lecture 5

lecture 5 - Fig PT3.5 Chapter 10 LU Decomposition and...

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Unformatted text preview: Fig PT3.5 Chapter 10 LU Decomposition and Matrix Inversion LU Factorization LU Factorization is an alternative to Gauss- Jordan. The A matrix (in the problem Ax=b) is factored into the product of two matrices L and U (A=LU) L is a lower triangular matrix and U is an upper triangular matrix L and U Matrices Lower Triangular Matrix Upper Triangular Matrix 44 34 33 24 23 22 13 13 12 11 u u u u u u u u u u U 44 34 42 41 33 32 31 22 21 11 l l l l l l l l l l L Another method for solving matrix equations Idea behind the LU Decomposition - start with We know (because we did it in Gauss Elimination) we can write 4 3 2 1 4 3 2 1 44 34 33 24 23 22 14 13 12 11 d d d d x x x x u u u u u u u u u u d x U LU Decomposition b x A or b x A LU Decomposition Assume there exists [ L ] Such that This implies b x A d x U L b d L & A U L 44 34 42 41 33 32 31 22 21 11 l l l l l l l l l l L The Steps of LU Decomposition LU Decomposition LU Decomposition * Based on Gauss elimination * Formally called Doolittle decomposition Direct Decomposition * Doolittle decomposition l ii = 1 * Crout decomposition u ii = 1 (omitted) * Cholesky decomposition u ii = l ii LU Decomposition (1) Factor (decompose) [ A ] into [ L ] and [ U ] (2) given { b }, determine { d } from [ L ]{ d } = { b } (3) using [ U ]{ x } = { d } and back-substitution, solve for { x } Advantage: Once we have [ L ] and [ U ], we can use many different { b }s without repeating the decomposition process LU decomposition / factorization [ A ] { x } = [ L ] [ U ] { x } = { b } Forward substitution [ L ] { d } = { b } Back substitution [ U ] { x } = { d } Forward substitutions are much more efficient than elimination LU Decomposition 1 4 2 3 5 F 14 F 23 F 12 F 24 F 45 H 1 F 35 F 25 V 2 V 1 Simple Truss W Exampe: Forces in a Simple Truss 100 F F F F F F F V H V 1 cos cos sin sin 1 cos cos sin sin cos 1 sin 1 cos cos 1 1 sin sin cos 1 1 sin 1 45 35 25...
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lecture 5 - Fig PT3.5 Chapter 10 LU Decomposition and...

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