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

Summary 2 1 exam, 1project and some exercises

Target 3 Solve problems with numerical methods

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

Why such methods? 5 Computer is stupid

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

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

What is the difference? 8

9 Human is logical (thinking)

10 Can do inference

11 Computer is procedural (executing)

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

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 Does computer have any advantage?

15 It is fast

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

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We know that 20

21 rubbish =.=

A systematic procedure 22

23 Bisection method

Bisection method 24

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

26 Computer works hard, so it could happen

Any Questions? 27

Algorithm 28 The heart of numerical analysis

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

30 How to implement?

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

With computer 32 in other words, do programming

Programming 33 Even scared!

34 Algorithm could be simple

An example from statistics 35

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In action 37

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

Not 40 A difficult sequence of steps

Any Questions? 41

Another example 42 Definite integral using trapezoidal rule

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In action 46

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Error 48

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Observations of the errors 50

Any Questions? 51

The third example 52

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Stopping condition 54

In action 55

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

Any Questions? 58

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

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

Poll 61 Programming ability

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

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

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

Pseudo-code 65 Not any real programming language

A pseudo-code example 66

Can You 67 Read/write pseudo-code?

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

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Rate of convergence 70

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Any Questions? 75

Which Is Better? 76

Using L'Hôpital's rule ( ) 77

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Rate of Convergence 79 There is another definition for function

Another definition of rate of convergence for function 80

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Rate of convergence 82

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

Order of Convergence Iterative Method 84

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

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