1. Become familiar with additional MATLAB functions and looping/conditional statements.

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1 MATLAB m-files and Flow Control Objectives 1. Become familiar with additional MATLAB functions and looping/conditional statements. 2. Learn how to create and use MATLAB m files. 3. Learn how to write and use MATLAB functions. The best way to use MATLAB is to use its programming facility. With sequences of MATLAB commands contained in files, it is easy to see what calculations were done to produce a certain result, and it is easy to show that the correct values were used in producing a graph. When the commands are in clear text files, with easily read, well-commented code, you have a very good idea of how a particular result was obtained. This way you will be able to reproduce it and similar calculations as often as you please. As the number of commands increases or when you wish to change the value of one or more variables and reevaluate a number of commands, typing at the MATLAB prompt quickly becomes tedious. MATLAB provides a logical solution to this problem it allows you to place MATLAB commands in a simple text file and then tell MATLAB to open the file and evaluate the commands exactly as it would if you had typed them at the MATLAB prompt. There are three kinds of files that MATLAB can use: 1. Script files (or m-files) 2. Function files (or m-files) 3. Data files. Script m-files We will begin with a script m-file, which is simply a text file that contains a list of valid MATLAB commands. To create an m-file, click on File at the upper left corner of your MATLAB window, then select New, followed by Script. A window will appear in the upper left corner of your screen with MATLAB's default editor. You are free to use an editor of your own choice, but for the brief demonstration here, let's stick with MATLAB's. It's easy to use and has some built-in color formatting that help identify certain code items. (See Figure 1.) As an alternative you can type editor at MATLAB prompt; >> editor Let s assume you are in editor. You should see a larger area than appears in Figure 2. Notice that there is a 1. Indicating where you should type the first line of code.

2 In this window, type the following lines (MATLAB considers everything following after a % sign as a comment to be and not to be executed): %NA_Example1 %Find the reduced row echelon form of a matrix A = fix(10*rand(3,4)) %randomly generate a 3 by 4 matrix of %integers rref(a) %compute the reduced row echelon form %If you consider A an augmented matrix, how many free variables %are there in the corresponding linear system? Saving and executing the script To save your script file you have several options. Keep in mind you must direct MATLAB to a folder and that folder should be in the MATLAB search path. You will need to choose a name for the script file. Keep it simple and meaningful. For example here use NA_Example1. The first character must be a letter; other letters and numbers can be used together with an underline. The name is case sensitive. 1. In Figure 3 if you click on the save icon MATLAB stores the script file in the Current Folder. See Figure 4 to see how to set the Current Folder.

3 2. In the editor you can click on File and chose to Save or Save as directing MATLAB to folder of your choice. However, the folder you must be in the MATLAB path in order for you to execute the m-file you just saved. Once you have saved your script in a folder in the search path to execute it in the Command Window at a MATLAB command prompt just enter the script s name. (Do not include the.m) Here that would be NA_Example1. To edit the script at a prompt in the Command Window type edit the name of the script. If you type help NA_Example1 in the Command Window, MATLAB will display the content of the first several continuous lines of comments. This will help you document your m-file. You can put all the necessary comments, instructions and explanations for the file in these lines (this will serve as your help file for the procedure). The name of your m-file should not be the name of a MATLAB command that is predefined in MATLAB or associated toolboxes. Function m-files The second type of m-file is called a function m-file and typically these will involve some variable or variables sent to the m-file to be processed. There may be output displayed by the function m-file, but often values are returned to as calling program further processing. We now define the function f(x) as a function m-file. As we did for script files click on File at the upper left corner of your MATLAB window, then select New, this time select Function. A window will appear in the upper left corner of your screen with MATLAB's default editor. You are free to use an editor of your own choice, but for the brief demonstration here, let's stick with MATLAB's. It's easy to use and has some built-in color formatting that help identify certain code items. (See Figure 1.) Figure 5 shows the template that appears in the editor. Every function m-file begins with the command function. The next expression is the value(s) the function returns (output args), while the m-file name on the right-hand side of the equality(untitled2) is the name of the function m-file (with.m omitted). Finally, the input variable appears in parentheses (Input args). The comment lines are to be filled in for the specifics of the m-file you are constructing. The function m-file terminates with an.

4 Figure 5. As an example, let's suppose we want to generate a set of ordered pairs for a function at various sets of domain values. Suppose the function is f(x) = e -0.1x cos(x). We will write a function m-file that will accept the domain values as input and generate a table of x and f(x) values. function [x,y] = NA_Example2( start,increment,value ) %NA_Example2 The objective is to generate ordered pairs for the %function f(x) = e-0.1x cos(x)where the x-values begin at start, %step by the value increment, and stop at value. %OUTPUT a table of x vs. f(x) x=start:increment:value; %defining the x-values k=length(x); %counting the x-values for jj=1:k y(jj)=exp(-0.1*x(jj))*cos(x(jj)); % of for loop [x' y'] %displays the table on the command screen On the Command Screen we type the following command (don t type >>) and the table is displayed. >> NA_Example2(0,0.25,2); Note the semicolon; this suppresses the display of the output [x y]. Without the semicolon an additional will be displayed containing the value of x used. ans =

5 Data files MATLAB also supports data files. The MATLAB save command will cause every variable in the workspace to be saved in a file called matlab.mat. You can also name the file with the command save filename that will put everything into a file named filename.mat. This command has many other options, and you can find more about it using the help facility. The inverse of the save command is load. Looping, Conditional Statements, and Flow Control Every time you create an m-file you are writing a computer program using the MATLAB programming language. Like any programming language, MATLAB has statements that allow for looping (for and while) as well as conditional execution (if). These statements each surround several Matlab statements with for, while or if at the top and at the bottom. It is highly recommed that you indent the statements between the for, while, or if lines and the line. This indentation strategy makes code much more readable. Your m-files will be expected to follow this convention. The For Loop One of the simplest and most fundamental structures is the for -loop, for example The output is x = 2; for n=1:5 v(n)=x^n; % saving the powers in vector v v' % displaying the transpose of v ans = Typically this kind of structure is typed into an m-file, not in the Command Window. However, you can type a for-loop directly into the command line. What happens is that after you type for n=1:5, MATLAB drops the prompt and lets you close the loop with before responding. If you type this series of commands in MATLAB's editor, it will space things for you to separate them and make the code easier to read. If you want to increment the value of n by something other than 1, you need only type, for example, for n = 1:2:5, which counts from n = 1 in steps of 2 until the last number; here n will use the values n = 1, n = 3, and n = 5. Note that for n = 1:2:6 also uses values n = 1, n = 3, and n = 5. The general form for a for loop is for control-variable=start : increment : value... Matlab statement(s)...

6 The While Loop One problem with for-loops is that they generally run a predetermined set of times. Whileloops, on the other hand, run until some criterion is no longer met. For example x=0; while x<7 x=fix(10*rand(1,1)) %randomly generating single integer if x > 50 break The output for this loop is given below. (If you executed this code your output could vary.) x = 3 x = 6 x = 1 Since while-loops don't necessarily stop after a certain number of iterations, they are notorious for getting caught in infinite loops. In the example above there is a break command in the loop, so that if I've done something wrong and x gets too large, the loop will be broken. The general form for a while loop is Matlab statement initializing a control variable while logical condition involving the control variable Matlab statement(s) Matlab statement changing the control variable Conditionals/Branching/ If-Else Statements We used a simple if statement in while loop example. Here we describe an if-else branching or conditional statement. A typical example, without output, is given below. x = fix(10*rand(1,1)); if x > 5 y = x; else

7 y = 0; The general form for a if-else statement is if logical condition Matlab statement(s) else Matlab statement(s)... As part of using looping/conditional statements, relational operators are used. The most commonly used relational operators are summarized in Table 1. Note that the equal relational operator (==) is not used for assigning values to a variable. To assign a value of 8 to the variable b, one would enter b = 8 in the MATLAB command window. Table 1.

8 Exercises: In each case include your m-file with your name in the comments and the output generated. (Copy/paste material from the MATLAB command screen.) 1. Create function m-file NA_Example2, save it, and then execute the file with command NA_Example2(0,0.1,1);. 2. Create a script m-file that computes and displays the first 15 Fibonacci numbers. Fibonacci numbers are defined by the expression x(n) = x(n 1) + x(n 2), n 3, where x(1) = x(2) = 1. Name the script m-file NA_Fibonacci. 3. A bank pays 3 percent interest each year on money in savings accounts. Determine a formula for the amounts of money a depositor would have after n years if it follows the investment strategy of: (a) Investing $1,000 and leaving it in the bank for n years. (b) Investing $100 at the of each year. Create a function m-file for each case (name them NA_Bank1and NA_Bank2, respectively, where the input parameters are the number of years and the initial deposit in case (a) and for case (b) the number of years and the yearly deposit. Use initial deposits of $1,000 for case (a) and $100 for case (b). Which strategy yields the most money after 20 years, assuming no withdrawals? 4. Create a function m-file called NA_Seqinteger that produces an m by n matrix S so that its elements are a sequence of integers arranged by columns. The function should have two inputs, m = the number of rows and n = number of columns. For example commands NA_Seqinteger(3,2) and NA_Seqinteger(2,4) should display the following matrices respectively The rise time of a function is a measure of the rate of change of the function. Numerically, the rise time is defined as the time it takes the function to rise from 10% to 90% of its value. Consider the family of functions f(t) = 1 e t/a where parameter a takes on values 0.5, , 2.0, 2.5, 3.0 (in MATLAB we would use the code 0.5:0.5:3.0. Construct a function m-file whose input is 0.5:0.5:3.0 which generates a table (a matrix with 2 columns) whose first column is the values 0.5, , 2.0, 2.5, 3.0 and the second column is the rise time for the corresponding function with parameter a replaced by the corresponding value in the first column. (a) The m-file is to consist of a single for loop that changes the value of parameter. This means we need a way to change the formula of the function that is evaluated to obtain the rise time since parameter a changes. We also need a way to determine the value of t which gives the 10% of the function value and also the value of t that gives 90% of the function value. Some help in this regard is contained in the code segments that appear below. If x = 0.5:0.5:3.0 then the number of entries in x is computed as xlen=length(x); We can define the set of function in the statement f='1-exp(-t/a)'; If our for loop is for jj=1:xlen then as we progress through the looping we set a = x(jj); and then substitute the value of a into the function expression using g=subs(f); Finally we need to use function g and determine when it is 10% of its value and also 90% of its value.

9 Here we can use a powerful function in MATLAB, but we must convert g to the correct class of variable. The command subs leaves g as a symbolic variable. To use MATLAB s solve command here we need to convert it to a character variable and modify it into an expression like [char(g),'-0.9'] which the solve command can then use to determine the value of t where char(g) = 0. The above hints should aid in developing the function m-file which you should name NA_Risetime. (b) Once you have your Rise Time table, discuss the following. Formulate an approximate relationship between the rise times vs. the values of the parameter a. 6. (OPTIONAL) The following expression, which arises in computability theory, is a totally computable function that is not a primitive recursive. (That is, it cannot be defined using basic recursion and composition.) (a) Determine numerical values for the blank cells in the following table. In addition explain why you can t compute the cells containing a question mark based on the numeric information in the table. n m x x 2 x x x x 3?? x x x x (b) If you tried to create an m-file to generate the table what modification would you need to change in the indexing of tabular locations? Note: A(3,4) = 125; the values in the cells grow rapidly.

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