{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

mws_gen_sle_ppt_ludecomp(1)

# mws_gen_sle_ppt_ludecomp(1) - LU Decomposition Major All...

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

1/21/2010 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

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

View Full Document
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.

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

View Full Document
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? If solving a set of linear equations If [ A ] = [ L ][ U ] then Multiply by Which gives Remember [ L ] -1 [ L ] = [ I ] which leads to Now, if [ I ][ U ] = [ U ] then Now, let Which ends with and [ A ][ X ] = [ C ] [ L ][ U ][ X ] = [ C ] [ L ] -1 [ L ] -1 [ L ][ U ][ X ] = [ L ] -1 [ C ] [ I ][ U ][ X ] = [ L ] -1 [ C ] [ U ][ X ] = [ L ] -1 [ C ] [ L ] -1 [ C ]=[ Z ] [ L ][ Z ] = [ C ] (1) [ U ][ X ] = [ Z ] (2)

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

View Full Document
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

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

View Full Document
http://numericalmethods.eng.usf.edu To find inverse of [A] Time taken by Gaussian Elimination Time taken by LU Decomposition ( ) + + = + = 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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 31

mws_gen_sle_ppt_ludecomp(1) - LU Decomposition Major All...

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

View Full Document
Ask a homework question - tutors are online