Download presentation

Presentation is loading. Please wait.

1
**Chapter 11 Gaussian Elimination (I)**

Speaker: Lung-Sheng Chien Reference book: David Kincaid, Numerical Analysis

2
OutLine Basic operation of matrix - representation - three elementary matrices Example of Gaussian Elimination (GE) Formal description of GE MATLAB usage

3
**Matrix notation in MATLAB**

6 -2 2 4 12 -8 6 10 3 -13 9 3 -6 4 1 -18

4
**Matrix-vector product**

Inner-product based 6 -2 2 4 x1 6 x1 + (-2) x2 + 2 x3 + 4 x4 12 -8 6 10 x2 12 x1 + (-8) x2 + 6 x x4 3 -13 9 3 x3 3 x1 + (-13) x2 + 9 x3 + 3 x4 -6 4 1 -18 x4 (-6) x1 + 4 x2 + 1 x3 + (-18) x4 outer-product based 6 -2 2 4 x1 6 -2 2 4 12 -8 6 10 x2 12 -8 6 10 x1 x2 x3 x4 3 -13 9 3 x3 3 -13 9 3 -6 4 1 -18 x4 -6 4 1 -18

5
**Matrix-vector product: MATLAB implementation**

matvec.m Inner-product based outer-product based Question: which one is better

6
**Column-major nature in MATLAB**

physical index : 1D 6 1 12 2 3 3 Question: how does column-major affect inner-product based matrix-vector product and outer-product based matrix-vector product? -6 4 Logical index : 2D -2 5 -8 6 6 -2 2 4 -13 7 12 -8 6 10 4 8 3 -13 9 3 2 9 -6 4 1 -18 6 10 9 11 1 12 4 13 10 14 3 15 -18 16

7
**Matrix representation: outer-product**

where is outer-product representation

8
**Elementary matrix [1] (1) The interchange of two rows in A:**

Define permutation matrix How to explain? Question 1: why Question 2: how to easily obtain

9
**Concatenation of permutation matrices**

such that such that Question: implies Direct calculation

10
**Elementary matrix [2] (2) Multiplying one row by a nonzero constant:**

Define scaling matrix How to explain? since

11
**Elementary matrix [3] (3) Adding to one row a multiple of another:**

Define GE (Gaussian Elimination) matrix How to explain? outer-product representation

12
Use MATLAB notation

13
**Concatenation of GE matrices**

such that Suppose such that Question:

14
**OutLine Basic operation of matrix**

Example of Gaussian Elimination (GE) - forward elimination to upper triangle form - backward substitution Formal description of GE MATLAB usage

15
6 -2 2 4 x1 12 12 -8 6 10 x2 34 3 -13 9 3 x3 27 -6 4 1 -18 x4 -38 12 -8 6 10 34 6 -2 2 4 x1 12 -4 2 2 x2 10 6 -2 2 4 12 3 -13 9 3 x3 27 -6 4 1 -18 x4 -38 -4 2 2 10 3 -13 9 3 27 6 -2 2 4 x1 12 -4 2 2 x2 10 6 -2 2 4 12 -12 8 1 x3 21 -6 4 1 -18 x4 -38 -12 8 1 21

16
-6 4 1 -18 -38 6 -2 2 4 x1 12 -4 2 2 x2 10 6 -2 2 4 12 -12 8 1 x3 21 2 3 -14 x4 -26 2 3 -14 -26 6 -2 2 4 x1 12 First row does not change thereafter -4 2 2 x2 10 -12 8 1 x3 21 2 3 -14 x4 -26 -12 8 1 21 6 -2 2 4 x1 12 -4 2 2 x2 10 -4 2 2 10 2 -5 x3 -9 2 3 -14 x4 -26 2 -5 -9

17
2 3 -14 -26 6 -2 2 4 x1 12 -4 2 2 x2 10 -4 2 2 10 2 -5 x3 -9 4 -13 x4 -21 4 -13 -21 6 -2 2 4 x1 12 -4 2 2 x2 10 2 -5 x3 -9 4 -13 x4 -21 4 -13 -21 6 -2 2 4 x1 12 -4 2 2 x2 10 2 -5 -9 2 -5 x3 -9 -3 -3 -3 x4 -3

18
**Backward substitution: inner-product-based**

6 -2 2 4 x1 12 6 -2 2 4 x1 12 -4 2 2 x2 10 -4 2 2 x2 10 2 -5 x3 -9 2 -5 x3 -9 -3 x4 -3 x4 6 -2 2 4 x1 12 6 -2 2 4 x1 12 -4 2 2 x2 10 x2 x3 x3 x4 x4

19
**Backward substitution: outer-product-based**

6 -2 2 4 x1 12 -4 2 2 x2 10 2 -5 x3 -9 -3 x4 -3 6 -2 2 x1 12 4 8 -4 2 x2 10 x4 2 8 2 x3 -9 -5 -4 6 -2 x1 8 2 12 x3 -4 x2 8 2 12 6 x1 12 x2 -2 6

20
**OutLine Basic operation of matrix Example of Gaussian Elimination (GE)**

Formal description of GE - component-wise and column-wise representation - recursive structure MATLAB usage

21
6 -2 2 4 x1 12 12 -8 6 10 x2 34 3 -13 9 3 x3 27 -6 4 1 -18 x4 -38 6 -2 2 4 x1 12 -4 2 2 x2 10 3 -13 9 3 x3 27 -6 4 1 -18 x4 -38 6 -2 2 4 x1 12 6 -2 2 4 x1 12 12 -8 6 10 x2 34 12 -8 6 10 x2 34 -12 8 1 x3 21 3 -13 9 3 x3 27 -6 4 1 -18 x4 -38 2 3 -14 x4 -26

22
**Eliminate first column: component-wise**

6 -2 2 4 x1 12 6 -2 2 4 x1 12 12 -8 6 10 x2 34 -4 2 2 x2 10 3 -13 9 3 x3 27 -12 8 1 x3 21 -6 4 1 -18 x4 -38 2 3 -14 x4 -26 where Exercise: check by MATLAB

23
**Eliminate first column: column-wise**

define and then Exercise: check by MATLAB Notation:

24
**Eliminate second column: component-wise**

6 -2 2 4 x1 12 -4 2 2 x2 10 -12 8 1 x3 21 2 3 -14 x4 -26 6 -2 2 4 x1 12 -4 2 2 x2 10 2 -5 x3 -9 2 3 -14 x4 -26 6 -2 2 4 x1 12 -4 2 2 x2 10 2 -5 x3 -9 4 -13 x4 -21

25
**Eliminate second column: column-wise**

define Exercise: check Exercise: check , why? Give a simple explanation. Notation:

26
**Eliminate third column: component-wise**

6 -2 2 4 x1 12 -4 2 2 x2 10 2 -5 x3 -9 4 -13 x4 -21 6 -2 2 4 x1 12 -4 2 2 x2 10 define 2 -5 x3 -9 -3 x4 -3 then

27
**LU-decomposition Notation: LU-decomposition Exercise: check**

Question: Do you have any effective method to write down matrix Question: why don’t we care about right hand side vector

28
**Problems about LU-decomposition**

Question 1: what condition does LU-decomposition fail? Question 2: does any invertible matrix has LU-decomposition? Question 3: How to measure “goodness of LU-decomposition“? Question 4: what is order of performance of LU-decomposition (operation count)? Question 5: How to parallelize LU-decomposition? Question 6: How to implement LU-decomposition with help of GPU?

29
**OutLine Basic operation of matrix Example of Gaussian Elimination (GE)**

Formal description of GE MATLAB usage - website resource - “help” command - M-file

30
MATLAB website

31
MATLAB: create matrix

32
**MATLAB: LU-factorization**

33
Start MATLAB 2008

34
**Command “help” : documentation**

!= in C

35
Loop in MATLAB

36
**Decision-making in MATLAB**

37
IO in MATLAB

38
**LU-factorization in MATLAB**

39
**M-file in MATLAB matvec.m**

Description of function matvec, write specificaiton of input parameter Consistency check Inner-product based

40
**Move to working directory**

M-file

41
**Execute M-file Construct matrix**

Execute M-file, like function call in C-language

42
**Exercise: forward / backward substitution**

Forward substitution since backward substitution since Write MATLAB code to do forward substitution and backward substitution

43
**Exercise: LU-decomposition**

You should find recursive structure of the decomposition

Similar presentations

OK

Using LU Decomposition to Optimize the modconcen.m Routine Matt Tornowske April 1, 2002.

Using LU Decomposition to Optimize the modconcen.m Routine Matt Tornowske April 1, 2002.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on product advertising photography Ppt on types of houses Ppt on second law of thermodynamics creationism Ppt on vitamin d deficiency in india Ppt on creativity and innovation Ppt on life of gautama buddha Ppt on current social evils of society Ppt on depreciation of indian rupee 2013 Ppt on natural disasters in india Chemistry ppt on water