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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 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 2

3 In this slide Error (motivation) Floating point number system –difference to real number system –problem of roundoff Introduced/propagated error Focus on numerical methods –three bugs 3

4 Let’s start from error Numerical methods are generally designed to determine approximation solutions 3 categories of error types –modeling: made when you decide the algorithm –discretization/truncation: conversion from continuous to discrete and/or truncation of an infinite series –roundoff/data: not due to the formulation of a numerical method, caused by the data representation (in computer) 4

5 Can be analyzed Numerical methods are generally designed to determine approximation solutions 3 categories of error types –modeling: made when you decide the algorithm –discretization/truncation: conversion from continuous to discrete and/or truncation of an infinite series –roundoff/data: not due to the formulation of a numerical method, caused by the data representation (in computer) 5

6 Should be prevented Numerical methods are generally designed to determine approximation solutions 3 categories of error types –modeling: made when you decide the algorithm –discretization/truncation: conversion from continuous to discrete and/or truncation of an infinite series –roundoff/data: not due to the formulation of a numerical method, caused by the data representation (in computer) 6

7 1.3 7 Floating Point Number Systems

8 8

9 9

10 Floating point vs. real number Discrete vs. continuous –continuous means that between any two numbers, there are infinitely many other numbers Finite vs. infinite –number of element and range of values –a floating point number system contains its smallest/largest element underflow/overflow 10

11 Any Questions? 11

12 Floating point vs. real number Nonuniform vs. uniform –real numbers are uniformly distributed –in a floating point number system, the elements **** *** **** are more closely spaced think about the difference between two adjacent elements while the exponent changes 12 hint

13 Floating point vs. real number Nonuniform vs. uniform –real numbers are uniformly distributed –in a floating point number system, the elements **** *** **** are more closely spaced think about the difference between two adjacent elements while the exponent changes 13

14 Floating point vs. real number Nonuniform vs. uniform –real numbers are uniformly distributed –in a floating point number system, the elements near the zero are more closely spaced think about the difference between two adjacent elements while the exponent changes 14

15 Floating point system is 15 discrete, finite and nonuniform

16 Roundoff error When the number is outside the system Select an element to represent the number –chop –round A number to its floating point equivalent – y → fl(y) 16

17 17

18 18

19 Roundoff error 19

20 Formal definition 20

21 An example 21

22 In general case (chopped) 22

23 In general case (chopped) 23

24 Machine precision/epsilon 24

25 Formal definition 25

26 Another term about precision 26

27 27

28 So far, 28 we talked about floating point number systems in abstract

29 Then, 29 what systems are we likely to encounter in practice?

30 Real floating point system 1970s –begun to develop a standard binary floating point numbers to eliminate inconsistencies 1985 –IEEE –Binary Floating Point Arithmetic Standard 754 The IEEE Standard –F(2,24,-125,128), single precision –F(2,53,-1021,1024), double precision 30

31 IEEE standard single precision 31

32 Floating Point Arithmetic

33 Motivation 33

34 Numerical methods 34 perform a sequence of calculations on computer, where each operation introduces some roundoff error

35 35 when they are accumulated

36 Typical arithmetic 36

37 37

38 Not associative 38 question

39 39 All intermediate results have been rounded

40 Any Questions? 40

41 Not associative 41

42 Not associative 42

43 In FP arithmetic, 43 always notice the number of significant digits and the least significant bits

44 Not distributive 44

45 45 Accumulation of roundoff error

46 46

47 Introduced/propagated error 47

48 Propagated error 48 can be large even if the introduced error is small

49 A notation in the analysis 49

50 In multiplication 50

51 In division 51

52 The relative error propagates slowly The absolute error can grow rapidly, when multiplying by a large number or dividing by a small number 52

53 Propagated error 53 in addition and subtraction

54 In addition and subtraction 54

55 Absolute vs. relative error Multiplication and division may result large absolute error Addition and subtraction may result large relative error –more crucial –cancellation error two nearly equal numbers are subtracted Algorithms should avoid the subtraction of nearly equal numbers 55

56 56 Recall that

57 Should be prevented Numerical methods are generally designed to determine approximation solutions 3 categories of error types –modeling: made when you decide the algorithm –discretization/truncation: conversion from continuous to discrete and/or truncation of an infinite series –roundoff/data: not due to the formulation of a numerical method, caused by the data representation (in computer) 57

58 To prevent, 58 we need to know the floating point system

59 59 Bug 1

60 60

61 ± 61 be careful

62 62 In action

63 In action 63

64 Analysis The larger root – (actual root: ) –is the floating point equivalent of the actual root The smaller root – 0.15 (actual root: ) –nearly 20% relative error 64

65 Any Questions? 65

66 An intuitive question How to solve the quadratic formula problem? Reformulate the calculation of the smaller root 66 hint

67 67

68 68

69 69 Bug 2

70 70

71 71

72 72

73 73 After one pass of Gaussian elimination

74 74

75 The next multiplier 75

76 76

77 77

78 Cascade of effects Cancellation error led to a small pivot element A small pivot led to a large multiplier A large multiplier led to loss of significant digits 78

79 79

80 80 Bug 3

81 Values of a function 81

82 82

83 83

84 How reformulate 84

85 Reforming with Taylor series 85

86 86

87 More precision 87

88 88

89 Need at least 37 digits 89

90 Any Questions? 90

91 Good, 91 that means we would like to have exercises

92 Exercise 92 Due at 2011/3/28 2:00pm to or hand over in class. You may arbitrarily pick one problem among the first three, which means this exercise contains only five problems. The picked problem should include a programming

93 93

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