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

Summary 2 2 exams, 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, “compile 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 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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19 t d is the time of death, which cannot be solved explicitly

We know that 20 t d is no earlier than PM 7:15, and t d is no later than PM 8:00. So…

t d could be PM 7:38 21 rubbish =.=

A systematic procedure 1. Let t d as PM 7:38 2. Evaluate the above formula 3. If the result exceeds 98.6, we use PM 7:27, otherwise, we use PM 7:49 instead 4. Repeat step 2 & 3 until the result is close to 98.6 enough 22

Bisection method 23

Bisection method The concept is –1) find the mid-point, 2) evaluate it, and 3) shrink the solution range It is stupid: just trial and error But it works, because t d is ascending And… 24

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

When #trails → ∞ 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 Mean and standard deviation on n values 35

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In action 37 input is { 1, 2, 3, 4, 5 }

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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

43 A partition a = x 0 < x 1 < x 2 < … < x n-1 < x n = b

44 where x i = a + i h, and h = ( b – a ) / n

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In action 46 f(x) = 1 / x, n = 4

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Error The analytic solution is ln2 The absolute error is 48

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Observations of the errors |e n |, the absolute error, is a decreasing function of n When n is doubled, |e n | is reduced by a factor (roughly 1/4) From the numerical evidence where c is independent of n 50

Any Questions? 51

The third example Square root – a is a nonnegative real number – x n+1 converges to sqrt(a) 52

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Stopping condition |x n+1 – sqrt(a)| <ε – |x n+1 – x n | <ε provides an estimate Prevent infinite loop –give a limit of the number of iterations 54

In action 55 sqrt(2), i.e., a=2 x 0 = 2, ε = 0.005, Nmax = 10

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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++ (??/23) Java (??/23) Other (??/23)

Learnt 63 Data structure (??/23) Algorithm (??/23)

Language vs. algorithm Two languages –The same concept, different patterns –e.g., Chinese and English – 想睡覺, 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

69 The sequence {x n } converges to the value L, and L is called the limit of the sequence

Rate of convergence Let {p n } converges to p, {β n } converges to 0, λ is a constant, and Rate of convergence of {p n } is O(β n ) O(β n ) is typically of the form – 1/n a – 1/a n 70

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Any Questions? 75

Which Is Better? 76 O(1/n 2 ), O(1/n 10 ), or O(1/2 n )

Using L'Hôpital's rule This is provided by a student 陳攀任 In its simplest form, l'Hôpital's rule states that for functions ƒ and g : To compare n 10 and 2 n 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 Let {p n } converges to p, {β n } converges to 0, λ is a constant, and Rate of convergence of {p n } is O(β n ) O(β n ) is typically of the form – 1/n a – 1/a n 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 An iterative method is said to be of order α if the sequence it generates converges of order α The most common values of α in practice are –α = 1 (linear convergence) –α = 2 (quadratic convergence) –α = 3 (cubic convergence) Non-integer values for α are possible 84

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

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