Download presentation

Presentation is loading. Please wait.

Published byLewis Lugar Modified over 2 years ago

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

2
In the previous 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 2

3
In this slide Fixed point iteration scheme –what is a fixed point? –iteration function –convergence Newtons method –tangent line approximation –convergence Secant method 3

4
Rootfinding Simple enclosure –Intermediate Value Theorem –guarantee to converge convergence rate is slow –bisection and false position Fixed point iteration –Mean Value Theorem –rapid convergence loss of guaranteed convergence 4

5
2.3 5 Fixed Point Iteration Schemes

6
6

7
7 There is at least one point on the graph at which the tangent lines is parallel to the secant line

8
Mean Value Theorem 8

9
9

10
Fixed points 10

11
11

12
Number of fixed points According to the previous figure, a trivial question is –how many fixed points of a given function? 12

13
13

14
14

15
Only sufficient conditions Namely, not necessary conditions –it is possible for a function to violate one or more of the hypotheses, yet still have a (possibly unique) fixed point 15

16
Fixed point iteration 16

17
Fixed point iteration 17

18
18 In action

19
19

20
20

21
Any Questions? 21 About fixed point iteration

22
Relation to rootfinding Now we know what fixed point iteration is, but how to apply it on rootfinding? More precisely, given a rootfinding equation, f(x)=x 3 +x 2 -3x-3=0, what is its iteration function g(x) ? 22 hint

23
Iteration function 23

24
24

25
25 In action

26
26

27
Analysis 27

28
Convergence 28

29
29

30
30

31
31

32
32

33
Order of convergence of fixed point iteration schemes 33

34
34

35
35

36
36

37
37

38
38

39
Stopping condition 39

40
40

41
Two steps 41

42
The first step 42

43
The second step 43

44
Any Questions? Fixed Point Iteration Schemes

45
Newtons Method

46
46

47
47

48
Newtons Method Definition 48

49
49 In action

50
50

51
In the previous example 51

52
52

53
Any Questions? 53

54
Initial guess 54 example answer

55
Initial guess 55 answer

56
Initial guess 56

57
57

58
Convergence analysis for Newtons method 58

59
59 The simplest plan is to apply the general fixed point iteration convergence theorem

60
Analysis strategy 60

61
61

62
62

63
63

64
64

65
Any Questions? 65

66
Newtons Method Guaranteed to Converge? 66 hint answer

67
Newtons Method Guaranteed to Converge? 67 answer

68
Newtons Method Guaranteed to Converge? 68

69
69 Oh no! After these annoying analyses, the Newtons method is still not guaranteed to converge!?

70
Dont worry Actually, there is an intuitive method Combine Newtons method and bisection method –Newtons method first –if an approximation falls outside current interval, then apply bisection method to obtain a better guess (Can you write an algorithm for this method?) 70

71
Newtons Method Convergence analysis 71

72
72 Recall that

73
73 Is Newtons method always faster?

74
74

75
75 In action

76
76

77
Any Questions? Newtons Method

78
Secant Method

79
Secant method Because that Newtons method –2 function evaluations per iteration –requires the derivative Secant method is a variation on either false position or Newtons method –1 additional function evaluation per iteration –does not require the derivative Lets see the figure first 79 answer

80
80

81
Secant method 81

82
82

83
83

84
84

85
85

86
Any Questions? Secant Method

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google