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LU Decomposition

# LU Decomposition - LUDecomposition Major: Authors:AutarKaw...

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05/01/11 http://numericalmethods.eng.usf.edu 1 LU Decomposition Major: All Engineering Majors Authors: Autar Kaw http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM  Undergraduates

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LU Decomposition      http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu LU Decomposition LU Decomposition is another method to solve a set of simultaneous linear equations Which is better, Gauss Elimination or LU Decomposition? To answer this, a closer look at LU decomposition is needed.

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Method For most non-singular matrix [ A ] that one could conduct Naïve Gauss Elimination forward elimination steps, one can always write it as [ A ] = [ L ][ U ] where [ L ] = lower triangular matrix [ U ] = upper triangular matrix                                             http://numericalmethods.eng.usf.edu LU Decomposition
http://numericalmethods.eng.usf.edu How does LU Decomposition work?

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http://numericalmethods.eng.usf.edu LU Decomposition How can this be used? Given [ A ][ X ] = [ C ] 1. Decompose [ A ] into [ L ] and [ U ] 2. Solve [ L ][ Z ] = [ C ] for [ Z ] 3. Solve [ U ][ X ] = [ Z ] for [ X ]
http://numericalmethods.eng.usf.edu When is LU Decomposition better than Gaussian Elimination? To solve  [ A ][ X ] = [ B ] Table.  Time taken by methods where T = clock cycle time and n = size of the matrix So both methods are equally efficient. Gaussian Elimination LU Decomposition + + 3 4 12 3 8 2 3 n n n T + + 3 4 12 3 8 2 3 n n n T

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http://numericalmethods.eng.usf.edu To find inverse of [A] Time taken by Gaussian Elimination  Time taken by LU Decomposition ( 29 + + = + = 3 4 12 3 8 | | 2 3 4 n n n T CT CT n BS FE + + = × + × + = 3 20 12 3 32 | | | 2 3 n n n T CT n CT n CT BS FS LU n 10 100 1000 10000 CT| inverse GE / CT| inverse LU 3.28 25.83 250.8 2501 Table 1 Comparing computational times of finding inverse of a matrix using  LU decomposition and Gaussian elimination.
http://numericalmethods.eng.usf.edu Method: [A] Decompose to [L] and [U] [ ] [ ][ ] = = 33 23 22 13 12 11 32 31 21 0 0 0 1 0 1 0 0 1 u u u u u u U L A [ U ] is the same as the coefficient matrix at the end of the forward elimination step. [ L

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LU Decomposition - LUDecomposition Major: Authors:AutarKaw...

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