1 >> Lecture 3 2 >> 3 >> -- Functions 4 >> Zheng-Liang Lu 172 / 225

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1 1 >> Lecture 3 2 >> 3 >> -- Functions 4 >> Zheng-Liang Lu 172 / 225

2 Functions The first thing of the design of algorithms is to divide and conquer. A large and complex problem would be solved by couples of methods for its subproblems. In order to reuse the algorithm without copying the codes, the best way is to make functions. 1 The idea is similar to the math function, which is typically written in form of y = f (x), where x is the input value and y is the output value. 1 Make bricks for your castle. Zheng-Liang Lu 173 / 225

3 A function is a piece of computer code that accepts an input argument from the caller and returns output argument for the specific job the caller dealing with. Functions allow you to modularize a program by separating its tasks into self-contained units. We can program more efficiently and avoid rewriting the computer code for calculations that are performed frequently. 2 2 Remember that the bug propagates when you copy and paste the codes. It is a serious problem especially when you are working on a medium- or huge-level project. Zheng-Liang Lu 174 / 225

4 Built-in Arithmetic Functions Zheng-Liang Lu 175 / 225

5 Built-in Rounding Functions Can you organize an algorithm for the function ceil and floor? How to check if the input is an integer? (Try.) Zheng-Liang Lu 176 / 225

6 Built-in Discrete Math Functions Zheng-Liang Lu 177 / 225

7 Built-in Max Functions Zheng-Liang Lu 178 / 225

8 Built-in Max Functions (Cont d) There are also min functions to find the minimal elements in arrays, similar to max. Zheng-Liang Lu 179 / 225

9 Built-in Average Functions Zheng-Liang Lu 180 / 225

10 Variance and Standard Deviation Zheng-Liang Lu 181 / 225

11 Built-in Size Functions Zheng-Liang Lu 182 / 225

12 Random Number Generators To generate random integers, you may use randi(n, m, n) to produce an m-by-n random integer matrix ranging from 1 to N. Zheng-Liang Lu 183 / 225

13 You can write a program to generate a random variable which follows a standard uniform distribution, say by a linear congruential generator. 3 Be aware that there is no true random number generator in the machines. 4 Widely used in Monte Carlo simulation 5 and random number generation of other distributions 6. 3 See 4 For now, the modern computers are all deterministic. Quantum computers share theoretical similarities with non-deterministic and probabilistic computers. 5 See Glasserman (2003). 6 See the acceptance-rejection method and Metropolis-Hastings algorithm. Zheng-Liang Lu 184 / 225

14 User-Defined Functions The syntax of a user-defined function is given by 1 function [outputvar] = funcname(inputvar) 2 % comment section 3 end The output variables, if there exist, are enclosed in square brackets. The input variables, if there exist, must be enclosed with parentheses. funcname should start with a letter, and be the same as the file name in which it is saved. Before this function can be used, it must be saved into the current folder 7. 7 If not, change the current folder or add to the path pool. Zheng-Liang Lu 185 / 225

15 Functions without Input and Output 1 function [] = star( ) % [] and () can be dropped. 2 theta = pi / 2 : 0.8 * pi : 4.8 * pi; 3 r = ones(1, 6); 4 polar(theta, r) % plot in polar coordinate 5 end Zheng-Liang Lu 186 / 225

16 Example: Addition of Two Numbers 1 function z = myadd(x, y) 2 % input: x, y (two numbers) 3 % output: z (sum of x and y) 4 z = x + y; 5 end It seems bloody trivial. Actually, the plus sign is a kind of syntactic sugar. 8 8 Recall how to use an addition instruction in assembly codes. Zheng-Liang Lu 187 / 225

17 Example: Mean of A Sequence 1 function y = mymean(x) 2 % input: x (array) 3 % output: y (mean) 4 5 sum = 0; 6 for i = 1 : length(x) 7 sum = myadd(sum, x(i)); % call myadd 8 end 9 y = sum / length(x); 10 end Zheng-Liang Lu 188 / 225

18 Numbers of Arguments The variable nargin determines the number of input arguments in a function when executed. Similarly, the variable nargout determines the number of output arguments from a function when executed. The variable varargin is a special word with two roles: The variable varargin declares a function with any number of arguments. 9 The variable varargin itself is a cell array containing the optional arguments to the function. The variable varargout is a special word similar to varargin but for outputs. 9 Note that varargin must be declared as the last input argument and collects all the inputs from that point onwards. Zheng-Liang Lu 189 / 225

19 Example 1 function ret = myadd(varargin) 2 switch nargin 3 case 0 4 disp('no input.') 5 case 1 6 x = varargin{1}; 7 ret = x; 8 case 2 9 x = varargin{1}; 10 y = varargin{2}; 11 ret = x + y; 12 case 3 13 x = varargin{1}; 14 y = varargin{2}; 15 z = varargin{3}; 16 ret = x + y + z; 17 otherwise Zheng-Liang Lu 190 / 225

20 18 error('\ntoo many inputs.\n'); 19 end 20 end The special words nargin, nargout, varargin and varargout provide flexibility for user-defined functions. Note that the special word varargin and varargout should be the last item in the argument list. Zheng-Liang Lu 191 / 225

21 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 between other functions is through input arguments and the outputs it returns. Zheng-Liang Lu 192 / 225

22 Example 1 clear; clc; 2 3 x = 0; 4 for i = 1 : 10 5 addone(x); 6 disp(x); % output? 7 end 1 function addone(x) 2 x = x + 1; 3 end Zheng-Liang Lu 193 / 225

23 Global Variables Unlike local variables, global variables are available to all parts of a computer program. Use global to declare x as a global variable. Note that it is not a good practice to define many global variables. 10 For example, the universal constant, say,. 11 However, you have to take the responsibly of not changing its value. 10 This violets the modularity. 11 Planck constant = in Quantum theories. Zheng-Liang Lu 194 / 225

24 Example 1 function h = falling(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 = falling(0 : 0.1 : 5); What if you take global out of the function? As a more robust alternative, redefine the function to accept GRAVITY as an input. Zheng-Liang Lu 195 / 225

25 Types of User-Defined Functions Primary functions and subfunctions Anonymous functions Nested functions Private functions Zheng-Liang Lu 196 / 225

26 Primary Functions and Subfunctions A function m-file may contain more than one user-defined function. The first defined function in the file is called the primary function, whose name is the same as the m-file name. All other functions in the file are called subfunctions. Subfunctions are normally visible only to the primary function and other subfunctions in the same file. Note that the order of the subfunctions does not matter, but function names must be unique within the m-file. Zheng-Liang Lu 197 / 225

27 Example 1 function [a c] = circle(r) % primary function 2 a = area(r); 3 c = perimeter(r); 4 end 5 6 function y = area(x) % subfunction 1 7 y = pi * x.ˆ 2; 8 end 9 10 function v = perimeter(u) % subfunction 2 11 v = 2 * pi * u; 12 end Zheng-Liang Lu 198 / 225

28 1 >> [a c] = circle(2) 2 3 a = c = Note that if only one variable is assigned as output of the function, herein, circle, then the area is returned while the perimeter is ignored. Zheng-Liang Lu 199 / 225

29 Example: Alternative to Infinite Loops 12 One subfunction can be called by other subfunctions in addition to the primary function. 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 Notice that two subfunctions calling each other can lead to a loop, resulting a fatal crash. 12 Thanks to a lively class discussion (MATLAB-238) on June 14, Zheng-Liang Lu 200 / 225

30 Anonymous Functions 14 Anonymous functions, which are the expression followed by operator, enable you to create a simple function without needing to create an m-file for it. A function handle is a data type that stores an association to a function. 13 For example, 1 >> f x.ˆ 2 + x + 1 % x.ˆ2 allows... vectorization. 2 >> f([1 10]) 3 4 ans = See 14 See anonymous-functions.html. Zheng-Liang Lu 201 / 225

31 Furthermore, you can make the existing function become a function handle, for example, y You can create anonymous functions having more than one input. One anonymous function can call another to implement function composition. 1 % f is a function handle 2 >> f y) sqrt(x.ˆ 2 + y.ˆ 2); 3 % g is a composite function 4 >> g y) f(x, y).ˆ 2; 5 >> g(3, 4) 6 7 ans = Zheng-Liang Lu 202 / 225

32 More Examples 16 Give input values, return a function handle. 15 For example, 1 function y = f(a, b, c) 2 y a * x.ˆ 2 + b * x + c; 3 end Acts like a parabolic function generator! 15 Contribution by Ms. Queenie Chang (MAT25108) on March 18, Thanks to a lively class discussion (MATLAB-244) on August 22, Zheng-Liang Lu 203 / 225

33 Give a function handle as an input, return a value. 17 For example, 1 function y = diff(f, x) 2 eps = 1e-9; 3 y = (f(x + eps) - f(x)) / eps; 4 end Give a function handle as an input, return a function handle. For example, 1 function y = diff(f) 2 eps = 1e-9; 3 y (f(x + eps) - f(x)) / eps; 4 end 17 Thanks to a lively class discussion (MATLAB-260) on September 16, Zheng-Liang Lu 204 / 225

34 Nested Functions Functions are said to be nested if the functions are defined within the parent function. A nested function can access the variables of its parent function. 1 function f = parabola(a, b, c) 2 f %f is a function handle of p 3 function y = p(x) % p shares a, b, and c 4 y = a * x.ˆ2 + b * x + c; 5 end 6 end Zheng-Liang Lu 205 / 225

35 1 >> y = parabola(1, 2, 1); 2 >> y(5) 3 4 ans = Note that the nested functions are defined anywhere within the main function. (Why?) Zheng-Liang Lu 206 / 225

36 Private Functions Private functions are useful when you want to limit the scope of the function. A private functions resides in subfolder with the special name private. The private function is visible only to functions in the parent directory. Note that you cannot call the private function from the command line or from functions outside the parent of the private folder. Zheng-Liang Lu 207 / 225

37 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 208 / 225

38 Example: Bisection Method The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. (Why?) It is often used to obtain an approximate solution. Zheng-Liang Lu 209 / 225

39 Zheng-Liang Lu 210 / 225

40 Outline for Bisection Method 1. At each step, the algorithm divides the interval in two by computing the midpoint c = (a + b)/2 of the interval and the value of the function f (c) at that point. 2. Unless c is itself a root (which is very unlikely, but possible), there are now two possibilities: either f (a) and f (c) have opposite signs and bracket a root, or f (c) and f (b) have opposite signs and bracket a root. Zheng-Liang Lu 211 / 225

41 3. The method selects the subinterval that is a bracket as a new interval to be used in the next step. 4. In this way the interval that contains a zero of f is reduced in width by 1/2 at each step. 5. The process is continued until the interval is sufficiently small How small? A certain number you give is to be the stop criterion. Zheng-Liang Lu 212 / 225

42 Problem Formulation Input - Target function f (x) - Endpoints of the interval [a, b] for any real numbers a < b - Minimal interval length ɛ = b a Output - the approximate root ˆr Note that a and b will be updated iteratively. Zheng-Liang Lu 213 / 225

43 Solution 1 clear; clc; format long; 2 3 a = input('a =?\n'); 4 b = input('b =?\n'); 5 f x.ˆ 3 - x - 2; % target function 6 eps = 1e-9; 7 iter = 0; % the number of iterations 8 9 if f(a) == 0 10 r = a; % lucky a 11 elseif f(b) == 0 12 r = b; % lucky b 13 else 14 while b - a > eps 15 iter = iter + 1; 16 c = (a + b) / 2; % middle point 17 if f(c) == 0 Zheng-Liang Lu 214 / 225

44 18 r = c; % lucky c 19 break; 20 elseif (f(a) * f(c) < 0) 21 b = c; 22 elseif (f(b) * f(c) < 0) 23 a = c; 24 else 25 error('failure: f(a) * f(c) > 0 and... f(c) * f(b) > 0.'); 26 end 27 fprintf('%d: %f\n', iter, c); 28 end 29 r = c % approximate solution 30 end Zheng-Liang Lu 215 / 225

45 c = Zheng-Liang Lu 216 / 225

46 Remarks Time complexity: O(log 2 n) What is n? In this case, n = b a. (Why?) ɛ So, it is an algorithm which runs in log time. This means that you need to make a trade-off between the numerical precision and the computation time. Be aware that this algorithm works well only with the premise that the behavior in [a, b] is mild. Approximate solutions may be significantly influenced by the initial interval [a, b]. 19 f (c) 0 but not equal to exactly 0. (Why?) 19 You may try another algorithm for the root finding problem, say the Newton-Raphson method. Zheng-Liang Lu 217 / 225

47 Exercise: Square Roots Using the aforesaid bisection algorithm, write a function which determines n for all n R. It is easy to see that n = x is equivalent to x 2 n = 0. So n is the positive root of x 2 n = 0, which is the input of the bisection algorithm. Intuitively, a = 0 and b = n. Zheng-Liang Lu 218 / 225

48 Recursive Functions Recursion is when something is defined in terms of itself. A recursive function is a function that calls itself. Recursion is an alternative form of program control. It is essentially repetition without a loop. Zheng-Liang Lu 219 / 225

49 Example The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 4! = = 4 3! So we can say f (n) = n f (n 1), for n > 0. Also, we need a base case such that the recursion can stop instead of being an infinite loop. In this example, 0! = 1. Zheng-Liang Lu 220 / 225

50 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 We can use a loop to calculate n!. (Try.) Zheng-Liang Lu 221 / 225

51 Call Stack Zheng-Liang Lu 222 / 225

52 Equivalence 1 function y = loopfactorial(n) 2 y = 1; 3 for i = n : -1 : 1 4 y = y * i; 5 end 6 end One more intriguing question is, Can we always turn a recursive method into an iterative one? Yes, theoretically. 20 Which is better? 20 The Church-Turing thesis proves it if the memory serves. Zheng-Liang Lu 223 / 225

53 Example: Fibonacci Numbers 21 The Fibonacci sequence is a sequence with the following pattern: 0, 1, 1, 2, 3, 5, 8, 13, 21, Alternatively, the Fibonacci sequence is defined by the recurrence relation F n = F n 1 + F n 2, where F 0 = 0, F 1 = 1, and n 2. Given a positive integer n, determine F n. Input: n 0 Output: F n 21 See Fibonacci numbers. Zheng-Liang Lu 224 / 225

54 Solution 1 function y = recursivefib(n) 2 if n == 0 3 y = 0; 4 elseif n == 1 5 y = 1; 6 else 7 y = recursivefib(n - 1) + recursivefib(n ); 8 end 9 end It is a binary recursion, because there are two calls in the function. Time complexity: O(2 n ) (Why?) Can we implement a linear recursion for this problem? Zheng-Liang Lu 225 / 225