1. Numerical Analysis 1 EE, NCKU Tien-Hao Chang (Darby Chang)

2. In the previous slide 2

3. In this slide 3

4. 3.3 4 Vector and matrix norms

5. Pivoting strategies are designed to reduce the impact roundoff error The size of a vector/matrix is necessary to measure the error 5

6. Vector norm 6

7. 7 The two most commonly used norms in practice

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10. Vector norm Equivalent 10

11. 11

12. 12 The Euclidean norm and the maximum norm are equivalent

13. Matrix norms Similarly, there are various matrix norms, here we focus on those norms related to vector norms –natural matrix norms 13

14. Matrix norms Natural matrix norms 14

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16. Natural matrix norms Computing maximum norm 16

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19. Natural matrix norms Computing Euclidean norm Euclidean norm, unfortunately, is not as straightforward as computing maximum matrix norms Requires knowledge of the eigenvalues of the matrix 19

20. Eigenvalue review 20 later

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22. Eigenvalue review 22

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24. 24 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg

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27. Any Questions? 27 3.3 Vector and matrix norms

28. 3.4 28 Error estimates and condition number

29. Error estimation 29

30. Any Questions? 30

31. 31 hint#3 hint#2 hint#1

32. 32 answer hint#4hint#3 hint#2

33. 33 answer hint#4hint#3

34. 34 answer hint#4

35. 35 answer

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41. Condition number 41

42. 42

43. Perturbations (skipped) 43......

44. Any Questions? 44 3.4 Error estimates and condition number

45. 3.5 45 LU decomposition

46. LU decomposition Motivation 46

47. LU decomposition 47

48. 48

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51. 51

52. Any Questions? 52

53. 53 answer hint

54. 54 answer

55. 55

56. Solving a linear system 56

57. 57

58. Solving a linear system In summary Anyway, the two-step algorithm (LU decomposition) is superior to Gaussian elimination with back substitution 58

59. Any Questions? 59 3.5 LU decomposition

60. 3.6 Direct Factorization 60

61. 61 Recall that http://www.dianadepasquale.com/ThinkingMonkey.jpg

62. Direct factorization 62

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67. Any Questions? 67 3.6 Direct factorization