Experiments with MATLAB

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28 Chapter 2. Fibonacci Numbers The sequence would be defined by g1 = 1, g2 = 1, g3 = 2 and, for n > 3, gn = gn−1 + gn−3 (a) Modify fibonacci.m and fibnum.m to compute this sequence. (b) How many pairs of rabbits are there after 12 months? (c) gn ≈ γ n . What is γ? (d) Estimate how long it would take your computer to compute fibnum(50) with this modified fibnum. 2.9 Mortality. What if rabbits took one month to mature, but then died after six months. The sequence would be defined by dn = 0, n 2, dn = dn−1 + dn−2 − dn−7 (a) Modify fibonacci.m and fibnum.m to compute this sequence. (b) How many pairs of rabbits are there after 12 months? (c) dn ≈ δ n . What is δ? (d) Estimate how long it would take your computer to compute fibnum(50) with this modified fibnum. 2.10 Hello World. Programming languages are traditionally introduced by the phrase ”hello world”. An script in exm that illustrates some features in Matlab is available with hello_world Explain what each of the functions and commands in hello_world do. 2.11 Fibonacci power series. The Fibonacci numbers, fn, can be used as coefficients in a power series defining a function of x. F (x) = ∞∑ fnx n n=1 = x + 2x 2 + 3x 3 + 5x 4 + 8x 5 + 13x 6 + ...
Our function fibfun1 is a first attempt at a program to compute this series. It simply involves adding an accumulating sum to fibonacci.m. The header of fibfun1.m includes the help entries. function [y,k] = fibfun1(x) % FIBFUN1 Power series with Fibonacci coefficients. % y = fibfun1(x) = sum(f(k)*x.^k). % [y,k] = fibfun1(x) also gives the number of terms required. The first section of code initializes the variables to be used. The value of n is the index where the Fibonacci numbers overflow. \excise \emph{Fibonacci power series}. The Fibonacci numbers, $f_n$, can be used as coefficients in a power series defining a function of $x$. \begin{eqnarray*} F(x) & = & \sum_{n = 1}^\infty f_n x^n \\ & = & x + 2 x^2 + 3 x^3 + 5 x^4 + 8 x^5 + 13 x^6 + ... \end{eqnarray*} Our function #fibfun1# is a first attempt at a program to compute this series. It simply involves adding an accumulating sum to #fibonacci.m#. The header of #fibfun1.m# includes the help entries. \begin{verbatim} function [y,k] = fibfun1(x) % FIBFUN1 Power series with Fibonacci coefficients. % y = fibfun1(x) = sum(f(k)*x.^k). % [y,k] = fibfun1(x) also gives the number of terms required. The first section of code initializes the variables to be used. The value of n is the index where the Fibonacci numbers overflow. n = 1476; f = zeros(n,1); f(1) = 1; f(2) = 2; y = f(1)*x + f(2)*x.^2; t = 0; The main body of fibfun1 implements the Fibonacci recurrence and includes a test for early termination of the loop. for k = 3:n f(k) = f(k-1) + f(k-2); y = y + f(k)*x.^k; if y == t return end 29
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 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: Exercises 2.1 Rabbits. Explain what
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
Page 63 and 64: 0.2588 0.9659 G45 = 0.7071 -0.7071
Page 65 and 66: Matrix arithmetic Matrix addition a
Page 67 and 68: % This is an executable program tha
Page 69 and 70: % and the complex conjugate transpo
Page 71 and 72: are mirror images of each other. 10
Page 73 and 74: Chapter 5 Linear Equations The most
Page 75 and 76: A = [3 12 1; 12 0 2; 0 2 3] Since a
Page 77 and 78: so b = 2.36 5.26 2.63 Now we do not
Page 79 and 80: Did that solution occur to you? The
Page 81 and 82: (c) (d) (e) (f) ⎛ ⎝ ⎛ ⎝ ⎛
Page 83 and 84: 5.7 Circuit. Figure 5.2 is the circ
Page 85 and 86: Chapter 6 Fractal Fern The fractal
Page 87 and 88: 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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