MATLAB Introductory Course Computer Exercise Session

Transcription

1 MATLAB Introductory Course Computer Exercise Session This course is a basic introduction for students that did not use MATLAB before. The solutions will not be collected. Work through the course within the session. Make sure you understand what you are doing and what the commands are doing. Collect your questions and ask for help. If you are already familiar with Matlab, this will finish quickly. After you have gone through this MATLAB introduction, start with the assignment exercises. If MATLAB should for some reasons not available, OCTAVE, a free MATLAB clone, is a good alternative. A documentation is given at The syntax is quite similar to MATLAB and most of the exercises given here apply without changes. QTOCTAVE even provides a nice user interface that is similar to the MATLAB window. OC- TAVE is mostly MATLAB compatible but there are some differences when it comes to plotting functionality and image handling. In these cases use the documentation. One difference is that the OCTAVE command image automatically scales the data to the full range, while the MATLAB command does not. 1 Welcome to MATLAB This session introduces some important parts of MATLAB but you will not get far without searching more information on your own. Therefore, use the MATLAB help system and other external sources to find out more. To log on to the computers you need a student account and a password. Having logged in, start MATLAB. The default user interface contains three parts: Workspace: Here the variables that you have entered will be listed (none so far). Command History: Recently run commands. Command Window: The most important window. This is where you enter the commands to run MATLAB. 2 The MATLAB help system The MATLAB help system is reached by entering >> helpdesk in the command window. A window containing the MATLAB help will open. The left part of the window contains a navigator to help you browse through the documentation. There are different ways to get help. A thorough introduction to MATLAB is found in Getting started with MATLAB. It contains the topics covered in this course and some additional material. Apart from Getting started..., the help system contains an index of the MATLAB commands, a search function, and demos. The command >> help command prints a short help text on the command command (command should be replaced by the command you would like to know more about). It is a good habit to always use help on new commands you run into. For instance, try it on itself! Another very useful command is doc. Find out what it does. 1

2 3 Matlab as a calculator Other related commands: By entering numerical expressions, MATLAB can be used as a calculator. Try >> 5+3*2-4/7 What (and why?) is the result of >> round(10*ans)/10 Other round off commands include floor and ceil. Find out what they do. What is ans? The up and down arrows of the keyboard can be used to navigate the command history. You may also enter the first letter(s) of a command that you have already entered. The arrow keys will then browse through all entered commands that start with these letters. Exponential functions can be entered by using. To use the base e there is a special command, exp. The natural logarithm is computed with log. The imaginary unit is denoted i or j. Some commands that are relevant to complex numbers are imag, real, abs, and angle. Display format Normally, MATLAB displays floating point numbers with 4 decimals. If you would like to see more decimals, you can use >> format long To switch back, use format short. If you do not want to see the result of a command, add a semicolon at the of the expression. Elementary functions Compute cos(π/3) = sin(π/4) In some cases, one has to take into account that MATLAB has a limited numerical precision. What should be the result of >> sin(pi) Variables For more complex computations, it is often convenient to store intermediate results in variables: >> a=3 >> b=6; >> c=b*a 2

3 Defined variables are visible in the Workspace window. You could also use the command whos. If you use the same variable name again, it gets a new value. For instance, you can write >> a = a+1 To delete variables, there is a command named clear. The risk of making mistakes often decreases if you clear variables when they are no longer needed. For large computations, it is sometimes also necessary, to avoid filling the entire memory. 4 Scripts and documentation Often, you would like to run a sequence of commands several times, perhaps with different values of the variables. This could be done through MATLAB - scripts, or m-files. An m-file is simply a text file with a sequence of MATLAB commands, with the file extension.m (the same extension is also used for functions, which will be studied later). The file name will become a command name that can be used in MATLAB like any other command. To remember what the script does, and thus be able to use it another day, comment your code. Comments start with the sign %. Start MATLAB s editor with the command edit. Write a solution for the following problem in your m-file: David has obtained a loan to buy something nice and expensive. He ints to pay 10% of his salary every month until he has payed his debt. The loan has a rate of interest of 1% per month. Write a script where David can enter the amount of the loan and his salary, and which computes how large his debt is next month. One suggestion for a solution could be: disp('the script computes the new debt after this month.'); debt = 2000; salary = 3000; rate = 0.01; % Rate of interest per month paymentsize = 0.1; % Percentage of salary to pay new_debt = debt + debt*rate - paymentsize*salary; disp('result:'); display(new_debt); Save the script by choosing Save or Save As in the File menu. Name the script myscript.m. The script is saved in the file myscript.m. Now, you can use it just like any other MATLAB command: >> myscript If the script is saved in a certain directory (folder), it might be necessary to change directory in MATLAB to be able to run it. You can also add the folder to the path using addpath or pathtool. The command dir prints the contents of the current directory (where you are), the command cd can be used to change directory, and pwd shows in which directory you are. 5 Matrices The matrix is the basic data type in MATLAB, and there are very powerful built-in functions for matrix manipulations and computations. A vector is a special case of a matrix. Construct some matrices: 3

4 >> A = [1 2 5;3 8 10] >> b = [7; 4; 5] >> c = [3 2 1] What is the function of semicolons here? What do the following commands do? >> b(3)? >> A(2,3)? (What is the meaning of 2 and 3?) >> A(:,2)? (Here : means all rows.) >> A(1,:)? To retrieve elements from a matrix in this way is called indexing. We can perform matrix calculations: >> A*b Why wouldn t A*c work? Use.' to get the matrix transpose. (Using ' gives the conjugate of the transposed matrix.) Try it out on some matrix! There are other ways to create certain special matrices. What do the following commands do: >> H = ones(3,2)? >> B = eye(3)? >> y = 3:9? >> x = 1:4:21? (What is the purpose of the 4 here?) >> z = 10:-0.5:7? A few more examples of indexing what happens here? >> ind = [1 4 5] >> y(ind) >> A(2,2:) Indexing submatrices also works for assignments: >> A = ones(4,4) >> A(1,1) = 9 >> A(2,3) = 4 >> A(3,2:4) = [7 3 9] Construct a larger matrix from the submatrices A and b: >> D = [A [b;20]] The brackets are used here to group a number of submatrices into a new matrix. As before, the semicolon indicates a line break (cf. the figure below). Matrix functions A Most elementary functions work with matrices, too, where they operate componentwise. Operations that have a special meaning in connection with matrices (such as *, /, and ^) can be forced to operate componentwise by adding a point before the operator. Try for instance b 20 4

5 >> x = 0:0.1:1 >> cos(x).^2./x What is the difference between >> A*B and >> A.*B A common application for matrices is to represent systems of linear equations. As is well known, the system Ax = b, where A and b are matrices and x unknown, can be solved by left multiplication of the inverse of A, i.e., x = A 1 b. However, in MATLAB this is not the best way to solve the equation system. More matrix commands The following commands are very useful. Try them out and use the MATLAB help system to understand how they work. Special matrices: zeros, rand, randn, eye, diag. Linear algebra: inv, det, eig, svd, null Processing data: sort, max, min, find, sum. 6 Writing and documenting your own functions Apart from saving scripts in m-files, that could be used to repeatedly invoke a command sequence, it is also possible to write your own functions in MATLAB. A function may take one or several arguments (inputs) and can return one or several return values. Functions always begin with a line of the form >> x = A\b computes the solution in a faster and numerically more reliable way. If A 1 exists, the solution will be the same as above. Otherwise (that is, for over- or underdetermined systems), the system is solved in least-squares sense. Solve the system of equations 5x 1 + 2x 2 = 3 3x 1 + 2x 2 = 5 Solution: function out = filename(in1,in2,in3) out specifies which variable is going to be returned from the function. filename is the name of the function. The function should be saved in a file named filename.m. in1, in2 and in3 are the arguments of the function. A function can have any number of arguments and return values. If two return values are to be given, the function header takes the form function [out1,out2] = filename(in1,in2,in3) Let us rewrite our script (that computed the new debt after the month) as a function: 5

6 function new_debt = loancomp(debt,salary,rate,paymentsize) % new_debt = loancomp(debt,salary,rate,paymentsize) % Computes the debt after this month. % Arguments: debt, salary, rate of interest, and % percentage of salary to pay. new_debt = debt + debt*rate - paymentsize*salary; Write the code in a file named loancomp.m. This file will be used for the next exercise as well. Comparing the function with the corresponding script, we can see that the debt and salary were entered directly in the script. Here we let those be arguments instead, which means that we let the person running the function decide for the values. The function is invoked with >> loancomp(davids_debt,davids_salary,... davids_rate,davids_paymentsize) where davids_debt, davids_salary, davids_rate and davids_paymentsize are numerical values, or variables containing these values. Compute the size of David s debt next month, if the loan is 1000 EUR, David earns 2500 EUR/month, the rate is 0.02, and he pays is 20% of his salary. There are two main differences between functions and scripts: 1. Functions may have arguments and return values. 2. A script shares the variables with the Workspace from where it was invoked. Functions, on the other hand, create their own workspaces, where only the arguments are defined to begin with. New variables can then be defined inside the function. When all commands of the function have been evaluated, the return value is sent to MATLAB s Workspace, and the workspace of the function disappears (together with its variables). Many (but not all) of the built-in functions in MATLAB are written in the same way as in the example above. For instance, to see how the function mean is implemented, write >> type mean 7 Plots What is drawn by the following code? >> t = 0:0.01:6*pi; >> y = cos(t); >> plot(t,y) What is the difference compared to >> plot(y) To plot several functions in the same figure, you can for instance use >> plot(t,y,t,sin(t)) 6

7 Another option is to use the command hold. Find out how it works, and plot cos and sin in the same figure. You can use zoom to magnify parts of the figure (or use the magnifying glass icon in the figure window). Another useful command to specify what to show is axis. axis equal will give the axis the same scale, so a circle will look like a circle. The grid command adds a grid to the figure. Title and axis labels can be added to the figure by using title, xlabel, and ylabel. You can print your figure with the command print. To include a MATLAB plot in a report (or similar) might be very illustrative. The figures can be saved using Save As in the File menu, or by giving extra flags to the print command. Deping on in which context the figure will be used, there are different file formats to choose between. For instance, the jpg format might be suitable for publishing figures on the Internet, while eps is useful for LATEX, and meta for files to be included in Word (only in Windows). 8 Images MATLAB represents images are represented as matrices. Colour images are three-dimensional matrices. The colour channels are in the third dimension. Use imread to load an image into a variable and look at the structure and data type of this variable in the MATLAB workspace. Do not forget to add a semicolon at the of the load command! Generate a cutout of the image using the indexing commands of Section 5. Use the MATLAB help system to find out how image works and use this command to visualise the cutout. Note that you might have to clip or scale the values appropriately. (Note that the R2008b release of MATLAB seems to have a bug in its imagesc command implementation, resulting in it not scaling the image data as claimed in the documentation.) A histogram can be generated easily with hist. Use imwrite to save the cutout in an image file. You can also save variables in a file. You can do this using the save command. Use load to make the variables available again. The variables are saved in MATLAB s own format (MAT- files), but you can also save them as text. The command >> save min_fil a b c saves the variables a, b and c in a MAT-file. Try it out with the cutout! 9 Control structures Conditions if clauses Often you would like to do different things deping on the value of some variable. In the loan example of Section 6, you might for instance want to check whether the debt is settled. In MATLAB (like in most other programming languages), there is a control structure called if clauses. The following code contains a test, which can be added at the of the loancomp function. It uses an if clause to check if the debt is settled, and in that case set the debt to zero: % Test if the debt is settled: if new_debt <= 0, new_debt = 0; 7

8 disp('the debt is settled!') else disp('the debt is not yet settled.') % Test if the rate of interest is positive: if rate<0 error('the rate of interest must be positive!') Three keywords: if, else and. Following if, we find our condition: new_debt<=0 (<= means less than or equal to). In this example, the condition consists of an inequality, but you could also check whether two variables are equal by ==, or if they are not equal by ~=. If the condition is true, the commands between if and else are executed. Otherwise the commands between else and are executed. In general, we can write if condition commands else commands MATLAB interprets nonzero values as true and 0 as false. Hence, the condition might be an expression that is evaluated to something which is either nonzero or zero. Use help relop or the MATLAB documentation to find the relational operators of MATLAB. The rest of the function is left unchanged. Now test what happens if you enter a negative rate of interest. The command warning can be used to warn about minor issues without interrupting the function. Repetitions As previously mentioned, often one would like to run the same code (at least approximately) a number of times. For this, there are better ways than just repeating the same code over and over again. for loops Suppose that David would like to know his debt, not only in one month, but in, for instance 12 months. Instead of running the function 12 times with different arguments, we can use a for loop. We also introduce a new input argument: number of months. Error handling Sometimes the if clause is used to find an error in the input. In the loan example above we might want to check that the rate of interest is positive. Otherwise we do not want to compute anything, but just print an error message and exit the function. This is obtained with the error command. Add the following test before new_debt is computed: function new_debt = loancomp(debt,salary,rate,... paymentsize,months) % new_debt = loancomp(debt,salary,rate,... % paymentsize,months) % Computes the debt after a number of months. % Arguments: debt, salary, rate of interest, % percentage of salary to pay and 8

9 % number of months. % Test if the rate of interest is positive: if rate<0 error('the rate of interest must be positive!') % Compute the new debt: for t = 1:months debt = debt + debt*rate - paymentsize*salary; new_debt = debt; % Test if the debt is settled: if new_debt <= 0, new_debt = 0; disp('the debt is settled!') else disp('the debt is not yet settled.') The keywords for and enclose the repetitive structure. The variable t will, step by step, take the values given after the = sign, and for each value, the lines between for and are executed. We recognize the colon notation, :, from Section 5. The first time the for loop is run, t will be 1, the second time t will be 2 etc., until t has reached the value of months (e.g. 12). Every time, debt gets a new value. Try removing the semicolon after the computation to see the debts for each month. In the example above, t is only used to keep track of the number of iterations, but it can also be used inside the loop. For instance, let us make debt a vector, where debt(t) gives the debt for month t. Before the for loop, add debt = [debt; zeros(months,1)]; to create a vector where we can save the size of the debt month by month. The line inside the for loop is replaced by debt(t+1) = debt(t) + debt(t)*rate - paymentsize*salary; Otherwise, the function is left as it is. Try this version of the function, and plot a figure of how the debt changes over the months. If you skip the semicolon in the loop, you can also see how the values of debt are entered one by one. Printouts like that may be useful when looking for bugs in the code. while You do not always know in advances how many times to repeat a computation. For instance, David might want to find out after how many months the debt will be settled. For this, one can use a while loop. while checks if a condition is satisfied (in the same way as if) and executes a number of commands if it is. Then the condition is checked again, and the whole procedure is repeated until the condition is not satisfied anymore. We can modify our example to compute at which month the debt is settled using while: function finalmonth = loancomp(debt,salary,rate,paymentsize) % finalmonth = loancomp(debt,salary,rate,paymentsize) % Computes at which month the debt is settled % Arguments: debt, salary, rate of interest, and % percentage of salary to pay % Test if the rate of interest is positive: if rate<0 9

10 error('the rate of interest must be positive!') month_no = 1; while debt(month_no)>0 % Compute the new debt: debt(month_no+1) = debt(month_no) +... debt(month_no)*rate - paymentsize*salary; month_no = month_no+1; finalmonth = month_no; When using a while loop, it is important to make sure that the condition is not always satisfied. In that case, the function would get stuck in an infinite loop 1. This should be checked before entering the loop. In our example, the loop will be infinite if the debt does not decrease every month. We can check this just like we test if the rate of interest is positive. Give a condition which ensures that the loop will be finite (and the debt will be settled): Modify the code so that we do not risk an infinite loop. Apart from the condition above, we could also allow a maximum number of iterations, i.e., we can require that month_no does not exceed 200 (for instance). The upper limit could also be an argument of the function. 1 If this would occur, you can stop the execution by pressing Ctrl-c. 10