Global Variables. ˆ Unlike local variables, global variables are available to all functions involved.

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1 Global Variables ˆ Unlike local variables, global variables are available to all functions involved. ˆ Use global to declare x as global. ˆ For example, the universal constant, say,. 1 ˆ However, it is not a good practice to define many global variables. 2 ˆ Besides, you have to take the responsibility of not changing its value. 1 Planck constant = in quantum theories. 2 This violets the principle of modularity. Zheng-Liang Lu 192 / 236

2 Example 1 function h = freefall(t) 2 global GRAVITY 3 h = 1 / 2 * GRAVITY * t.ˆ 2; 4 end 1 clear; clc; 2 global GRAVITY 3 GRAVITY = 9.8; 4 y = freefall(0 : 0.1 : 5); ˆ As a more robust alternative, redefine this function to accept GRAVITY as an input. Zheng-Liang Lu 193 / 236

3 Relationships among User-Defined Functions ˆ Primary functions and subfunctions ˆ Anonymous functions ˆ Nested functions ˆ Private functions Zheng-Liang Lu 194 / 236

4 Primary Functions and Subfunctions ˆ An m-file may contain more than one user-defined function. ˆ The function whose name is identical to the file name is called the primary function, while the other functions are called subfunctions. ˆ Subfunctions are normally visible only to the primary function and other subfunctions. 3 ˆ Note that the order of the subfunctions does not matter, but function names must be unique within the m-file. 3 This is an implementation of namespaces which avoid name collisions between multiple identifiers that share the same name. Zheng-Liang Lu 195 / 236

5 Example: Alternative to Infinite Loops 4 1 function test1 2 fprintf('hi, there. This is test1.\n'); 3 test2; 4 end 5 6 function test2 7 fprintf('hi, there. This is test2.\n'); 8 test1; 9 end ˆ Notice that the aforesaid program becomes an infinite loop and is terminated by Matlab since the depth of call stack is limited. 4 Thanks to a lively class discussion (MATLAB238) on June 14, Zheng-Liang Lu 196 / 236

6 Handling Anonymous Functions 6 ˆ You can use function handles for anonymous functions, which are the expression followed by operator. 5 ˆ This enables you to create a simple function without creating an m-file for it. ˆ For example, 1 >> f x.ˆ 2 + x + 1 % x.ˆ2 allows... vectorization. 2 >> f([1 10]) 3 4 ans = See 6 See anonymous-functions.html. Zheng-Liang Lu 197 / 236

7 ˆ Furthermore, you can assign an existing function to a function handle, for example, y 7 ˆ You can create anonymous functions having more than one input. ˆ One anonymous function can call another to implement function composition. 1 >> f y) sqrt(x.ˆ 2 + y.ˆ 2); 2 >> g y) f(x, y).ˆ 2; 3 >> g(3, 4) 4 5 ans = You may refer to The truth is that every function name is also an alias of the function address! Zheng-Liang Lu 198 / 236

8 More Examples 9 ˆ Given input values, return a function handle. 8 ˆ For example, 1 function y = paragen(a, b, c) 2 y a * x.ˆ 2 + b * x + c; 3 end ˆ Acts like a parabolic function generator! 8 Contribution by Ms. Queenie Chang (MAT25108) on March 18, Thanks to a lively class discussion (MATLAB244) on August 22, Zheng-Liang Lu 199 / 236

9 ˆ Given a function handle as an input, return a value. 10 ˆ For example, 1 function y = df(f, x) 2 eps = 1e-9; 3 y = (f(x + eps) - f(x)) / eps; 4 end ˆ Given a function handle as an input, return a function handle. ˆ For example, 1 function y = df(f) 2 eps = 1e-9; 3 y (f(x + eps) - f(x)) / eps; 4 end 10 Thanks to a lively class discussion (MATLAB260) on September 16, Zheng-Liang Lu 200 / 236

10 Nested Functions ˆ A function is said to be nested if the function is defined within another function. ˆ A nested function can access the variables of its parent function. 1 function f = paragen(a, b, c) 2 f %f is a function handle of p 3 function y = p(x) 4 % p shares a, b, and c with parabola 5 y = a * x.ˆ2 + b * x + c; 6 end 7 end Zheng-Liang Lu 201 / 236

11 Private Functions ˆ Private functions are useful when you want to limit the scope of the functions. ˆ A private functions resides in a subfolder with the special name private. ˆ The private function is visible only to functions in its parent directory. ˆ Note that you cannot call the private function from the command line or from functions outside the parent folder of the private. Zheng-Liang Lu 202 / 236

12 Precedence When Calling Functions 1. Nested functions 2. Subfunctions within the same file 3. Private functions 4. Local functions in the same directory 5. Built-in functions 6. Standard m files in PATH Zheng-Liang Lu 203 / 236

13 Recursive Functions ˆ Recursion is when something is defined in terms of itself. ˆ A recursive function is the function which calls itself. ˆ Recursion is an alternative form of program control. ˆ It is essentially repetition without a loop. Zheng-Liang Lu 204 / 236

14 Example ˆ The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. ˆ For example, 4! = = 4 3! ˆ So we can say f (n) = n f (n 1), for n > 0. ˆ Also, we need a base case such that the recursion can stop instead of being an infinite loop. ˆ In this example, 0! = 1. Zheng-Liang Lu 205 / 236

15 Exercise Write a program which determines return a factorial of n. 1 function y = recursivefactorial(n) 2 if n > 0 3 y = n * recursivefactorial(n - 1); 4 else 5 y = 1; % base case 6 end 7 end ˆ We can use a loop to calculate n!. (Try.) Zheng-Liang Lu 206 / 236

16 Call Stack Zheng-Liang Lu 207 / 236

17 Equivalence 1 function y = loopfactorial(n) 2 y = 1; 3 for i = n : -1 : 2 4 y = y * i; 5 end 6 end ˆ One more intriguing question is, Can we always turn a recursive method into an iterative one? ˆ Yes, theoretically. 11 ˆ Which is better? 11 The Church-Turing thesis proves it if the memory serves. Zheng-Liang Lu 208 / 236

18 Example: Fibonacci Numbers 12 ˆ The Fibonacci sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, ˆ Equivalently, the Fibonacci sequence follows the recurrence relation F n = F n 1 + F n 2, where n 2, and F 0 = 0, F 1 = 1. ˆ Given integer n 0, determine F n. 12 See Fibonacci numbers. Zheng-Liang Lu 209 / 236

19 Solution 1 function y = recurfib(n) 2 if n == 0 3 y = 0; 4 elseif n == 1 5 y = 1; 6 else 7 y = recurfib(n - 1) + recurfib(n - 2); 8 end 9 end ˆ It is a binary recursion because this function calls itself twice. ˆ Time complexity: O(2 n ) (Why?) Zheng-Liang Lu 210 / 236

20 Fibonacci Numbers by Loops 1 function y = loopfib(n) 2 if n == 0 3 y = 0; 4 elseif n == 1 5 y = 1; 6 else 7 x = 0; 8 y = 1; 9 for i = 2 : n 10 tmp = x + y; 11 x = y; 12 y = tmp; 13 end 14 end 15 end ˆ Can we implement a linear recursion for this problem? Zheng-Liang Lu 211 / 236

21 Fibonacci Numbers by Linear Recursion 1 function ret = linearfib(n) 2 ret = fib(0, 1, n); 3 function c = fib(a, b, n) % nested function 4 if n > 0 5 c = fib(b, b + a, n - 1); 6 else 7 c = a; 8 end 9 end 10 end ˆ Time complexity: O(n)! Zheng-Liang Lu 212 / 236

22 Remarks ˆ Recursion bears substantial overhead. ˆ So the recursive algorithm may execute a bit more slowly than the iterative equivalent. 13 ˆ Moreover, a deeply recursive method depletes the call stack, which is limited, and causes a exception fast In modern compiler design, recursion elimination is automatic if the recurrence occurs in the last statement in the recursive function, so called tail recursion. However, only some of programming languages support the elimination of tail recursion. Unfortunately, Matlab does not support it. 14 Aka stack overflow. Zheng-Liang Lu 213 / 236

23 More Interesting Recursions ˆ Sum of numbers ˆ Product of numbers ˆ Greatest common divisor ˆ Towers of Hanoi ˆ Check whether a string is palindrome 15 using recursion ˆ N-Queens problem ˆ Quick sqrt ˆ Depth-first search ˆ Reversing a string ˆ Computing n mod m without using the function mod() 15 For example, noon, level, and radar. Zheng-Liang Lu 214 / 236

24 1 >> Lecture 4 2 >> 3 >> -- Graphics 4 >> Zheng-Liang Lu 215 / 236

25 Introduction ˆ Engineers use graphic techniques to make the information easier to understand. ˆ With graphs, it is easy to identify trends, pick out highs and lows, and isolate data points that may be measurement or calculation errors. ˆ Graphs can also be used as a quick check to determine whether a computer solution is yielding expected results. ˆ A set of ordered pairs is used to identify points on a two-dimensional graph. ˆ The data points are then connected by straight lines in common cases. Zheng-Liang Lu 216 / 236

26 2D Line Plot ˆ The function plot(x, y, CLM ) creates a 2-D line plot of the data in y versus the corresponding values in x. ˆ (x, y): data set, ˆ C: specify the color, ˆ L: specify the line type, ˆ M: specify the data marker. ˆ A figure is created automatically as soon as the function plot is called. Zheng-Liang Lu 217 / 236

27 Optional Parameters for CLM 16 ˆ Try help for further details about plot. 16 See Zheng-Liang Lu 218 / 236

28 Example 1 clear; clc; close all; 2 3 x = 0 : 0.1 : 2 * pi; 4 y = sin(x); 5 figure(1); % create Figure 1 6 plot(x, y, 'r-'); % plot a red and solid line 7 grid on; % show the grid ˆ Note that you can simply call figure without specifying a number. ˆ Use close to close all figures or specific one. ˆ You can try clf and cla. Zheng-Liang Lu 219 / 236

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30 Example 2: Multiple Curves 1 clear; clc; 2 x = linspace(0, 2 * pi, 100); 3 figure(2); % create Figure 2 4 plot(x, sin(x), '*', x, cos(x), 'o', x, sin(x) +... cos(x), '+'); 5 grid on; ˆ You can also use plot(x, A), where A is an m-by-n matrix and x is an array of size n, draws a figure for m lines. (Try.) Zheng-Liang Lu 221 / 236

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32 Exercise ˆ Plot sin(x), x 2, log 10 (x), and 2 x where x [0, 1] with 100 points in a single figure. 1 clear; clc; close all; 2 3 x = linspace(0, 1, 100); 4 y = [sin(x); x.ˆ 2; log10(x); 2.ˆ x]; 5 C = 'rgbc'; 6 M = '.od+'; 7 L = {'-', '-.', ':', '--'}; 8 figure; hold on; grid on; 9 for i = 1 : size(y, 1); 10 plot(x, y(i,:), [C(i) M(i) L{i}]); 11 end Zheng-Liang Lu 223 / 236

33 Zheng-Liang Lu 224 / 236

34 Annotating Plots ˆ title( string ) adds a title to the plot. ˆ xlabel( string ) adds a label to the x axis of the plot. ˆ ylabel( string ) adds a label to the y axis of the plot. ˆ legend({ string1, string2,...}) allows you to add a legend for the multiple lines. ˆ The legend shows a sample of the line and lists the string you have specified. ˆ text(x, y, string ) allows you to add a text box for the string and located at (x, y) in the figure. ˆ gtext is similar to text. ˆ The box is placed at a location determined interactively by the user by clicking in the figure window. Zheng-Liang Lu 225 / 236

35 Example 1 clear; clc; 2 3 x = linspace(0, 1, 100); 4 y = [sin(x); x.ˆ 2; log10(x); 2.ˆ x]; 5 color = 'rgbk'; 6 figure(1); hold on; grid on; 7 for i = 1 : 4 8 plot(x, y(i, :), [color(i) 'o:']); 9 end 10 title('demonstration'); 11 xlabel('time (Year)'); 12 ylabel('this is Y label. (Unit)'); 13 legend({'curve A', 'xˆ{2}', 'log {10}(x)',... 'eˆ{x}'}, 'Interpreter', 'tex'); 14 text(0.4, -1.5, 'This is a text.'); 15 gtext('matlab supports the LaTex symbols.'); Zheng-Liang Lu 226 / 236

36 Demonstration Curve A x 2 log 10 (x) e x This is Y label. (Unit) Matlab supports the LaTex symbols. 1.5 This is a text Time (Year) Zheng-Liang Lu 227 / 236

37 L A TEX ˆ TEX is a technical typesetting language, created by Donald Knuth, which is widely used for formatting mathematical expressions. ˆ L A TEXis a set of macros created by Leslie Lamport, which expand the power of TEX and made it easier to use. ˆ By default, several Matlab text-related commands, including xlabel, ylabel, title, and text, automatically interpret a subset of TEX commands. ˆ See and for further information. Zheng-Liang Lu 228 / 236

38 Greek Symbols 17 ˆ Replace the first letter by the upper case one in the L A TEXcode for the upper case Greek letters. ˆ For example, use \Omega for Ω. 17 See Table B.1 in Lent, p Zheng-Liang Lu 229 / 236

39 Binary Operators See Table B.3 in Lent, p Zheng-Liang Lu 230 / 236

40 Example: Use L A TEX 1 clf 2 axis([ ]); 3 title('testing Latex Math:... $\sqrt{1 - \chi(x) ˆ2}$',... 4 'Interpreter','latex') 5 text(0.2,0.25,'$$ \int 0ˆ\infty f(x) dx $$',... 6 'Interpreter','latex','FontSize',14); 7 text(0.2,0.45,'$$ \sum {k=0}ˆn \frac{1}{kˆn} $$',... 8 'Interpreter','latex','FontSize',14); 9 text(0.2,0.65,'$$ \left(\frac{1}{a}... \right)ˆ{2k+1} $$', 'Interpreter','latex','FontSize',14); 11 text(0.2,0.80,'$ \sum {k=0}ˆn \frac{1}{kˆn} $', 'Interpreter','latex','FontSize',14); 13 text(0.6,0.80,'$ \frac{dˆ2y}{dxˆ2} $', 'Interpreter','latex','FontSize',14); 15 text(0.6,0.65,'$$ \frac{dˆ2y}{dxˆ2} $$',... Zheng-Liang Lu 231 / 236

41 16 'Interpreter','latex','FontSize',14); 17 text(0.6,0.45,'$$ \frac{-b\pm\sqrt{bˆ2-4ac}}{2a}... $$', 'Interpreter','latex','FontSize',14); 19 text(0.6,0.25,'$$ \frac{1-\beta}{1+\alpha} $$', 'Interpreter','latex','FontSize',14); 21 text(0.4,0.9,'$$ $$', 'Interpreter','latex','FontSize',14); 23 xlabel('\alpha 1ˆ2'); 24 ylabel('$$\frac{1-x}{1+x}$$','interpreter','latex'); Zheng-Liang Lu 232 / 236

42 1 Testing Latex Math: 1 χ(x) 2 1 x 1+x N k=0 1 k n ( ) 2k+1 1 a N k=0 0 1 k n f(x)dx d 2 y dx 2 d 2 y dx 2 b± b 2 4ac 2a 1 β 1 +α α 1 Zheng-Liang Lu 233 / 236

43 Exercise: Black-Scholes Model 19 ˆ The Black-Scholes model is one of the most famous models for derivatives. ˆ In this model, a geometric Brownian motion is used to model the price dynamic of the underlying asset, given by ds t = rsdt + σs t dw t, where r is the risk-free interest rate, σ is the annualized volatility, and dw is also called Wiener process, following the standard normal distribution. ˆ We can approximate ds by calculating S i S i 1 where i is the time step. ˆ This numerical scheme is called finite difference. 19 Black and Scholes (1972). Zheng-Liang Lu 234 / 236

44 1 clear; clc; 2 3 N = 252; % 252 trading days typically 4 r = 0.01; % annual risk-free interest rate 5 vol = 0.3; % annual volatility 6 dt = 1 / N; 7 S = zeros(1, N + 1); 8 S(1) = 100; 9 10 for i = 2 : N ds = S(i - 1) * (r * dt + vol * sqrt(dt) *... randn(1)); 12 S(i) = S(i - 1) + ds; 13 end figure; plot(1 : N, S, '.:'); grid on; 16 ylabel('stock Price (TWD)'); 17 xlabel('trading Day (Day)'); Zheng-Liang Lu 235 / 236

45 110 MATLAB Stock Price (TWD) Trading Day (Day) Zheng-Liang Lu 236 / 236

Scope of Variables. In general, it is not a good practice to define many global variables. 1. Use global to declare x as a global variable.

Scope of Variables. In general, it is not a good practice to define many global variables. 1. Use global to declare x as a global variable. Scope of Variables The variables used in function m-files are known as local variables. Any variable defined within the function exists only for the function to use. The only way a function can communicate