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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 5 LU Factorization 3.9 Applications of PA = LU 3.9.1 Linear System with Multiple RHS Ax (1) = b (1) , Ax (2) = b (2) ,..., Ax ( p ) = b ( p ) AX = B where X = h x (1) x ( p ) i is n p , B = h b (1) b ( p ) i is n p . 1. Compute G.E. with p.p on A : PA = LU . PAx ( i ) = Pb ( i ) LUx ( i ) = Pb ( i ) 2 3 n 3 flops 2. Solve Ly ( i ) = Pb ( i ) for y ( i ) . n 2 p flops 3. Solve Ux ( i ) = y ( i ) for x ( i ) n 2 p flops CSE/MATH 6643: Numerical Linear Algebra p.1/13 LU Factorization Special case, when B = I n = h e 1 e n i , X is A 1 . 2 3 n 3 + 2 n 3 flops? No, less. 1. PA = LU 2 3 n 3 flops 2. Ly ( i ) = P e i e.g. 2 6 6 6 6 6 4 x x x x x x x x x x x x x x x 3 7 7 7 7 7 5 2 6 6 6 6 6 4 x x 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 1 3 7 7 7 7 7 5 . No computation required for rows above 1 . It takes n P j =1 ( n j )( n j + 1) , i.e. n 3 3 flops 3. Solve Ux ( i ) = y ( i ) . Same usual case, n 3 flops. Totally takes 2 3 n 3 + 1 3 n 3 + n 3 = 2 n 3 flops. CSE/MATH 6643: Numerical Linear Algebra p.2/13 LU Factorization 3.9.2 Want = c T A 1 b method 1: 1. Compute A 1 : 2 n 3 flops 2. Compute d = A 1 b : 2 n 2 flops 3. Compute = c T d : 2 n flops Totally 2 n 3 flops. method 2: 1. Subsystem: d = A 1 b . We solve Ad = b for d ....
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This note was uploaded on 02/04/2012 for the course CS 8801 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
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