Experiments with MATLAB

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iv Preface We expect that you already know something about high school level material in geometry, algebra, and trigonometry. We will introduce ideas from calculus, matrix theory, and ordinary differential equations, but we do not assume that you have already taken courses in the subjects. In fact, these experiments are useful supplements to such courses. We also expect that you have some experience with computers, perhaps with word processors or spread sheets. If you know something about programming in languages like C or Java, that will be helpful, but not required. We will introduce Matlab by way of examples. Many of the experiments involve understanding and modifying Matlab scripts and functions that we have already written. You should have access to Matlab and to our exm toolbox, the collection of programs and data that are described in Experiments with MATLAB. We hope you will not only use these programs, but will read them, understand them, modify them, and improve them. The exm toolbox is the apparatus in our “Laboratory”. You will want to have Matlab handy. For information about the Student Version, see http://www.mathworks.com/academia/student_version For an introduction to the mechanics of using Matlab, see the videos at http://www.mathworks.com/academia/student_version/start.html For documentation, including “Getting Started”, see http://www.mathworks.com/access/helpdesk/help/techdoc/matlab.html For user contributed programs, programming contests, and links into the world-wide Matlab community, check out http://www.mathworks.com/matlabcentral To get started, download the exm toolbox, use pathtool to add exm to the Matlab path, and run exmgui to generate figure 1. You can click on the icons to preview some of the experiments. You will want to make frequent use of the Matlab help and documentation facilities. To quickly learn how to use the command or function named xxx, enter help xxx For more extensive information about xxx, use doc xxx We hope you will find the experiments interesting, and that you will learn how to use Matlab along the way. Each chapter concludes with a “Recap” section that is actually an executable Matlab program. For example, you can review the Magic Squares chapter by entering magic_recap
Preface v Better yet, enter edit magic_recap and run the program cell-by-cell by simultaneously pressing the Ctrl-Shift-Enter keys. A fairly new Matlab facility is the publish command. You can get a nicely formatted web page about magic_recap with publish magic_recap If you want to concentrate on learning Matlab, make sure you read, run, and understand the recaps. Cleve Moler October 4, 2011
Page 1 and 2: Experiments with MATLAB Cleve Moler
Page 3 and 4: Contents Preface iii 1 Iteration 1
Page 5 and 6: Experiments with MATLAB R⃝ Copyri
Page 7: Preface Figure 1. exmgui provides a
Page 11 and 12: Chapter 1 Iteration Iteration is a
Page 13 and 14: x = 2 x = 1.732050807568877 x = 1.6
Page 15 and 16: produce a figure that has three com
Page 17 and 18: provide brief descriptions of comma
Page 19 and 20: format compact %% Names and assignm
Page 21 and 22: Express one barn-megaparsec in teas
Page 23 and 24: 1.11 cos(x). Find the numerical sol
Page 25 and 26: 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.
Page 27 and 28: Chapter 2 Fibonacci Numbers Fibonac
Page 29 and 30: f = zeros(n,1); f(1) = 1; f(2) = 2;
Page 31 and 32: The fibnum function is recursive. I
Page 33 and 34: 2.00000000000000 1.50000000000000 1
Page 35 and 36: and ends with Recap 2 3 5 8 13 21 3
Page 37 and 38: Exercises 2.1 Rabbits. Explain what
Page 39 and 40: Our function fibfun1 is a first att
Page 41 and 42: There is no array of coefficients.
Page 43 and 44: Chapter 3 Calendars and Clocks Comp
Page 45 and 46: and datenum(2011,8,2) datenum(’Au
Page 47 and 48: period of 23 days, the emotional cy
Page 49 and 50: xmas = datenum(c(1),12,25) days_til
Page 51 and 52: 3.9 Your birthday. On which day of
Page 53 and 54: (b) Make clockex run counter-clockw
Page 55 and 56: Chapter 4 Matrices Matlab began as
Page 57 and 58: This implies that points near x are
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10 8 6 4 2 0 −2 −4 −6 −8 3
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R = 2*rand(2,2) - 1 generates a 2-b
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0.2588 0.9659 G45 = 0.7071 -0.7071
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Matrix arithmetic Matrix addition a
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% This is an executable program tha
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% and the complex conjugate transpo
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are mirror images of each other. 10
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Chapter 5 Linear Equations The most
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A = [3 12 1; 12 0 2; 0 2 3] Since a
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so b = 2.36 5.26 2.63 Now we do not
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Did that solution occur to you? The
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(c) (d) (e) (f) ⎛ ⎝ ⎛ ⎝ ⎛
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5.7 Circuit. Figure 5.2 is the circ
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Chapter 6 Fractal Fern The fractal
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the point into the lower subfern on
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stop = uicontrol(’style’,’tog
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% fern % finitefern %% fern.jpg F =
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Chapter 7 Google PageRank The world
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The best way to compute PageRank in
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alpha beta gamma delta sigma rho Fi
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We see that alpha has a higher Page
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0.02 0.018 0.016 0.014 0.012 0.01 0
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Exercises 7.1 Use surfer and pagera
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% PAGERANKPOW PageRank by power met
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Chapter 8 Exponential Function The
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0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55
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The result gives us the numerical v
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then log e(y) = t The function log
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17 2292.02 2335.52 2339.65 18 2406.
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Figure 8.3. Two plots of e iθ . 10
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= s; n = n + 1; term = (t/n)*term;
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format compact format long Explain
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Chapter 9 T Puzzle A classic puzzle
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Figure 9.4. The arrow and the rhomb
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z = 3.0000 + 4.0000i r = 5 phi = 0.
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% % puzzle_recap % edit puzzle_reca
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9.8 Rotation. Reproduce figure 9.6.
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Chapter 10 Magic Squares With origi
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produces sum(diag(flipud(A))) 15 Th
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.. image(X) colormap(map) axis imag
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odd, n is odd. singly-even, n is di
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M = n*A+B+1 131 produces a matrix w
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Further Reading 133 The reasons why
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%% Rank figure for n = 3:20 r(n) =
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gbplot(hot) This is what TV movie c
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for n = 3:20 r(n) = rank(magic(n));
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Chapter 11 TicTacToe Magic Three si
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Figure 11.4. Typical output from a
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145 Briefly, then, the green moves
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147 You see that 4 and 5 are on the
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% function [i,j] = winningmove(X,p)
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Chapter 12 Game of Life Conway’s
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Figure 12.3. A glider gliding. Figu
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generate index vectors that increas
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Further Reading The Wikipedia artic
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12.4 Glider collisions. What happen
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R = P(:,2)./P(:,1); hist(P(:,1),30)
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Chapter 13 Mandelbrot Set Fractals,
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Figure 13.3. Two trajectories. z0 =
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167 Use the up arrow and backspace
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n = length(x); e = ones(n,1); z0 =
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grid^2 * depth So the statement cou
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4.0e-11,1024,2048,2) 173 Taking wid
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Exercises 13.1 Explore. Use the man
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Figure 13.8. Region #4. “Valley o
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Figure 13.12. Region #8. “Nebula
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Figure 13.16. Region #12. “Geode
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Chapter 14 Sudoku Humans look for p
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8 3 4 1 5 9 6 7 2 8 3 4 1 5 9 6 7 2
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1 1 3 2 3 4 Figure 14.8. diag(1:4)
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189 A number of operations on a Sud
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191 The following statement creates
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%% Disclaimer % Our Sudoku chapter
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14.6 Try this. Try to use sudoku_as
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H C D A A E I B F G B C A H C D A E
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Chapter 15 Ordinary Differential Eq
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Figure 15.2. Spacewar, the world’
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203 two different values of the tim
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1 0.8 0.6 0.4 0.2 0 −0.2 −0.4
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will also produce figure 15.5. Use
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plot(y(:,1),y(:,2),’-o’) axis s
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15.5 Linear system Write the system
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Chapter 16 Predator-Prey Model Mode
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y = mu*eta*exp(k*t)./(eta*exp(k*t)
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217 There are two tricky parts of t
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eta = 1 mu = 20 ydot = @(t,y) k*(1-
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Chapter 17 Orbits Dynamics of many-
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Figure 17.1. Initial, and final, po
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plot(x,y,’.-’) s = 2*sqrt(m); a
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227 interface controls. In particul
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1.5 1 0.5 0 −0.5 −1 −1.5 −1
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1 0 −1 −1 0 1 1 0 −1 −1 0 1
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0.3836 -0.4836 233 The three masses
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0.3 0.2 0.1 0 −0.1 −0.2 −0.3
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%% Snapshot of three dimensional Br
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Exercises 239 17.1 Bouncing ball. (
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Chapter 18 Shallow Water Equations
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243 represents a three component ve
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%% Create a two dimensional grid. m
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Chapter 19 Morse Code Morse code de
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produces a cell array that contains
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And finally one assignment statemen
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. . E T I A N M S U R W . . Figure
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255 Encoding is a little more work
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257 A computer byte is 8 bits and s
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% publish morse_recap % % Related E
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19.3 dash. Why didn’t I use an un
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Chapter 20 Music What does 12√ 2
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ange is -12:12. The statement piano
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Figure 20.4. x = sin t, y = sin 3/2
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269 a and b by entering values in e
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5 1.334840 4/3 1.333333 0.0011 6 1.
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Figure 20.10. C major fifth with eq
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http://en.wikipedia.org/wiki/Music_
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tfinal = 12*pi; t = 0:pi/512:tfinal
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