1. Dr. Mubashir Alam King Saud University

2. Outline LU Factorization (6.4) Solution by LU Factorization Compact Variants of Gaussian Elimination, (Doolittle’s Method ) (6.4.1) Stability in Solving Linear Systems (6.5.2)

7. LU Factorization

8. Gaussian Elimination based on LU Factorization

9. Step:1 Step:2 Solve for g Solve for x

10. Example: 6.4.3 (Page 284)

13. Compact Variants of Gaussian Elimination Doolittle’s Method

14. Dolittle’s Method

16. Example: 6.4.4 (Page:285)

18. Stability in Solving Linear Systems Rounding Errors in Gaussian Elimination lead to errors in computed solution of Ax=b

19. Examine the degree of sensitivity of the solution ‘x’ to these rounding errors. Direct and complete analysis is difficult. Effect on the solution ‘x’ is done by making a small change in the right side ‘b’ of the linear system Stability in Solving Linear Systems How to measure the sizes of vectors and matrices?

21. Different Definition of Vector Norm Define and Use only One

22. Will use only || x || as notation

25. Matrix ‘A’ is the square matrix of size ‘n x n’

28. Vector Norm

29. (Same vector norm which we define in previous slides)

32. Condition Number

35. Example: 6.5.5 (Page 300)