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lecture3 - CSE/MATH 6643 Numerical Linear Algebra Haesun...

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Unformatted text preview: CSE/MATH 6643: Numerical Linear Algebra Haesun Park { hpark } @cc.gatech.edu School of Computational Science and Engineering College of Computing Georgia Institute of Technology Atlanta, GA 30332, USA Lecture 3 LU Factorization 3.2 LU Factorization How to find L and U ? e.g. n = 2 , " a 11 a 12 a 21 a 22 # = " 1 l 21 1 #" u 11 u 12 u 22 # = " u 11 u 12 l 21 u 11 l 21 u 12 + u 22 # a 21 = l 21 u 11 = l 21 a 11 , so a 11 can not be zero in general case, A = " A 11 v w T a nn # = " L 1 l T 1 #" U 1 u α # = " L 1 U 1 L 1 u l T U 1 l t u + α # Assume we have L 1 and U 1 s.t. A 11 = L 1 U 1 . Note v = L 1 u ⇒ u , w = U T 1 l ⇒ l , a nn = l T u + α ⇒ α a 11 can not be zero. How about A 11 ? CSE/MATH 6643: Numerical Linear Algebra – p.1/13 LU Factorization Another way: Gauss transformation: I + spike e.g. 2 6 6 6 4 1 l 21 1 l 31 1 l 41 1 3 7 7 7 5 2 6 6 6 4 11 2 3 4 3 7 7 7 5 = 2 6 6 6 4 11 3 7 7 7 5 , l 21 =- 2 / 11 , l 31 =- 3 / 11 , l 41 =- 4 / 11 2 6 6 6 4 1 1- 3 / 2 1- 4 / 2 1 3 7 7 7 5 2 6 6 6 4 11 2 3 4 3 7 7 7 5 = 2 6 6 6 4 11 2 3 7 7 7 5 CSE/MATH 6643: Numerical Linear Algebra – p.2/13 LU Factorization A (1) = A = 2 6 6 6 4 a 11 x x x a 21 x x x a 31 x x x a 41 x x x 3...
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lecture3 - CSE/MATH 6643 Numerical Linear Algebra Haesun...

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