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

2. Summary 2 1 exam, 1project and some exercises http://zoro.ee.ncku.edu.tw/na/

3. Target 3 Solve problems with numerical methods

4. In this slide Why numerical methods? –differences between human and computer –a very simple numerical method What is algorithm? –definition and components –three problems and three algorithms Convergence –compare rate of convergence 4

5. Why such methods? 5 Computer is stupid

6. x-2=0 6 Human says, x=2, easy!

7. { x-2=0; } 7 Computer says, compilation error!

8. What is the difference? 8

9. 9 Human is logical (thinking) http://www.wallcoo.com/paint/Donald_Zolan_Early_Childhood_02/wallpapers/1280x1024/painting_children_kjb_DonaldZolan_68TheThinker_sm.jpg

10. 10 Can do inference http://files.myopera.com/conansakura/albums/31567/thumbs/2.jpg_thumb.jpg

11. 11 Computer is procedural (executing) http://www.aclibrary.org/eventkeeper/Graphics/SLZ/computer.jpg

12. An example (((x+3)-2)+6)=0 –Human requires only the rules (in this case, arithmetic), –and can inference the steps for the solution 12

13. Computer (((x+3)-2)+6)=0 –Requires the exact procedure (steps) { x0=0–6; } { x1=x0+2; } { x=x1–3; } –These steps is numerical method 13

14. 14 Does computer have any advantage?

15. 15 It is fast http://www.masternewmedia.org/images/fast_snail_id86636_size350.jpg

16. So, why numerical methods? Computer is stupid Computer is fast (and works hard) Sometimes, stupid methods can solve difficult problems 16

17. 17

18. 18

19. 19

20. We know that 20

21. 21 rubbish =.=

22. A systematic procedure 22

23. 23 Bisection method http://www.leda-tutorial.org/en/unofficial/Pictures/BisectionMethod.png

24. Bisection method 24

25. And very accurate 25 Actually, it is getting accurate after every trial

26. 26 Computer works hard, so it could happen

27. Any Questions? 27

28. Algorithm 28 The heart of numerical analysis

29. Algorithm Definition –A precisely defined sequence of steps In this course –design; –implement; and –examine the performance 29

30. 30 How to implement?

31. By hand 31 too painful (but you might need to)

32. With computer 32 in other words, do programming

33. Programming 33 Even scared!

34. 34 Algorithm could be simple

35. An example from statistics 35

36. 36

37. In action 37

38. 38

39. It is also an algorithm 39 (a precisely defined sequence of steps)

40. Not 40 A difficult sequence of steps

41. Any Questions? 41

42. Another example 42 Definite integral using trapezoidal rule

43. 43

44. 44

45. 45

46. In action 46

47. 47

48. Error 48

49. 49

50. Observations of the errors 50

51. Any Questions? 51

52. The third example 52

53. 53

54. Stopping condition 54

55. In action 55

56. 56

57. So far 57 a statistics problem, the integral problem, and the square root problem

58. Any Questions? 58

59. 59 What is the differences among them? (hint: the concepts of the output)

60. Type of methods The statistics algorithm –generates an exact (analytic) solution The integral algorithm –generates an approximate (numerical) solution –many numerical methods work in much the same way The square root algorithm –generates a sequence of approximations which converge to the solution –another typical class of numerical methods 60

61. Poll 61 Programming ability

62. Learnt 62 C/C++ (??/24) Java (??/24) Other (??/24)

63. Learnt 63 Data structure (??/24) Algorithm (??/24)

64. Language vs. algorithm Two languages –The same concept, different patterns –e.g., Chinese and English –, feel sleepy English vs. C –Increase i by 1 –{ ++i; } Language is/defines the pattern Algorithm is/describes the concept 64

65. Pseudo-code 65 Not any real programming language

66. A pseudo-code example 66

67. Can You 67 Read/write pseudo-code?

68. Convergence 68 When several numerical methods are available, choose the fastest one

69. 69

70. Rate of convergence 70

71. 71

72. 72

73. 73

74. 74

75. Any Questions? 75

76. Which Is Better? 76

77. Using L'Hôpital's rule ( ) 77

78. 78

79. Rate of Convergence 79 There is another definition for function

80. Another definition of rate of convergence for function 80

81. 81

82. Rate of convergence 82

83. Order of Convergence A different measure of convergence speed than rate of convergence Examines the relationship between successive error values 83

84. Order of Convergence Iterative Method 84

85. 85 Note the dramatic difference between 1 and 2, and the slight difference between 2 and 3

86. 86