty Snce the determnnts of mtr A nd A re equl, then In generl, A A A A ( n) nn In the cse where pvotng s performed, the determnnt chnges sgn every tme two rows re nterchnged. Therefore the determnnt of the mtr becomes where p s the number of row nterchnges. ( n) nn ( A A ) p stes.google.com/ste/zydmsoud/numercl 9