lecture07

lecture07 - INTRODUCTION TO NUMERICAL SIMULATION LECTURES 7...

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I NTRODUCTION TO N UMERICAL S IMULATION L ECTURES 7 & 8. QR and Krylov-Subspace Matrix Solution Methods 1 T ODAY S O UTLINE : ± Minimization View of QR Singular Matrix Basic Minimization Approach Orthogonalized Search Directions QR and Length Minimization Produce Identical Results ± Arbitrary Subspace Algorithm Orthogonalization of Search Directions ± Generalized Conjugate Residual Algorithm Krylov-subspace Simplification in the symmetric case Leaky and insulating examples QR F ACTORIZATION = 2 1 0 1 2 1 0 1 2 A Example. U L 3 4 2 3 3 1 2 1 0 0 1 0 0 1 2 1 0 1 0 0 1 R Q 1 . 1 0 0 9 . 1 7 . 1 0 45 . 0 8 . 1 2 . 2 80 . 0 60 . 0 0 53 . 0 72 . 0 44 . 0 27 . 0 35 . 0 89 . 0
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I NTRODUCTION TO N UMERICAL S IMULATION L ECTURES 7 & 8. QR and Krylov-Subspace Matrix Solution Methods 2 Matrix is Singular, column of Q is zero ± Zero Column If a column is zero + = = 1 1 i j j j i M w M r r { } t independen linearly not ,..., 1 i M M r r What if a column becomes zero? 0 0 0 0 0 0 0 0 0 0 0 0 ~ ~ 0 1 13 12 11 3 1 L M O M M M L L L r L r r N N r r r r M M Q NN N N N r r r r r r r Q Q Q L M O M M M L L L r L r r 0 0 0 0 0 0 0 0 0 0 3 33 1 13 12 11 3 1 Matrix MUST be Singular!! 1. Do not try to normalize the column. 2. Do not use the column as a source for orthogonalization. 3. Perform backward substitution as well as possible. b x b x T r r r r Q R QR = = = b Q Q Q x x x x r r r r r r r N N NN N N r r M r L r M M O L L L 3 1 3 2 1 3 33 1 13 12 11 0 0 0 0 0 0
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I NTRODUCTION TO N UMERICAL S IMULATION L ECTURES 7 & 8. QR and Krylov-Subspace Matrix Solution Methods 3 QR Factorization : Problem b x r r = M { Solve ) 3 ( Solve ) 2 ( (1) x y x b y b y b x T y r r r r r r r r r r = = = = R Q Q R Q ± Singular Example ⎡− 0 0 0 2 2 1 0 0 0 0 0 0 2 1 2 1 2 M r 3 M r 1 Q r 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 2 1 2 1 2 Q r 3 M r 1 Q r i = 1 i = 1 v 2 v 1 v 3 = 1 1 1 1 0 0 0 1 1 0 1 1 3 2 1 x x x 2 M r 1 M r 3 M r 1 M r 3 M r 2 M r = =
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This note was uploaded on 09/26/2010 for the course AERO 16.910 taught by Professor Daniel during the Spring '10 term at MIT.

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lecture07 - INTRODUCTION TO NUMERICAL SIMULATION LECTURES 7...

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