Experiments with MATLAB

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2 Chapter 1. Iteration previous sessions, and the other arrows keys allow you to revise these commands. Use the up-arrow key, followed by the enter or return key, to iterate, or repeatedly execute, this statement: x = sqrt(1 + x) Here is what you get when you start with x = 3. x = 3 x = 2 x = 1.7321 x = 1.6529 x = 1.6288 x = 1.6213 x = 1.6191 x = 1.6184 x = 1.6181 x = 1.6181 x = 1.6180 x = 1.6180 These values are 3, √ 1 + 3, √ 1 + √ √ 1 + 3, 1 + √ 1 + √ 1 + 3, and so on. After 10 steps, the value printed remains constant at 1.6180. Try several other starting values. Try it on a calculator if you have one. You should find that no matter where you start, you will always reach 1.6180 in about ten steps. (Maybe a few more will be required if you have a very large starting value.) Matlab is doing these computations to accuracy of about 16 decimal digits, but is displaying only five. You can see more digits by first entering format long and repeating the experiment. Here are the beginning and end of 30 steps starting at x = 3. x = 3
x = 2 x = 1.732050807568877 x = 1.652891650281070 .... x = 1.618033988749897 x = 1.618033988749895 x = 1.618033988749895 After about thirty or so steps, the value that is printed doesn’t change any more. You have computed one of the most famous numbers in mathematics, ϕ, the Golden Ratio. In Matlab, and most other programming languages, the equals sign is the assignment operator. It says compute the value on the right and store it in the variable on the left. So, the statement x = sqrt(1 + x) takes the current value of x, computes sqrt(1 + x), and stores the result back in x. In mathematics, the equals sign has a different meaning. x = √ 1 + x is an equation. A solution to such an equation is known as a fixed point. (Be careful not to confuse the mathematical usage of fixed point with the computer arithmetic usage of fixed point.) The function f(x) = √ 1 + x has exactly one fixed point. The best way to find the value of the fixed point is to avoid computers all together and solve the equation using the quadratic formula. Take a look at the hand calculation shown in figure 1.1. The positive root of the quadratic equation is the Golden Ratio. ϕ = 1 + √ 5 . 2 You can have Matlab compute ϕ directly using the statement phi = (1 + sqrt(5))/2 With format long, this produces the same value we obtained with the fixed point iteration, phi = 1.618033988749895 3
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 and 8: Preface Figure 1. exmgui provides a
Page 9 and 10: Preface v Better yet, enter edit ma
Page 11: Chapter 1 Iteration Iteration is a
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
Page 59 and 60: 10 8 6 4 2 0 −2 −4 −6 −8 3
Page 61 and 62: 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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