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# 04 gaussmethods

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### 04 gaussmethods

1. 1. 1 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Linear Algebraic Equations Gauss Elimination ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Objectives • Knowing how to solve a system of linear equations • Understanding how to implement Gauss elimination method • Understanding the concepts of singularity and ill-conditioning
2. 2. 2 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik A System of Linear Equations 1823 21 =+ xx 22 21 =+− xx ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Solution Subtract second equation from first 16*04 21 =+ xx 41 =x 32 =x
3. 3. 3 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik In Matrix Form       =             − 2 18 21 23 2 1 x x 1823 21 =+ xx 22 21 =+− xx ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Elimination       =             − 2 18 21 23 2 1 x x Using row operations       =             − 2 16 21 04 2 1 x x
4. 4. 4 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Elimination Using row operations       =             − 2 16 21 04 2 1 x x       =             6 16 20 04 2 1 x x ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Naïve Elimination Routine!
5. 5. 5 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Back Substitution ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Special Cases
6. 6. 6 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Conclusions • In this lecture, we revised the process of Gauss elimination • A clear algorithm for the elimination process and the back substitution was presented • The different cases of no solution, infinite number of solutions, and ill conditioning were graphically presented ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Linear Algebraic Equations Iterative Solutions
7. 7. 7 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Objectives • Recognize the need for iterative solutions • Understand the difference between different iterative methods for solving systems of linear equations • Apply iterative methods to solve a system of linear equations ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Why Iterative Methods? • When system of equations is sparse; too many zero elements. Such systems are produced when approximately solving differential equations; finite difference, finite element, etc… • We already have sources of error in the solution; model errors, approximation errors, truncation errors, etc…, so why not use approximate method any way • Saves on time!
8. 8. 8 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Gauss-Jacobi ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Gauss-Jacobi: Example 1642 302 1124 321 21 321 =++ =++− =++ xxx xx xxx ( ) ( ) ( ) 4/216 2/3 4/211 213 12 321 xxx xx xxx −−= += −−= ( ) ( ) ( ) 4/216 2/3 4/211 21 1 3 1 1 2 32 1 1 kkk kk kkk xxx xx xxx −−= += −−= + + +
9. 9. 9 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Is that really going to work?!!! ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Let’s try it! { }           = 1 1 1 0 x ( ) ( ) ( ) 25.34/1216 22/13 24/1211 1 3 1 2 1 1 =−−= =+= =−−= x x x ( ) ( ) ( ) 5.24/22*216 5.22/23 9375.04/25.32*211 2 3 2 2 2 1 =−−= =+= =−−= x x x ( ) ( ) ( ) 9063.24/5.2875.116 9688.12/9375.03 875.04/5.2511 3 3 3 2 3 1 =−−= =+= =−−= x x x
10. 10. 10 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik In General! ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik A system of linear equations               =                           nnnnnn n n b b b x x x aaa aaa aaa MM L MOMM L L 2 1 2 1 21 22221 11211 Let’s examine one equation! 11212111 ... bxaxaxa nn =+++ ( ) 11 12121 1 ... a xaxab x nn++− =
11. 11. 11 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik For any equation       −−= ∑∑ += − = + n ij k jij i j k jiji ii k i xaxab a x 1 1 1 1 1 ( ) ii niniiiiiiii i a xaxaxaxab x +++++− = ++−− ...... 11,11,11 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Can it be any better?
12. 12. 12 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Gauss-Seidel       −−= ∑∑ += − = + n ij k jij i j k jiji ii k i xaxab a x 1 1 1 1 1       −−= ∑∑ += − = ++ n ij k jij i j k jiji ii k i xaxab a x 1 1 1 11 1 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Gauss-Seidel: Example 1642 302 1124 321 21 321 =++ =++− =++ xxx xx xxx ( ) ( ) ( ) 4/216 2/3 4/211 213 12 321 xxx xx xxx −−= += −−= ( ) ( ) ( ) 4/216 2/3 4/211 1 2 1 1 1 3 1 1 1 2 32 1 1 +++ ++ + −−= += −−= kkk kk kkk xxx xx xxx
13. 13. 13 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Let’s try it! { }           = 1 1 1 0 x ( ) ( ) ( ) 375.24/5.22*216 5.22/23 24/1211 1 3 1 2 1 1 =−−= =+= =−−= x x x ( ) ( ) ( ) 0586.34/9531.1906.0*216 9531.12/90625.03 90625.04/375.2511 2 3 2 2 2 1 =−−= =+= =−−= x x x ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Convergence Condition • For the iterative solutions presented to converge, the matrix must be diagonally dominant. ∑ ≠ = > n ij j ijii aa 1
14. 14. 14 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Error! { } { } { }kkk xxe −= −1 k i k ik a x e max=ε ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Algorithm 1. If the system is not diagonally dominant; end 2. Start with any initial solution {x} 3. Apply the steps for Gauss-Seidal method to evaluate the next iteration 4. If the maximum approximate relative error < εs; end 5. Let the old solution vector equal the new solution vector 6. Goto step 3
15. 15. 15 ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Homework #3 • For the given system of simultaneous equations • Write down the system of equation in a form that can be used for iterative methods for solving systems of equations • Use four iterations using Gauss-Jacobi method to find an approximate solution using initial values {0,0,0} • Use four iterations using Gauss-Seidel method to find an approximate solution using initial values {0,0,0} 1642 1124 32 321 321 21 =++ =++ =+− xxx xxx xx ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Homework #3 cont’d • Chapter 11, p 303, numbers: 11.8,11.9,11.10 • Due Next week