Lecture19 - Naive Gauss Elimination Begin with a11 x1 + a12...

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Naive Gauss Elimination Begin with Multiply the first equation by a 21 / a 11 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 a x a b x a ... x a x a = + + + = + + + = + + + n n nn 2 2 n 1 1 n 1 11 21 2 n n 1 11 21 n 2 2 12 11 21 22 1 1 n n 2 12 1 11 b x a ... x a x a b a a b x a a a a ... x a a a a x b x a ... x a x a = + + + - = - + + - + = + + +
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Gauss Elimination Reduce to Repeat the forward elimination to get n n nn 2 2 n 1 1 n 2 n n 2 2 22 1 n n 1 2 12 1 11 b x a ... x a x a b x a ... x a b x a ... x a x a = + + + = + + = + + + n n nn 2 2 n 2 n n 2 2 22 1 n n 1 2 12 1 11 b x a ... x a b x a ... x a b x a ... x a x a = + + = + + = + + +
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Gauss Elimination First equation is pivot equation a 11 is pivot element Now multiply second equation by a ' 32 / a 22 and subtract from third equation - = - + + - = + + + = + + + + 2 22 32 3 n n 2 22 32 n 3 3 23 22 32 33 2 n n 2 3 23 2 22 1 n n 1 3 13 2 12 1 11 b a a b x a a a a x a a a a b x a x a x a b x a x a x a x a
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Gauss Elimination Repeat the elimination of a i2 and get Continue and get n n nn 3 3 n 3 n n 3 3 33 2 n n 2 3 23 2 22 1 n n 1 3 13 2 12 1 11 b x a x a b x a x a b x a x a x a b x a x a x a x a = + + = + + = + + + = + + + + ) ( ) ( 1 n n n 1 n nn 3 n n 3 3 33 2 n n 2 3 23 2 22 1 n n 1 3 13 2 12 1 11 b x a b x a x a b x a x a x a b x a x a x a x a - - = = + + = + + + = + + + +
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Now we can perform back substitution to get { x } By simple division ) ( ) ( , ) ( , 2 n 1 n n 2 n n 1 n 1 n 2 n 1 n 1 n b x a x a - - - - - - - - = + Back Substitution ) ( ) ( 1 n nn 1 n n n a b x - - =
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Back Substitution a x a b x a b x 1 i ii n 1 i j j 1 i ij 1 i i i 1 n nn 1 n n n ) ( ) ( ) ( ) ( ) ( - + = - - - - - = = for i = n - 1, n 2, …, 1 0 a 1 i ii - ) ( Naive Gauss Elimination
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Elimination of first column ) ( ) ( ) ( ) ( ) ( ) ( 1 f 4 1 f 3 1 f 2 b a a a 0 b a a a 0 b a a a 0 b a a a a 41 31 21 4 44 43 42 3 34 33 32 2 24 23 22 1 14 13 12 11 × - × - × - 11 41 41 11 31 31 11 21 21 4 44 43 42 41 3 34 33 32 31 2 24 23 22 21 1 14 13 12 11 a a f a a f a a f b a a a a b a a a a b a a a a b a a a a / / / = = =
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Elimination of second column ) ( ) ( ) ( ) ( / / 2 f 4 2 f 3 b a a 0 0 b a a 0 0 b a a a 0 b a a a a a a f a a f b a a a 0 b a a a 0 b a a a 0 b a a a a 42 32 4 44 43 3 34 33 2 24 23 22 1 14 13 12 11 22 42 42 22 32 32 4 44 43 42 3 34 33 32 2 24 23 22 1 14 13 12 11 × - × - = =
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Elimination of third column Upper triangular matrix ) ( ) ( / 3 f 4 b a 0 0 0 b a a 0 0 b a
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Lecture19 - Naive Gauss Elimination Begin with a11 x1 + a12...

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