EE103 Gauss Elimination

EE103 Gauss Elimination - The Basic Idea of Gauss...

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EE103 SLIDES 4B (SEJ) 1 The Basic Idea of Gauss Elimination or Pivoting , , Ax b x NAx Nb x = = nonsingula ,x, 1. , 2. , 3. Nnni s i f N has an inverse OR The rows of N are linearly independent OR The columns of N are linearly indep r endent EE103 SLIDES 4B (SEJ) 2 Example of Complete Gauss Elimination 12 3 1 2 3 1 2 3 32 1 4 1 0 0 622 8 0 1 0 312 9 0 0 1 aaa b e e e x x x Ab I ⎛⎞ ⎡⎤ ⎜⎟ ⎢⎥ −= −− ⎣⎦ ⎝⎠ 1123 123 36 3 48 9 e g aeee be e e =−+ =++ .. , 03 3 3 x Be e e I abasis matrix for R == [, , ]
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EE103 SLIDES 4B (SEJ) 3 11 3 , Pivot on a = 1,1 1 0 0 2 1 0 0 0 1 E ⎡⎤ ⎢⎥ = ⎣⎦ 1 1 0 0 -2 1 0 0 0 1 E = 1,2 1 0 0 0 1 0 -1 0 1 E = 1 1,2 1 0 0 0 1 0 1 0 1 E = 1,3 1/3 0 0 0 1 0 0 0 1 E = 1 1,3 3 0 0 0 1 0 0 0 1 E = 1,3 1,2 1 1/3 0 0 2 1 0 -1 0 1 E EE E Δ = = 32 1 622 312 EE103 SLIDES 4B (SEJ) 4 12 3 1 2 3 1 3 1 4 1 0 0 8 0 1 0 9 0 0 1 aaa b e e e x Ex x ⎛⎞ ⎜⎟ −= = −− ⎝⎠ 3 3 1 2 3 1 12313 43 1300 06 4 1 6 21 0 033 5 1 0 1 aa a b ee e x x x Ab E ⎜⎟⎜⎟ // / /
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EE103 SLIDES 4B (SEJ) 5 31 2 3 12 3 11 2 3 1 43 3 4 16 5 3 1 2 3 aa e e ba ee ea e e =+ =++ 123 0 [, , ] B e e e hasbeenreplaced by = 1 [, , ] B aee = EE103 SLIDES 4B (SEJ) 6 Fact 3 states: 1 1 1 1 3 1 3 06 4 033 AB A B E A B E A BE I E B == ⎡⎤ ⎢⎥ −− ⎣⎦ ⇒= = // 1 , ] B =
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EE103 SLIDES 4B (SEJ) 7 2,2 Now, pivot on 6 a = 2,3 2,1 2 1 -1/9 0 0 1/6 0 0 1/2 1 E EE E Δ ⎡⎤ ⎢⎥ == ⎣⎦ 1 12313 06 4 033 A = −− // 12 3 3 1 22 3 1 11 43 1300 4 1 6 21 0 5 1 0 1 aa a b ee e x Ex x Ab E ⎛⎞ ⎜⎟ ⎜⎟⎜⎟ ⎝⎠ / / EE103 SLIDES 4B (SEJ) 8 3 1 2 3 2 1 1 0 19 49 19 19 0 01 2 3 8 3 1 3 1 6 0 00 1 1 3 0 1 2 1 /// / / / / a b e e e E E 123 1 2 has been replaced by [, ,] Ba e e a e = = 3 31 2 3 For instance, 93 48 13 ba ae a e =+ +
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EE103 SLIDES 4B (SEJ) 9 As before, Fact 3 states: 22 221 1 21 2 AB A B E E A BEE I EE B = = =⇒ = Ba a e 2 123 = [, , ] EE103 SLIDES 4B (SEJ) 10 Finally, pivot on a 3,3 1 = − EEE E 3,3 3,2 3,1 3 1 = L N M M M O Q P P P = 0 -1/ 9 0 1 2 / 3 0 0 -1 Δ 12 3 1 2 3 3 2 1 1 0 19 49 19 19 0 01 2 3 8 3 1
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EE103 Gauss Elimination - The Basic Idea of Gauss...

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