Numerical Analysis EE NCKU TienHao Chang Darby Chang

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

In the previous slide n Why numerical methods? – differences between human and computer – a very simple numerical method n What is algorithm? – definition and components – three problems and three algorithms n Convergence – compare rate of convergence 2

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

Let’s start from error n n 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

Can be analyzed n n 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

Should be prevented n n 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

1. 3 Floating Point Number Systems 7

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

Any Questions? 11

Floating point vs. real number n 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 hint adjacent elements while the exponent changes 12

Floating point vs. real number n 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

Floating point vs. real number n 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

Floating point system is discrete, finite and nonuniform 15

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

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Roundoff error n 19

Formal definition 20

An example 21

In general case (chopped) 22

In general case (chopped) 23

Machine precision/epsilon n 24

Formal definition 25

Another term about precision 26

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

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

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

IEEE standard single precision 31

1. 4 Floating Point Arithmetic 32

Motivation n 33

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

http: //www. radgraphics. net/images/main/atomic%20 explosion%20 -%204. jpg when they are accumulated 35

Typical arithmetic n 36

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Not associative n question 38

All intermediate results have been rounded 39

Any Questions? 40

Not associative n 41

Not associative n 42

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

Not distributive 44

Accumulation of roundoff error 45

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Introduced/propagated error 47

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

A notation in the analysis 49

In multiplication 50

In division 51

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

Propagated error in addition and subtraction 53

In addition and subtraction 54

Absolute vs. relative error n n 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 n Algorithms should avoid the subtraction of nearly equal numbers 55

http: //www. dianadepasquale. com/Thinking. Monkey. jpg Recall that 56

Should be prevented n n 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

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

http: //rinat. relcom. net/Gallery/slides/bug. jpg Bug 1 59

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± be careful 61

http: //thomashawk. com/hello/209/1017/1024/Jackson%20 Running. jpg In action 62

In action 63

Analysis n The larger root – 239. 4 (actual root: 239. 4246996) – is the floating point equivalent of the actual root n The smaller root – 0. 15 (actual root: 0. 1253003555) – nearly 20% relative error 64

Any Questions? 65

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

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http: //rinat. relcom. net/Gallery/slides/bug. jpg Bug 2 69

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http: //i 5. tinypic. com/4 yqudc 7. jpg After one pass of Gaussian elimination 73

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The next multiplier 75

http: //www. radgraphics. net/images/main/atomic%20 explosion%20 -%204. jpg 76

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Cascade of effects n n n 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

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http: //rinat. relcom. net/Gallery/slides/bug. jpg Bug 3 80

Values of a function n 81

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How reformulate 84

Reforming with Taylor series 85

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More precision n 87

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Need at least 37 digits 89

Any Questions? 90

Good, that means we would like to have exercises 91

Exercise Due at 2011/3/28 2: 00 pm Email to darby@ee. ncku. edu. tw 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 work. 92

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