lecture04

lecture04 - INTRODUCTION TO NUMERICAL SIMULATION - L....

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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T. L ECTURE 4. Linear Systems – LU Decomposition 1 Pivot Multipliers T ODAY S O UTLINE : Solution of Linear Systems Gaussian Elimination Basics LU factorization Computational Complexity Pivoting for Growth Control S OLUTION OF L INEAR S YSTEMS Gaussian Elimination Basics ± LU Factorization o Important Properties Gaussian Elimination Method for solving b x r v = M ² A “Direct” Method Finite Termination for exact result (ignoring roundoff error) ² Produces accurate results for a broad range of matrices ² Computationally expensive o Reminder by Example 3 ¯ 3 Example = 3 2 1 3 2 1 33 32 31 23 22 21 13 12 11 b b b x x x M M M M M M M M M 3 3 33 2 32 1 31 2 3 23 2 22 1 21 1 3 13 2 12 1 11 b x M x M x M b x M x M x M b x M x M x M = + + = + + = + + Use equation 1 to eliminate x 1 from equation 2 and 3 1 11 31 3 3 13 11 31 33 2 12 11 31 32 1 11 11 31 31 1 11 21 2 3 13 11 21 23 2 12 11 21 22 1 11 11 21 21 1 3 13 2 12 1 11 b M M b x M M M M x M M M M x M M M M b M M b x M M M M x M M M M x M M M M b x M x M x M = + + = + + = + + = 1 11 31 3 1 11 21 2 1 3 2 1 13 11 31 33 12 11 31 32 13 11 21 23 12 11 21 22 13 12 11 0 0 b M M b b M M b b x x x M M M M M M M M M M M M M M M M M M M Pivot should NOT BE = 0. Simplify the notation = 3 2 1 3 2 1 33 32 23 22 13 12 11 ~ ~ ~ ~ 0 ~ ~ 0 b b b x x x M M M M M M M
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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T. L ECTURE 4. Linear Systems – LU Decomposition 2 Pivot Multiplier Use updated equation 2 to eliminate x 2 from the updated equation 3 = 2 22 32 3 2 1 3 2 1 23 22 32 33 23 22 13 12 11 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 ~ ~ 0 b M M b b b x x x M M M M M M M M M Right-hand side = 2 22 32 1 11 31 3 1 11 21 2 1 3 2 1 ~ ~ ~ b M M b M M b b M M b b y y y = 3 2 1 3 2 1 22 32 11 31 11 21 1 ~ ~ 0 1 0 0 1 b b b y y y M M M M M M b x r r = M = = b y y x r r r r L U LU M { b x y r r r = U L } b x r r = M LU M =
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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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lecture04 - INTRODUCTION TO NUMERICAL SIMULATION - L....

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