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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