{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture7

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}