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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 Error (motivation) Floating point number system –difference to real number system –problem of roundoff Introduced/propagated error Focus on numerical methods –three bugs 2

3 Any Questions? 3 About the exercise

4 In this slide Rootfinding –multiplicity Bisection method –Intermediate Value Theorem –convergence measures False position –yet another simple enclosure method –advantage and disadvantage in comparison with bisection method 4

5 Rootfinding 5

6 6 Is a rootfinding problem

7 7

8 8

9 9

10 Multiplicity 10

11 Definition 11

12 Multiplicity for polynomials For polynomials, multiplicity can be determined by factoring the polynomial Thats easy, but 12

13 For non-polynomials 13 answer

14 14

15 15

16 For non-polynomials 16

17 Rootfinding methods 2 categories –simple enclosure methods –fixed point iteration schemes Simple enclosure –bisection and false position –guaranteed to converge to a root, but slow Fixed point iteration –Newtons method and secant method –fast, but require stronger conditions to guarantee convergence 17

18 18

19 A pathological example 19

20 The Bisection Method

21 Bisection method The most basic simple enclosure method All simple enclosure methods are based on Intermediate Value Theorem 21

22 22 Drawing proof for Intermediate Value Theorem

23 In Plain English Find an interval of that the endpoints are opposite sign Since one endpoint value is positive and the other negative, zero is somewhere between the values, that is, at least one root on that interval 23

24 Bisection method The objective is to systematically shrink the size of that root enclosing interval The simplest and most natural way is to cut the interval in half Next is to determine which half contains a root –Intermediate Value Theorem, again Repeat the process on that half 24

25 Bisection method 25

26 In action 26

27 27

28 28

29 Any Questions? 29

30 30 You know what the bisection method is, but so far it is not an algorithm, why?

31 31 An algorithm requires a stopping condition

32 32 Convergence of {p n }

33 33

34 Note 34

35 35

36 36 We are now in position to select a stopping condition

37 Convergence measures 37

38 Which is the Best? 38 No one is always better than another answer

39 39

40 Which is the Best? 40 No one is always better than another

41 Algorithm 41

42 42

43 Note 43

44 Summary of bisection method Advantage –straightforward –inexpensive (1 evaluation per iteration) –guarantee to converge Disadvantage –error estimation can be overly pessimistic –(drawing for a extreme case of bisection method) 44

45 Any Questions? The Bisection Method

46 The Method of False Position

47 False position 47

48 48

49 49

50 Which method is better? 50

51 Which method is better From another aspect to only the convergence rate –bisection method provides a theoretical bound of error, but no error estimate –false position provides computable error estimate –(the only one advantage of false position) Thus, we can have a more appropriate stopping condition for false position –(we will use this advantage in Section 2.6) 51

52 52 Since false position has no theoretical bound of error, it requires effort to prove the convergence

53 53

54 54

55 Convergence analysis One observation to proceed the convergence analysis –one of the endpoints remains fixed –the other endpoint is just the previous approximation Namely – a n =a n-1, b n =p n-1 or – b n =b n-1, a n =p n-1 55 observation

56 56 The first problem

57 57 The second problem

58 58 The third problem

59 59

60 Convergence analysis 60

61 Go back to the equation (4) 61

62 62

63 Any Questions? 63

64 Guarantee to convergence 64 answer

65 Guarantee to convergence 65

66 The first condition 66 The remaining three conditions can be proved in a similar fashion

67 67 Now its time to select a stopping condition

68 Stopping condition 68

69 69

70 70 The first problem

71 71 The second problem

72 72 The third problem

73 Any Questions? The Method of False Position


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