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lecture7 - CSE/MATH 6643 Numerical Linear Algebra Haesun...

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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 7
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Symmetric Positive Definite & Cholesky Decomp. 5.3 Cholesky Decomposition in Banded Systems Definition: A has symmetric bandwidth p if a ij = 0 when | i j | > p . A has upper bandwidth p if a ij = 0 when j > i + p . A has lower bandwidth p if a ij = 0 when i > j + p A is tridiagonal if a ij = 0 when | i j | > 1 If A is S.P.D with sym. bandwidth p , then its Cholesky decomposition is A = GG T where G has lower bandwidth p . CSE/MATH 6643: Numerical Linear Algebra – p.1/8
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Symmetric Positive Definite & Cholesky Decomp. e.g. p = 1 , A = 2 6 6 6 6 6 6 6 6 4 a 1 c 1 c 1 a 2 c 2 c 2 a 3 . . . . . . . . . c n - 1 c n - 1 a n 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 4 g 11 g 21 g 22 g 32 g 33 . . . . . . g n,n - 1 g nn 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 g 11 g 21 g 22 g 32 g 33 . . . . . . g n,n - 1 g nn 3 7 7 7 7 7 7 5 T For banded matrix A : n × n , we don’t need O ( n 2 ) storage. For sym. cases, ( p + 1) n storage is enough to store A .
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