LU Decomposition - LUDecomposition...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
LU Decomposition      http://numericalmethods.eng.usf.edu
Background image of page 2
                                            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.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
                                            http://numericalmethods.eng.usf.edu How does LU Decomposition work?
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
                                            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 ]
Background image of page 6
                                            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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
                                            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.
Background image of page 8
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
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/01/2011 for the course CHBE 2120 taught by Professor Gallivan during the Spring '07 term at Georgia Institute of Technology.

Page1 / 31

LU Decomposition - LUDecomposition...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online