Lecture+19--Gauss+Elimination

Lecture+19--Gauss+Elimination - Gauss Elimination...

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Unformatted text preview: Gauss Elimination Manipulate equations to eliminate one of the unknowns Develop algorithm to do this repeatedly The goal is to set up upper triangular matrix Back substitution to find solution (root) = nn n 3 33 n 2 23 22 n 1 13 12 11 a a a a a a a a a a U Gauss Elimination Method Solve x in: = Soln. i. Augmented matrix : [A | b] = ii. Reduction (to reduce A to an upper triangular matrix) * Now A has become an upper-triangular matrix pivot element - 2 1 2 1 4 1 1 1 3 3 2 1 x x x 3 2 1 10 12 2--------- - 10 2 1 2 12 1 4 1 2 1 1 3 1 1' row row 3 1 1 2 ....1' 1.3333 11.3333 ....2' 0.3333 2.6667 8.6668 ....3'- 3.6667 ....1 ....2 ....3 " 3 ) 3.6667 0.3333 2'*( 3'- " 2 ' 2 " 1 ' 1 R R R R R R R " 3 .... " 2 .... " 1 .... 6365 . 7 5455 . 2 3333 . 11 3333 . 1 6667 . 3 2 1 1 3 - 3- 1'*(2/3) 3' row row row 2 1*(1/ 3) 2' row row row- Gauss Elimination Method iii. Back substitution * Solve for x 3: x 3= 7.6365 / 2.5454 = 3.0 (from 3") * Solve for x 2: x 2= (11.3333-1.3333*3.0)/3.6667=2.0 * Solve for x 1: x 1= (2+ x 3 - x 2)/3=1.0 (from 1") iv. Determinant : det ( A ) = 3*3.6667*2.5455 = 28.0 " 3 ) 3.6667 0.3333 2'*( 3'- " 2 ' 2 " 1 ' 1 R R R R R R R " 3 .... " 2 .... " 1 .... 6365 . 7 5455 . 2 3333 . 11 3333 . 1 6667 . 3 2 1 1 3 - Determinant of a triangle matrix Determinant : det ( U ) = a11*a22**ann = nn n 3 33 n 2 23 22 n 1 13 12 11 a a a a a a a a a a U Naive Gauss Elimination Begin with Multiply the first equation by a21 / a11 and subtract from second equation n n nn 2 2 n 1 1 n 2 n n 2 2 22 1 21 1 n n 1 2 12 1 11 b x a ... x a x a b x a ... x...
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Lecture+19--Gauss+Elimination - Gauss Elimination...

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