1. Chapter 2 System of Linear Equations Sensitivity and Conditioning (2.3) Solving Linear Systems (2.4) January 19, 2010

2. Vector Norms

3. Example: Vector Norm

4. Properties of Vector Norms

5. Matrix Norm Since A is linear, we can evaluate the max for ||x||=1

6. Illustration

7. Matrix Norms

8. Properties of Matrix Norms

9. Example

10. Condition Number

11. Properties of Condition Number

12. Error Bounds

14. Error Bounds, cont’d For example, if ε = 10 -6, cond(A) = 100 X = [1345.45677, 1643.56346]

15. Computing Matrix Condition Number

17. Residual

19. Solving Linear System (2.4) What should the form be for MA ?

20. Triangular Linear System

21. Triangular Matrices

22. Back-Substitution

23. Forward-Substitution

24. Example: Permutations

25. Example: Diagonal Scaling