Introduction To MATLAB Introduction to Programming GENG 200

Transcription

1 Introduction To MATLAB Introduction to Programming GENG 200, Prepared by Ali Abu Odeh 1

2 Table of Contents M Files Execution Control 3 Vectors User Defined Functions Expected Chapter Duration: 6 classes. 6 Matrices 2-D Plotting 2

3 1. Introduction MATLAB, which stands for MATrix LABoratory, is a state-of-the-art mathematical software package, which is used extensively in both academia and industry. It is high- level technical computing language and interactive environment for algorithm development, data visualization, data analysis, and numeric computation. 3

4 Graphical User Interface MATLAB uses several display windows (see Figure 1.1). The default view includes command window, current directory, workspace, and command history windows. Command Window: This window offers an environment similar to scientific calculator, which will help you in performing fast experiments of your commands and finds out the results of fragments of your code. 4

5 Command History: This window will save a record of your commands you issued in the command window. Current Directory: MATLAB accesses files and saves information to a location on your hard drive given by the current directory. Workspace Window: This window will show the variables you have defined, and it will keep track of them. When you open MATLAB every time this window will be empty because you aren t defining any variables yet. 5

6 Current Directory Command Window Workspace Window Command History Window Figure 1.1 6

7 Help Menu The MATLAB help menu is very powerful. It contains detailed description of every command used in MATLAB along with illustrative examples to show the user the explanation of the command and how it can be used. To access the help menu, from the help tab select Product Help. Then you can search for any command to get detailed information. See Figure 1.2 which shows the details of the sin function. 7

8 Type the phrase you want to search for Figure 1.2 8

9 Simple variables and data assignments MATLAB presents very easy environment for defining simple variables. Remember that any simple variable (scalar) is considered as a 1- by-1 matrix. Now, if you want to define a variable in MATLAB just go directly to the command window where you will see the prompt (>>) and type the variable name and assign a numerical value and press Enter button. 9

10 When your command is executed, the MATLAB will respond by showing you the result of calculation. Note 1.1: If you are doing calculation without assigning the output to a variable (see example 1.1), the output will be assigned to a variable called ans created by MATLAB itself. Note 1.2: MATLAB is case sensitive. That is: x + y is not the same as X + y 10

11 Creating Integer Data MATLAB stores numeric data as double precision floating point (double) by default. To store data as an integer, you need to convert from double to the desired integer type. Use one of the following functions: >> x=int8(25.4) >> x=int16(25.4) >> x=int32(25.4) >> x=int64(25.4) Note: 8 or 16 or 32 or 64 bits. 11

12 Example 1.1: Calculate: 5*2*(6^4+2). >> 5*2*(6^4+2) ans = Example 1.2: Calculate: x*y+3; where x = 3, y = 4. >> x = 3 x = 3 >> y = 4 y = 4 >> x*y+3 ans = 15 12

13 After doing the previous examples, look at the workspace window. You will see the variables and results there as shown in figure 1.3 Figure

14 Note 1.3: The following commands are useful clc: clears the command window clear: removes the variables from the workspace» To delete a particular variable in freemat, type: clear variable_name» clc command will clear the temporary variable ans. Note 1.4:the symbol (%) is used for documentation and the constant π is defined in MATLAB as pi. 14

15 Example 1.3: Calculate the volume of a cylinder if height = 10 cm, radius = 5 cm. v = height*π*radius 2 >> h = 10 % height h = 10 >> r = 5 % radius r = 5 >> v = h*pi*r^2 % volume v =

16 2. Scripts (M-Files) MATLAB uses text files for saving scripts (set of instructions) and executing them rather than just entering the commands in the command window. It uses its own editor to create those text files with the extension.m, and referred to as m-files. You can create m- file by choosing: 16

17 File>New>M-File (or by clicking on the new icon on the far left of the MATLAB s toolbar). Once you created your m-file, you need to save it so that you can run and refer to it in future. However, in order to run the m-file it should be in the current directory of the MATLAB.» Example 2.1: redo example 1.3 but this time using m-file. 17

18 Figure

19 To execute the volume.m file, press F5 or from debug menu select run or from toolbar of the m-file press on the green play button (or you can write the name of the m-file on the command line and press Enter key, so in this example: >>volume). Note 2.1: Terminate any line with semicolon (;) to suppress the output 19

20 Example 2.2: Write m-file to calculate the roots of the quadratic equation. f(x) = ax 2 +bx+c where: a, b, and c are constants and a 0. Roots will be: What will be the answer if a = 2, b = 4, and c = 1? 20

21 clc clear a=2; b=4; c=1; x1=(-b-sqrt(b^2-4*a*c))/(2*a); x2=(-b+sqrt(b^2-4*a*c))/(2*a); Output: x1 = x2 =

22 3. Vectors Creating Vectors A vector is one-dimensional matrix (array) as shown in Figure 3.1 which contains data items such as: numbers. Individual items in a vector are usually referred to as elements. Vector elements have two properties: their numerical values and position (index) making them unique in a specific vector. 22

23 Element Index n-1 n There are many different ways to create a vector. The following shows how to create a vector: 23

24 3.1 Entering the values directly vector_name = [type vectors elements here] For example: a = [2, 4, 6, 8]. The commas are optional and can be omitted, that is you can write: a = [ ]. 24

25 Example 3.1: Entering values directly >> x = [ ] x = >> a = [4, 10, 12, 20] a =

26 3.2 Creating a vector with constant spacing by using colon operator: vector_name = [first element: spacing : Last element] For example: x = [1:2:10]. Note that the brackets are optional and the spacing can be omitted if the increment you need is 1. 26

27 Example 3.2: Using the colon operator >> m = [0:2:10] % vector of 1-by-6 m = >> w = 1:2:10 % vector of 1-by-5 w =

28 3.3 Using the linspace function to create a fixed number of values between two limits. vector_name = linspace(xi, xf, n). Where xi: initial limit, xf: final limit and n: number of elements in the vector. For example: x = linspace(0,10,6)» The space will be computed according to (xf-xi)/(n-1) 28

29 Example 3.3: Using linspace( ) function >> x = linspace(0,10,6) x = >> y = linspace(0,10,3) y = >> z = linspace(0,10,4) z =

30 Interesting Function! Check it out: --> x=linspace(2,8,3) x = > x=linspace(8,2,3) x = > x=linspace(-5,-1,3) x = > x=linspace(-1,-5,3) x =

31 3.4 Using built-in function: such as: ones(1,n),zeros(1,n), and rand(1,n). We will discuss them in the following examples. 31

32 Example 3.4: Creating vectors using zeros, ones and rand functions >> a=rand(1,4) a = >> b=ones(1,4) b = >> c=zeros(1,4) c =

33 >> d=6;e=3;h=4; %Three variables are defined >> x = [e d*h cos(pi/3) h^2 sqrt(h^2/d)] x =

34 Generating Random Numbers Sometimes there is a need to generate random numbers that are distributed in an interval other than (0, 1),or to have numbers that are only integers. Random numbers that distributed in a range (a, b) can be obtained by the following equation vector_name = (b-a)*rand(1,n)+a 34

35 Example 3.5: Generate a 1-by-5 vector random numbers between 0 and 10 >> a=0;b=10; >> x = (b-a)*rand(1,5)+a x = Example 3.6: Generate a 1-by-10 vector of integer random numbers from 1 to 100 >> a=1;b=100; >> x = round((b-a)*rand(1,10)+a) x =

36 Indexing and Accessing a Vector The elements of a vector can be accessed by enclosing the index of the required elements in parenthesis. If you attempt to read beyond the length of the vector or below index 1, an error will result. Example 3.7: >> A = [ ] A =

37 >> A(3) ans = 2 >> A(1) ans = 3 >> A(0)??? Attempted to access A(0); index must be a positive integer or logical. 37

38 >> A(7)??? Attempted to access A(7); index out of bounds because numel(a) = 6. Note that you can change the value of any element in a vector.» You can add to the vector but you can t assign to more than the index. A(7) = new_value (correct) A(7) (wrong) 38

39 Operations on Vectors Arithmetic Operations Since all variables in MATLAB are considered as arrays (scalar:1 1, vector: 1 n, array: n m), one should take care of how to perform: multiplication, division and exponentiation. On the other hand addition and subtraction have the syntax exactly as one would expect. 39

40 The first set of operations with vectors is element-by-element operations, not array operations, and hence, linear algebra rules do not apply in this case. Therefore, new set of symbols is required. For multiplication we will use (.*), for division we will use (./) and for exponentiation we will use (.^). Note 5:Using the dot to represent that the operation is element-by-element operation. 40

41 Note 3.1: vector by vector operation requires that both vectors should be of the same length. That is, they have the same number of elements. Example 3.8: >> A = [ ] A =

42 >> B = [ ] B = >> A + 5 ans = >> A*2 ans =

43 >> A*B??? Error using ==> mtimes Inner matrix dimensions must agree. >> A.*B ans = >> A^2??? Error using ==> mpower Matrix must be square. 43

44 Logical Operations We can perform logical operations using relational operators (we will discuss logical operators in the coming sections) listed in table 3.1. Those operations can produce numerical or logical results. Operator Description < Less than <= Less than or equal to > Greater than >= Greater than or equal to Table 3.1 == Equal to ~= Not equal to 44

45 Example 3.9: >> A = [ ] A = >> B = [ ] B = >> A >= 5 ans =

46 >> A(A >= 5) ans = Return elements that are greater than or equal to 5. >> A>=B ans = Return where each element of A that is not less than the corresponding elements of B 46

47 Note 3.2: An alternative to using indexing into the vector using the logical expression is to use the find function. For example: find(a>=5) will return where index A is greater than or equal to 5. Use mod(x,y) or rem(x,y) to find the modulus, e.g. mod(5,4) equal 1. 47

48 Example 3.10: >> A = [ ] A = >> sum(a) ans = 26 >> mean(a) ans =

49 To find also the mean, you can use these library functions: >> sum(a) / length(a) >> sum(a) / numel(a) 49

50 >> [w i] = max(a) w = 10 i = 7 >> [d z] = min(a) d = -1 z = 1 Note 3.3: w, i, d, and z can be replaced by any variable. 50

51 4. MATLAB Arrays (Matrices) In the previous section we saw that vector is a simple way to group a collection of similar data items. Let us now extend the idea to include arrays confined to two dimensional arrays. Typical two dimensional array A with m rows and n columns and A a transpose array of A with n rows and m columns. 51

52 >> A = [9 9 6; 7 1 2] A = >> A ans =

53 Creating an Array As with vectors, you can create arrays in MATLAB using many different ways. The following summaries most common techniques: - You can enter the values directly using semicolon to indicate the end of a row. For example: A = [2 4 6; 1 3 5], A is 2 x 3 array. - The functions zeros(m, n) and ones(m, n) create creates arrays with m rows and n columns filled with zeros and ones respectively. 53

54 - The function rand(m, n) create an array filled with random numbers in the range from 0 to 1. -The function diag(a) where A an array, returns its diagonal as a column vector. -The function magic(m), which creates a square array of size m x m filled with numbers from 1 to m 2 organized in such a way that its rows, columns, and diagonals all add up to the same value. Note: magic doesn't work with FreeMat. 54

55 Example 4.1: >> A = [2 4 6; ] A = >> A = [2 4; 6 8] A =

56 >> B=zeros(3,3) B = >> C = [ones(2,2) zeros(2,2)] C =

57 >> rand(3,4) ans = Example 4.2: >> A = [2 3 4; 7 8 9] A =

58 >> diag(a) ans = 2 8 To access a specific element in an 2D array, just indicate the row and column of the specified element, i.e., A(row,col). Also the colon operator can be used to access array elements. 58

59 The following notations illustrate some of the different cases: A(:,n) Refers to all elements in all rows of a column n of the array A. A(n,:) Refers to all elements in all columns of a row n of the array A. A(:,m:n) Refers to all elements in all rows between columns m and n of the array A. 59

60 Operations on Arrays Addition and Subtraction: Standard addition or subtraction can be done on arrays of identical size (the same number of rows and columns). Also can be done when we want to add (subtract) a scalar value to an array. When two arrays are involved, the sum or difference is performed by adding or subtracting their corresponding elements. 60

61 But when a scalar (number) is involved, that number is added (or subtracted from) all the elements of the array. Example 4.3: >> A = [2 4 1; 3 9 5] A =

62 >> B = [1 2 3; 4 5 6] B = >> A - B ans = >> A + B ans = 62

63 >> A + 1 ans = >> B -1 ans =

64 Multiplication The multiplication * of arrays is executed by MATLAB according to the rules of linear algebra (This is known as matrix multiplication). This means that if A*B is to be executed. Then the number of columns in matrix A should equal to the number of rows in matrix B. The result is a matrix that has the same number of rows as A and the same number of columns as B. 64

65 For example: if A4x3*B3x2 will result in a new matrix C4x2. This means that A*B B*A. C4x2 = A4x3*B3x2 65

66 Example 4.4: >> A=[1 2 3; 4 5 6;7 8 9] %3x3 A = >> B=[2 3 ;-4 7;5 5] %3x2 B =

67 >> A*B ans = >> B*A??? Error using ==> mtimes Inner matrix dimensions must agree. 67

68 Identity Matrix and Inverse of Matrix Identity Matrix is a square matrix in which the diagonal elements are 1 s and the rest of the elements are 0 s. When the identity matrix multiplies another matrix (multiplication should be done according to linear algebra), that matrix is unchanged. That is: A*I = I*A=A. The identity matrix can be generated in MATLAB using eye(n) which will create n x n identity matrix. 68

69 Inverse of a matrix is best explained using the following illustration. The matrix B is the inverse of matrix A if when the two matrices are multiplied product is the identity matrix. Both matrices must be square and the order can be A*B or B*A. The inverse of matrix A can be written as A -1. In MATLAB you can find the inverse using the two commands: inv(a) or raise to the power of -1, that is: A^-1. The following example illustrates both identity and inverse matrices. 69

70 For a 2 x 2 matrix A = a c b d The matrix inverse is A 1 = 1 A d c b a 1 ad bc d c b a 70

71 Example 4.5: >> A=[2 1 4;4 1 8;2-1 3] %Create the Matrix A A = >> B=inv(A) %use inv() to find the inverse of A B =

72 >> A*B %multiplication of A and B gives I ans = >> A^-1 % use power of -1 to find the inverse of A ans = >> B- A^-1 %what the answer for this command? 72

73 Right Division & Left Division Solve the following linear equations: x + 2y = 4 2x y =3 By solving x and y, you will find that x=2 and y=1. 73

74 Left Division >> a=[1 2;2-1] a = >> b=[4;3] b = 4 3 >> x=a\b x =

75 Right Division >> a=[1 2;2-1] a = >> b=[4 3] b = 4 3 >> x=b/a x =

76 Element-By-Element Operations These operations are carried out on each element of the array. Note that these operations are performed on arrays of the same size. Example 4.6: >> A=[1 2 3;4 5 6] % Define 2x3 array A A =

77 >> B=[5 7 4;6 2 9] % Define 2x3 array B B = >> A*B??? Error using ==> mtimes Inner matrix dimensions must agree. >> A.*B ans =

78 >> A./B ans = >> A.^2 ans =

79 Applying Library Functions Example 4.7: >> A=[1 5 70;pi 2*pi 3*pi] % Define 2x3 array A = >> sin(a) % Apply the sin function ans =

80 >> sqrt(a) % Apply the sqrt function ans = MATLAB has a large built-in Library functions. For example: sum(a): treats the columns of A as vectors, returning a row vector of the sums of each column. 80

81 std(a): returns a row vector containing the standard deviation of the elements of each column of A. det(a):return the determinant of a square array Example 4.8: >> A=[2 4 6;3 5 7;1 2 3] A =

82 >> sum(a) % Apply the sum function ans = >> std(a) % Apply the standard deviation ans = >> sort(a) % Sorting array in an ascending order ans =

83 >> max(a) % finding the maximum value for every column ans = >> mean(a) ans = >> max(mean(a)) ans =

84 5. Execution Control Relational and Logical Operators & AND Operates on two operands (A&B). If both results are true, the result is true. Otherwise the result is false. OR Operates on two operands (A B). If either one or both are true, the result is true, Otherwise, the result is false. 84

85 ~ NOT Operates on one operand (~A). The result is true if the operand is false and false if the operand is true 85

86 if statements if evaluates a logical expression and executes a block of statements based on whether the logical expressions evaluates to true (logical 1) or false (logical 0). The general structure is as follows. 86

87 if logical_expression 1 statements elseif logical_expression 2 statements.. elseif logical_expression n statements else default statements end» Note: elseif is one word without space. 87

88 The function disp(... ) is used to display a text. Actually, this function can also be used to display an array without printing its name, for example: disp(a), where A is an array. The function input( ) will ask the user to enter a number, i.e. n=input( Enter an Integer: ), then this message will appear on the command line waiting for the user to enter a number and then assigns the value to n. 88

89 » Important Note: If you use input command in your m file then, run the program by typing its name in the command window and not by pressing the run button.» Important Note : There will be no difference whether we surround the condition with parentheses or not in if statements. if (round(n/2) == n/2) if round(n/2) == n/2 89

90 Example 5.1: Write an m file to find if the number is Even or Odd. clear;clc; n = input('enter an integer number: '); if round(n/2) == n/2 disp('even') % disp('...') used to display text else disp('odd') end 90

91 while Loop while logical_expression statements end Example 5.2: Write m file to Generate sequence of even numbers clc;clear; x=2 while x>=2 & x<=10 x=x+2 end 91

92 for Loop for i = initial_value : step_size : final_value statements end Example 5.3: Write an m file to Generate sequence of numbers. clc;clear; for i=1:3:10 x=i^2 end 92

93 function fprintf( ) is similar to the function printf that used in c language and it will do the same job. The general format of this function is: fprintf(format, variable) Example 5.4: clc;clear; for i=1:3:10 x=i^2; fprintf('x = %d\n',x) % Try to use disp(x) end % or use disp('x=',x) 93

94 Example 5.5: a vector is given by A = [-5,-17,- 3,8,0,-1,12,-4,-6,6,-7,17]. Write an m-file that doubles the negative odd elements. clc;clear A = [-5,-17,-3,8,0,-1,12,-4,-6,6,-7,17]; disp('a Before = ') disp(a) for i=1:length(a) if (A(i)<0) & (A(i)/2 ~= round(a(i)/2)) A(i) = A(i)*2; end end disp('a After = ') disp(a) 94

95 6. User Defined Functions User defined functions must be stored in a separate m-file and in your directory of work. And the m-file that contains your function should be saved with same name as the function. The general definition of a function is as follows: function [output_vars] = function_name (input_vars) function code body 95

96 Note that if you have one output variable, then omit the brackets [ ]. Those only used if you have more than output variable. Example 6.1: Write a MATLAB function file for y= x 2 +2x+3. Then calculate f(0) and f(3). Solution: create a new m-file, save it as f.m and then write your code. Figure below shows the required function. 96

97 The name of the function and the name of the file must be the same. 97

98 You can call the function in the command line using its name as follows: >> f(0) ans = 3 >> f(6) ans = 51 Example 6.2: Write MATLAB function to calculate the area and circumference of a circle for given radius r. 98

99 function [area,cercom]=circle(radius) % Function to compute area and % circumference of a circle area = pi*radius^2; cercom = 2*pi*radius; >> [a,c]=circle(4) a = c =

100 >> circle(4) ans = >> [area,cercomference]=circle(4) area = cercomference =

101 7. 2-D Plots plot(xvalues, yvalues, style-option ) Where xvalues and yvalues are vectors of the same length containing x and y coordinates of points on the graph. style-option is an optional arguments that specifies the color, the line style (e.g., solid, dashed, dotted,etc.) and the point-marker style (e.g., o, +, *, etc.). 101

102 Style Option The style-option in the plot command is a character string that consists of one, two or three characters that specify the color and/or line style as shown below. Color style-option Line style-option Marker style-option y yellow - solid + plus sign m magenta -- dashed o circle c cyan : dotted * asterisk r red -. dash-dot x x-mark g green. point b blue ^ up triangle w white s square k black d diamond 102

103 Examples plot(x,y) plots y vs. x with a blue solid line (default). plot(x,y,'r') plots y vs. x with a red solid line. plot(x,y,':') plots y vs. x with a dotted line. plot(x,y,'b--') plot y vs. x with a blue dashed line. plot(x,y,'+') plot y vs. x as unconnected points. marked by

104 Example 7.1: Plot the function y = x cos(6x), where -2 x 4 using different line styles, the step size is 0.01 clc;clear x=-2:0.01:4; y=3.5.^(-0.5*x).*cos(6*x); figure(1) plot(x,y) figure(2) plot(x,y,'r') 104

105 figure(3) plot(x,y,':') figure(4) plot(x,y,'b--') figure(5) plot(x,y,'+') 105

106 106

107 107

108 Example 7.2: clc;clear x= -pi:1/pi:pi; y= cos(x); z= sin(x); plot(x,y,'g--',x,z,'r-.') % Add grid lines grid on 108

109 Formatting a Plot A figure that contains a plot needs to be formatted to have a specific look and to display information in addition to the graph itself. Consider the following commands: xlabel( x-axis label ) ylabel( y-axis label ) title( title of the figure ) axis([xmin xmax ymin ymax]) 109

110 Example 7.3: clc;clear t = 0:0.001:1; f = 15; y = cos(2*pi*f*t); plot(t,y,'--') xlabel('time (sec)') ylabel('cos(2\pift)') title('t vs. cos(2\pift)') grid on 110

111 Subplots The command subplot( ) can be used to set and design the required layout. subplot(rows,columns,index) This function divides the graphics window into rows x columns sub-windows and puts a certain plot in a position specified by the index. For example the two commands subplot(2,2,3), plot(x,y) divides the graphics window into 4 sub-windows and places the plot of y vs. x in the third sub-window. 111

112 clc;clear t = -0.5:0.001:0.5; f = 15; x = cos(2*pi*f*t); y = sin(2*pi*f*t); z = sinc(2*pi*f*t); w = 0.5*t-0.5; subplot(2,2,1) plot(t,x) grid on title('cosine vs. time') 112

113 subplot(2,2,2) plot(t,y) grid on title('sine vs. time') subplot(2,2,3) plot(t,z) grid on title('sinc vs. time') subplot(2,2,4) plot(t,w) grid on title('w vs. time') 113

114 114

115 References: Slides, ERU team of instructors. 115