Arrays. ˆ An array, is a linear data structure consisting of a collection of elements, each identified by one array index. ˆ For math, arrays could be

Transcription

1 Arrays ˆ An array, is a linear data structure consisting of a collection of elements, each identified by one array index. ˆ For math, arrays could be ˆ row vectors: u R 1 n for any positive integer n ˆ column vectors: u R n 1 for any positive integer n ˆ matrices: A R n m for any positive integers n, m ˆ Arrays can be said the simplest one of data structures. Zheng-Liang Lu 55 / 124

2 Example 1 >> rowvector = [1 2 3] % [... ] is used in arrays 2 3 rowvector = >> colvector = [1; 2; 3] % semicolon used to... change rows colvector = Zheng-Liang Lu 56 / 124

3 17 18 >> A = [1 2 3; 4 5 6; 7 8 9] A = ˆ You can create an n m k matrix for n, m, k,... N. ˆ For example, a matrix takes MBytes in memory. 1 ˆ The number of dimensions of arrays is unlimited without regarding to the limit of the memory space. 1 More explicit, / 2 20 = Zheng-Liang Lu 57 / 124

4 Alternatives for Vector Creation ˆ The function linspace(start, end, numpoints) generates a vector of uniformly incremented values. 2 1 >> x = linspace(0, 10, 5) 2 3 x = Also try logspace. Zheng-Liang Lu 58 / 124

5 ˆ You can also use the colon notation, which is to create a vector. start : stepsize : end 1 >> x = 0 : 2 : x = Zheng-Liang Lu 59 / 124

6 ˆ This is widely used to create index arrays to manipulate data arrays. 1 >> x = 1 : 10 % default step size = x = ˆ You cannot create a vector by a contradiction. 1 >> x = 1 : -1 : x = 4 5 Empty matrix: 1-by-0 Zheng-Liang Lu 60 / 124

7 ˆ Some special matrices can be initialized by the following functions: Zheng-Liang Lu 61 / 124

8 Array Addressing ˆ Subscripted indexing ˆ Linear indexing: column major order. 1 >> A(2, 3) % using subscripted indexing 2 3 ans = >> A(8) % using linear indexing 8 9 ans = Zheng-Liang Lu 62 / 124

9 1 >> A(2, [1 3]) % selecting 2 3 ans = >> A(:, 1) = [] % deleting the first column vector A = >> A = [[1; 4; 7], A] % combining two matrices ˆ In Line 7, we can use the colon operator (:) to select all elements in the 1st column. Zheng-Liang Lu 63 / 124

10 Cells and Cell Arrays 3 ˆ An array cannot contain data of different types correctly! ˆ A cell array is a data type with indexed data containers called cells, and each cell can contain any type of data. ˆ Cell arrays commonly contain either lists of text strings, combinations of text and numbers, or numeric arrays of different sizes. 3 Basically, cell arrays do not require completely contiguous memory because Matlab does not allocate any memory for the contents of each cell when the cell array is created. However, each cell requires contiguous memory, as does the cell array header that Matlab creates to describe the array. Zheng-Liang Lu 64 / 124

11 Example 4 1 >> A{1} = 'first cell'; 2 >> A{2} = [1 2 3; 4 5 6; 7 8 9]; 3 >> A{3} = {'Tim'; 'Chris'}; 4 >> A 5 6 A = 7 8 'first cell' [3x3 double] {2x1 cell} ˆ However, the arithmetic operators cannot be applied to the cell arrays! 4 Thanks to a lively class discussion (Matlab263) on March 3, Zheng-Liang Lu 65 / 124

12 Referring to Cell Arrays ˆ Using curly braces, { } will reference the contents of a cell. 1 >> A{3}{1} 2 3 ans = 4 5 Tim ˆ More details can be found in here. Zheng-Liang Lu 66 / 124

13 Structures and Structure Arrays ˆ A structure groups related data by fields, which can contain any type of data. ˆ You can access data of the structure by the period operator. ˆ For example, 1 >> Student.university = 'NTU'; 2 >> Student.department = 'csie'; 3 >> Student.grades = [80, 90, 100]; ˆ More details can be found here. ˆ Widely used in object-oriented applications, such as GUI and databases. ˆ The arithmetic operators cannot be applied to structures, either! Zheng-Liang Lu 67 / 124

14 Symbols for Common Data Structures Zheng-Liang Lu 68 / 124

15 Summary 5,6 ˆ Parentheses ( ) ˆ Arithmetic, e.g. (x + y)/z. ˆ Input arguments of functions, e.g. sin(1), exp(1). ˆ Array addressing, e.g. A(1) refers to the first element in array A. ˆ Square brackets [ ]: only used in array operations ˆ e.g. x = [ ]. ˆ Curly brackets { }: only used in cells ˆ e.g. A = { This is Matlab class., x}. 5 Thanks to a lively class discussion (Matlab237) on April 16, You can refer to this page for more details of special characters. Zheng-Liang Lu 69 / 124

16 More Data Structures ˆ See data-types_data-types.html. 7 7 Some of them may not be available if your version is out-of-date. Zheng-Liang Lu 70 / 124

17 Vectorization 8 ˆ Matlab favors array operations. ˆ When two arrays have the same dimensions, addition, subtraction, multiplication, and division apply on an element-by-element basis. ˆ For example, 1 >> x = [1 2 3]; 2 >> y = [4 5 6]; 3 >> x + y 4 5 ans = More about vectorization. Zheng-Liang Lu 71 / 124

18 Element-By-Element Operations ˆ We will see the difference between left and right divisions in matrix computation. Zheng-Liang Lu 72 / 124

19 1 >> Lecture 2 2 >> 3 >> -- Programming Basics 4 >> Zheng-Liang Lu 73 / 124

20 If debugging is the process of removing software bugs, then programming must be the process of putting them in. Edsger W. Dijkstra ( ) Zheng-Liang Lu 74 / 124

21 Introduction ˆ Decision making facilitates the usefulness of programs. ˆ For example, Self Driving Cars by Google. 9 ˆ Also, some actions (functions) are repeated for a specified number of times or until the stopping condition is satisfied. ˆ An algorithm must have the ability to alter the order of its instructions using what is called a control structure. ˆ The first principle in designing algorithms is to divide and conquer. ˆ Recall the algorithm for max(a). ˆ This feature enables engineers to solve problems of great complexity or requiring numerous calculations. 9 See Zheng-Liang Lu 75 / 124

22 Building Blocks ˆ Sequential operations: be executed in order. ˆ Selections: check which condition is satisfied and then execute the actions accordingly. ˆ Repetitions: repeat some instructions and stop while the termination condition is satisfied. Zheng-Liang Lu 76 / 124

23 Pseudo Code ˆ The pseudo code is widely used to describe algorithms. ˆ They are a mixture of natural language and mathematical expressions with common structures of programming languages. ˆ For example, 1 if one student's grade is greater than 60 2 print 'pass' 3 else 4 print 'fail' 5 end Zheng-Liang Lu 77 / 124

24 Relational Operators 10 ˆ Matlab provides 6 relational operators to make comparisons between two arrays of equal size. 10 See Table 8.1 in Moore, p Zheng-Liang Lu 78 / 124

25 Boolean (Logical) Values ˆ Boolean values are only true and false. 11 ˆ The function true and false represent logical true and logical false, respectively. 12 ˆ Boolean variable contains a boolean value. ˆ For example, 1 >> x = 1; y = 2; 2 >> w = (y > x) ~= true 3 4 w = Note that the numeric number 0 can be regarded as false in general, and nonzero numbers are regarded as true in Matlab and some programming languages. 12 The usage of true and false is similar to zeros. Zheng-Liang Lu 79 / 124

26 Mask: A Filter ˆ Boolean arrays are often used as masks to manipulate arrays as an alternative to index arrays 1 >> scores = 50 : 10 : 70; 2 >> haspassed = scores >= haspassed = >> scores(haspassed) 9 10 ans = Zheng-Liang Lu 80 / 124

27 Logical Operators ˆ Assume x = 0; ˆ How about 1 < x < 3? (Surprising!) Zheng-Liang Lu 81 / 124

28 Truth Table ˆ Let A and B be two Boolean variables. ˆ Then the truth table for logical operators is as follows: A B A A&B A B xor(a, B) T T F T T F T F F F T T F T T F T T F F T F F F ˆ Note that the instructions of computers, such as arithmetic operations, are implemented by logic gates See any textbook for digital circuit design. Zheng-Liang Lu 82 / 124

29 & vs. == 14 1 >> x = [ ]; 2 >> y = [ ]; 3 >> x == y 4 5 ans = >> x & y 11 ans = Thanks to a lively class discussion (Matlab-237) on April 16, Zheng-Liang Lu 83 / 124

30 Quantifiers 15 ˆ The function all determines if all elements are true. 1 >> x = [ ]; 2 >> all(x >= 60) 3 ans = ˆ The function any determines if there is any true element in the array. 1 >> any(x >= 60) 2 ans = See Zheng-Liang Lu 84 / 124

31 Precedence of Operators See Table 1.2 Operator Precedence Rules in Attaway, p. 25. Zheng-Liang Lu 85 / 124

32 More Logical Functions ˆ NaN: Not A Number, caused by and See NaN. Zheng-Liang Lu 86 / 124

33 Digress: Set Membership 19 ˆ Matlab provides useful functions such as intersect, union, unique, and setdiff. 18 ˆ For example, 1 >> x = [ ]; y = [ ]; 2 >> intersect(x, y) 3 4 ans = See 19 Thanks to a lively class discussion (Matlab263) on March 9, Zheng-Liang Lu 87 / 124

34 Logic is the anatomy of thought. John Locke ( ) This sentence is false. I know that I know nothing. anonymous Plato (In Apology, Plato relates that Socrates accounts for his seeming wiser than any other person because he does not imagine that he knows what he does not know.) Zheng-Liang Lu 88 / 124

35 Selections ˆ Selection enables us to write programs that make decisions on. ˆ Selection structures contain one or more of the if, else, and elseif statements. ˆ The end statement denotes the end of selection structures. Zheng-Liang Lu 89 / 124

36 Example: if-else ˆ If x is a nonnegative real number, then y = x. ˆ The function input(prompt) shows the prompt (in a string) and then waits for a numeric input from the keyboard. 1 clear; clc; 2 3 x = input('enter a number? '); % input from keyboard 4 if x >= 0 5 sqrt(x) 6 else 7 disp([num2str(x),' is a negative number.']); 8 end ˆ Note that one if should be paired with one end. ˆ input(prompt, s ) waits for a string input. Zheng-Liang Lu 90 / 124

37 Exercise: Nested Conditional Statements 1 clear; clc; 2 x = input('enter x?', 's'); 3 % Contribution by Mr. Curtis Yen on Sep. 10, w = str2num(x); 5 if isempty(w) % w is not empty if w is a number 6 disp([x, ' is not a number.']); 7 else 8 if w >= 0 9 sqrt(w) 10 else 11 disp([x ' is not a positive real number.']); 12 end 13 end ˆ Do not write a deeply nested conditional statement. Zheng-Liang Lu 91 / 124

38 Example: if-elseif-else 1 clear; clc; 2 x = input('enter x?', 's'); 3 w = str2num(x); 4 if isempty(w) % w is not empty if w is a number 5 disp([x ' is not a number.']); 6 elseif w >= 0 7 sqrt(w) 8 else 9 disp([x ' is not a positive real number.']); 10 end ˆ Can we change the order of the conditions? ˆ You can put w >= 0 in the first condition. 20 ˆ The empty set can be evaluated as false by the if statement. 20 Thanks to a lively class discussion (Matlab-260) on August 29, Zheng-Liang Lu 92 / 124

39 Example: GPA Conversion Problem 21 ˆ Write a program which converts centesimal points to GPA. ˆ Let x be the grade input. ˆ For simplicity, ˆ If 90 x 100, then x is converted to 4. ˆ If 80 x < 90, then 3. ˆ If 70 x < 80, then 2. ˆ If 60 x < 70, then 1. ˆ If x < 60, then See GPA. Zheng-Liang Lu 93 / 124

40 1 clear; clc; 2 x = input('enter points? '); 3 if x >= 90 && x <= disp('4'); 5 elseif x >= 80 && x < 90 6 disp('3'); 7 elseif x >= 70 && x < 80 8 disp('2'); 9 elseif x >= 60 && x < disp('1'); 11 else 12 disp('0'); 13 end ˆ Can we reverse the conditions? ˆ Can we place the conditions at your will? Zheng-Liang Lu 94 / 124

41 & vs. && 1 clear; clc; 2 3 x = [ ]; 4 y = [ ]; 5 6 x > 0 & y > 0 % return [ ] 7 all(x > 0) && any(y > 0) % return 0 ˆ Consider A && B. ˆ If A returns false, then B won t be evaluated. ˆ This is so-called the short-circuit operator, which facilitates time-saving. ˆ Note that the argument for if statements can be a scalar only! Zheng-Liang Lu 95 / 124

42 Another Selection Structure: switch-case ˆ The switch-case statements allow you to choose between multiple outcomes by checking the value of the target variable. ˆ For example, 1 clear; clc; 2 city = input('enter the name of a city: ', 's'); 3 switch city 4 case {'Taipei', 'New Taipei'} 5 disp('price: $100'); 6 case 'Taichung' 7 disp('price: $200'); 8 case 'Tainan' 9 disp('price: $300'); 10 otherwise 11 disp('not an option.'); % default option 12 end Zheng-Liang Lu 96 / 124

43 Why Not if? if strcmp(city, 'Taipei') strcmp(city, 'New... Taipei') 3 disp('price: $100'); 4 elseif strcmp(city, 'Taichung') 5 disp('price: $200'); 6 elseif strcmp(city, 'Tainan') 7 disp('price: $300'); 8 else 9 disp('not an option.'); 10 end ˆ Can you implement the strcmp function? 22 Thanks to a lively class discussion (Matlab-244) on August 20, Zheng-Liang Lu 97 / 124

44 Exercise ˆ We can try to replicate the functionality of strcmp by not calling it. ˆ You may use the function length to calculate the size of the string (or any arrays later) istaipei = length(city) == length('taipei') &&... 3 all(city == 'Taipei'); 4 isnewtaipei =... % Do the same thing below. 5 if istaipei isnewtaipei 6... % Show the ticket price. 7 elseif % Do the same thing below. 9 end Zheng-Liang Lu 98 / 124

45 Error and Error Handling ˆ You can issue an error if you do not allow the callee for some situations >> error('this is an error.'); ˆ Also, you can use a try-catch statement to handle errors. 1 try 2 % normal operations 3 catch 4 % handler operations 5 end 23 In other words, you can decide if the action can be performed. Zheng-Liang Lu 99 / 124

46 Example: Combinations ˆ For all nonnegative integers n k, ( n k) is given by ( ) n n! = k k!(n k)!. ˆ Note that factorial(n) returns n!. 1 clear; clc; 2 3 n = input('n =? '); 4 k = input('k =? '); 5 y = factorial(n) / (factorial(k) * factorial(n - k)) 6 disp('end of program.'); Zheng-Liang Lu 100 / 124

47 ˆ Try n = 2, k = 5. ˆ factorial( 3) is not allowed! ˆ So your program stops in Line 5, and terminates abnormally. ˆ The reason is that Line 5 produces an error and the program is not designed to handle it. ˆ Then Matlab takes over and kills this program try 3 y = factorial(n) / (factorial(k) *... factorial(n - k)) 4 catch e % capture the thrown exception 5 disp(['error: ' e.message]); 6 end 7 disp('end of program.'); Zheng-Liang Lu 101 / 124

48 Exercise: Divided by Zero ˆ Write a program which calculates x/y for any two real numbers x, y. ˆ The program restricts the user to do division by zero. ˆ Note that we don t use this error-handling mechanism for normal executions in regard to the performance! 1... % assume x = 1 any y = 0 2 if y == 0 3 error('divided by zero!'); 4 else 5 ans = x / y; 6 end 7... Zheng-Liang Lu 102 / 124

49 Repetitions ˆ If a group of instructions is potentially repeated, you should wrap those in a repetition structure, called loops. ˆ All loops consist of 3 parts: ˆ Find the repeated pattern for each iteration. ˆ Warp them by a loop. ˆ Set the continuation condition by defining a loop variable with some criterion. ˆ Matlab provides two different types: the for loops and the while loops. ˆ Use for loops if you know the number of iterations. ˆ Otherwise, use while loops. Zheng-Liang Lu 103 / 124

50 for Loops ˆ A for loop is the easiest choice when you know how many times you need to repeat the loop. 1 for loopvar = somearray 2 % body 3 end ˆ Particularly, we often use for loops to manipulate arrays! Zheng-Liang Lu 104 / 124

51 Zheng-Liang Lu 105 / 124

52 Example 1 for i = 1 : 10 2 disp(i); 3 end ˆ Can you show the odd integers from 1 to 9? 1 for student = {'Arthur', 'Alice'} 2 disp(cell2mat(student)); 3 % remember to take things out of cells 4 end ˆ Clearly, Matlab has for-each loops, which is an enhanced one compared to the naive one in C. ˆ Note that we use the function cell2mat so that a string stored in the cell can be extracted as a string. Zheng-Liang Lu 106 / 124

53 Example: Find Maximum (Revisited) 1 clear; clc; 2 3 x = [ ]; % input list 4 max = x(1); 5 for i = 2 : 7 6 if max < x(i) 7 max = x(i); 8 end 9 end 10 max ˆ Can you find the location of the maximum element? ˆ Try to find the minimum element and its location. Zheng-Liang Lu 107 / 124

54 Example: Running Sum ˆ Write a program which calculates the sum of array elements. 1 clear; clc; 2 3 x = 1 : 1 : 100; 4 s = 0; 5 for i = 1 : length(x) 6 s = s + x(i); % running sum 7 end 8 s ˆ Use length(x) instead of 100 in Line 5. ˆ Then the loop will do as many times as the number of elements in x. Zheng-Liang Lu 108 / 124

55 Exercise: Estimating π by Monte Carlo Simulation 24 ˆ Write a program which estimates π by generating a certain number of sample points (x, y) and calculating ˆπ = 4 m n, where m is the number of points falling in the region of the quarter circle shown in the next page, n is the total number of points, and ˆπ is the estimate of π. 24 Monte Carlo was a code word used in World War II to describe sampling-based methods for neutron scattering computations associated with Manhattan project. Zheng-Liang Lu 109 / 124

56 Zheng-Liang Lu 110 / 124

57 1 clear; clc; 2 3 n = 1e5; 4 m = 0; 5 for i = 1 : n 6 x = rand(1); 7 y = rand(1); 8 if x ˆ 2 + y ˆ 2 < 1 9 m = m + 1; 10 end 11 end 12 estimated pi = 4 * m / n ˆ ˆπ π as n by the law of large numbers (LLN). 25 ˆ Try to vectorize this program. 25 See Zheng-Liang Lu 111 / 124

58 More Exercises ˆ Write a program to estimate 5 by Monte Carlo. ˆ Write a program to estimate 1 0 xdx by Monte Carlo. ˆ Write a program to estimate the call price of European option by Monte Carlo. Zheng-Liang Lu 112 / 124

59 while Loops ˆ The while loops are preferred when you need to keep repeating the instructions until a continuation criterion is not met. 1 while criterion 2 % body 3 end ˆ Note that before entering the while loop, the criterion will be checked. ˆ Also note that the if statement is similar to the while loop except that the if statement executes only once. Zheng-Liang Lu 113 / 124

60 Zheng-Liang Lu 114 / 124

61 Example: Compounding ˆ Let x be the initial amount of some investment, and r be the annual interest rate. ˆ Write a program which calculates the holding years n so that this investment doubles it value. Zheng-Liang Lu 115 / 124

62 Solution ˆ In this case, we don t know how many iterations we need before the loop. 1 clear; clc; 2 3 curr = 100; 4 goal = 200; 5 r = 0.01; 6 n = 0; 7 8 while curr <= goal 9 curr = curr * (1 + r); 10 n = n + 1; 11 end 12 n ˆ Note that the criterion is to continue the loop. Zheng-Liang Lu 116 / 124

63 Infinite Loops 1 while true 2 disp('press ctrl+c to stop me!!!'); 3 end ˆ Note that your program can terminate the program by pressing ctrl+c. Zheng-Liang Lu 117 / 124

64 More Exercises ˆ Let a > b be two any positive integers. ˆ Write a program which calculates the remainder of a divided by b. ˆ Do not use mod(a, b). ˆ Write a program which determines the greatest common divisor (GCD) of a and b. ˆ Do not use gcd(a, b). Zheng-Liang Lu 118 / 124

65 Numerical Example: Bisection Method for Root-finding Zheng-Liang Lu 119 / 124

66 Problem Formulation Input - Consider f (x) = x 3 x 2 - Initial search interval [a, b] for any real numbers a < b - Error tolerance ɛ = 1e 9 Output - ˆr which approximates the exact solution r ˆ Note that a and b will be updated iteratively. Zheng-Liang Lu 120 / 124

67 Solution 1 clear; clc; 2 3 a = 0; b = 2; eps = 1e-9; 4 iter = 0; % the number of iterations 5 6 while b - a > eps 7 8 c = (a + b) / 2; 9 fa = a * a * a - a - 2; 10 fc = c * c * c - c - 2; if fa * fc < 0 13 b = c; 14 else 15 a = c; 16 end 17 Zheng-Liang Lu 121 / 124

68 18 iter = iter + 1; 19 end 20 r = c ˆ You may try another algorithm for the root finding problem, say the Newton-Raphson method. Zheng-Liang Lu 122 / 124

69 c = Zheng-Liang Lu 123 / 124

70 Exercise: Square Roots 26 ˆ Using the aforesaid bisection algorithm, write a function which determines n for all n R +. ˆ It is easy to see that n = x is equivalent to x 2 n = 0. ˆ So n is the positive root of x 2 n = 0, which is the input of the bisection algorithm. ˆ Trivially, a = 0. ˆ b = n if n > 1; b = 1, otherwise. 26 Also see Best-Square-Root-Method-Algorithm-Function-Precisi. Zheng-Liang Lu 124 / 124