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

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Presentation on theme: "Numerical Analysis 1 EE, NCKU Tien-Hao Chang (Darby Chang)"— Presentation transcript:

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

8 8

9 9

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

15 15

16 Natural matrix norms Computing maximum norm 16

17 17

18 18

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

21 21

22 Eigenvalue review 22

23 23

24 24 In action

25 25

26 26

27 Any Questions? Vector and matrix norms

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

36 36

37 37

38 38

39 39

40 40

41 Condition number 41

42 42

43 Perturbations (skipped)

44 Any Questions? Error estimates and condition number

45 LU decomposition

46 LU decomposition Motivation 46

47 LU decomposition 47

48 48

49 49

50 50

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? LU decomposition

60 3.6 Direct Factorization 60

61 61 Recall that

62 Direct factorization 62

63 63

64 64

65 65

66 66

67 Any Questions? Direct factorization


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