An Introductory Guide to MATLAB - UBC Department of Computer
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1. can learn more about the format command by entering help format The remainder of the examples in this guide assume that the short e format has been chosen 6 MATLAB Functions In addition to the standard arithmetic operators MATLAB provides an extensive collection of built in functions For ex ample most elementary mathematical functions sin cos log sqrt are available gt gt cos pi 4 ans 7 0711e 01 pi is an example of a function that does not require parameters and simply returns a commonly used constant gt gt pi ans 3 1416 Other functions are available in libraries of M files grouped into toolboxes Section 11 briefly describes how to create your own functions So far we have only seen functions that return a single matrix but some functions return two or more matrices To save these matrices we surround the output variables by brackets and separate them by commas For example gt gt V D eig A returns the eigenvectors and eigenvalues of A in matrices V and D Several additional MATLAB functions are described in the remainder of this guide For extensive lists of MATLAB functions consult the MATLAB primer or explore the list of topics provided by entering the command help with no arguments In addition the MATLAB demos provide examples of the use of many interesting functions 7 Matrix Operations We have already seen the matrix operator prime for transposing matrices The arithme2. left division x A B solves Ax X B right division x A B solves X x A B In CPSC 303 we will typically use left division Whenever B has as many rows as A left division is defined If A is a square nonsingular matrix and b is a vector with as many rows MATLAB evaluates the expression x A b the solution to Ax b by factoring A with Gaussian elimination and then solving two triangular systems to compute x When A is not square MATLAB factors A using Householder orthogonalization and the factors are used to solve the under determined or over determined system of equations in the least squares sense This can lead to surprising results if the wrong slash is used or if the dimensions of your matrices are wrong Finally the expression A p raises A to the pt power This operation is only defined if A is square and p is an scalar For example A 2 is equivalent to A A although MATLAB does not always compute powers with simple matrix multiplication 8 Array Operations The previous section described standard linear algebra matrix operations Alternatively element by element matrix arith metic is provided by array operations An array operator is formed by preceding one of the symbols or by a period Of course the matrix and array operators for addition and subtraction are equivalent and and are used in either case The operator denotes array multiplication and is defined whenever the two operands have the s
3. CPSC 303 99W T2 An Introductory Guide to MATLAB Ian Cavers Department of Computer Science University of British Columbia 1 Introduction MATLAB provides a powerful interactive computing environment for numeric computation visualization and data analy sis Its wide range of commands functions and language constructs permit users to solve and analyze difficult computa tional problems from science and engineering without programming in a general purpose language This document is not intended to be a complete manual for MATLAB users Instead it provides a brief introduction to MATLAB outlining features that might be of particular use to CPSC 303 students Additional sources of MATLAB documentation and examples to be used in conjunction with this guide include On Line Help MATLAB provides access to extensive on line documentation with the commands help and lookfor To learn more about these commands enter help help lt rtn gt and help lookfor lt rtn gt at the MATLAB prompt Section 2 describes how to initiate a MATLAB session You will find help an valuable supplement to the information in this guide MATLAB Manuals Two sets of MATLAB manuals are available for reference use in the CICSR CS reading room CI CSR CS 262 Please note that these manuals may NOT be taken out of the reading room In addition a third set of MATLAB manuals is on course reserve in the Sedgewick library MATLAB Primer The second edition of the MATLAB Primer
4. FAQ Each 303 assignment will require some use of MATLAB The extent to which you use MATLAB beyond these re quirements is up to you For those who would like to explore the full potential of MATLAB its programming language permits one to largely avoid languages such as C Pascal or FORTRAN for example One warning however You may discover built in numerical functions that you think implement a portion of one or more of your assignments You may not use these function unless I approve them Of course you could always verify your solution if you discover an appropriate MATLAB function Please check with me well before assignment due dates if you need this rule clarified for special MATLAB functions We have approximately 25 or 30 MATLAB licenses available to Rick Lab users Please respect the needs of other users and exit from MATLAB rather than leaving it idle for extended periods of time To save the current state of a MATLAB session and initialize a subsequent MATLAB session with the same variables use the commands save and load While reading the remainder of this document you are strongly encouraged to go to Rick s Lab and experiment with the different features described Be sure to explore some of the MATLAB Expo demos too 2 MATLAB Basics To initiate a MATLAB session in Rick s Lab simply enter mat lab at the shell prompt of an xterm MATLAB will display some introductory information before printing the MATLAB prompt gt gt H
5. ame dimensions For example gt gt A 1 2 37 4 56 y B 2 2 2 3 3 S e CHA B results in Cla 2 4 6 12 15 30 Similarly A B and A B provide the left and right element by element division Raising each element of a matrix to the same power is accomplished by the operator Most standard MATLAB functions operate on a matrix element by element For example gt gt cos C ans 4 1615e 01 6 5364e 01 9 6017e 01 8 4385e 01 7 5969e 01 1 5425e 01 If you create your own functions See Section 11 you should keep in mind that MATLAB assumes that a matrix or a vector can be passed to it 9 Vector and Matrix Manipulation 66 99 Vectors are easily generated with MATLAB s colon notation For example the expression 1 5 creates the following row vector 1 2 3 4 5 You can also create a vector using an increment other than one For example 1 2 7 results in the vector I 3 5 7 The increment may be negative and need not be an integer It is very easy to create a table using the colon notation Experiment with the commands gt gt x O0 pi 4 pi y cos x AA x y for a demonstration of this technique MATLAB permits users to easily manipulate the rows columns submatrices and individual elements of a matrix The subscripts of matrices can be vectors themselves If x and v are vectors then x v is equivalent to the vector x v 1 x v 2 Subscripting a matrix with vectors extracts a submatrix f
6. by Kermit Sigmon University of Florida is available in the file ugradO cs303Nib primer35 ps You can browse through this document with any postscript previewer for example ghostview Sigmon s primer is similar in nature to this guide but provides a more detailed discussion and additional examples of MATLAB features MATLAB Demos Enter demo lt rtn gt at the MATLAB prompt to start up the MATLAB Expo and then use your mouse to select and interact with the different demos Demonstrations are available for most of MATLAB s wide ranging features To start I suggest visiting the MATLAB option of the MAIN MAP to check out the demonstrations on Matrices Numerics and Visualization If you have the opportunity to use color X terminal many demonstrations are especially interesting In addition you can also enter intro lt rtn gt atthe MATLAB prompt for a tour of MATLAB lasting a few minutes This demonstration is controlled by the keyboard rather than the mouse MATLAB FAQ The URL of MATLAB s FAQ most frequently asked questions for Matlab 4 is http www mathworks com faq html Choose Matlab 4 FAQ at the top right of the page This extensive FAQ contains some interesting notes but you may have to sift through answers to questions of little interest to you at the moment More advanced MATLAB users however may find some of the information very helpful I have also provided an unofficial MATLAB FAQ in the text file ugrad0 cs303 lib
7. decimal notation to enter numbers The leading minus sign and decimal point are optional and numbers may be specified using scientific notation The following examples are valid numbers in MATLAB 4 189 032 2 2le 23 0 972E18 0 000001 WARNING If you try to import data produced by a FORTRAN routine that prints the values of double precision variables MATLAB will not understand the use of D in place of E in scientific notation You can use the Unix command t r to replace all D s with E s To build expressions MATLAB provides the usual arithmetic operators addition subtraction multiplication rightdivision left division power MATLAB s use of two division operators will be explained in Section 7 To change the normal precedence of these operators enclose portions of an expression in parentheses in the usual manner On the Rick Lab s HP servers MATLAB uses IEEE double precision floating point arithmetic to perform its compu tations Although MATLAB stores the full precision of all computations by default it displays results in a 5 digit fixed point format The output format can be changed using the format command The following example prints the result of entering the vector 5 7 3 7 7432e 7 under different format settings format short 0 6849 0 0000 format short e 6 8493e 01 7 7432e 07 format long 0 68493150684932 0 00000077432000 format long e 6 849315068493150e 01 7 743200000000001e 07 You
8. e unless explicit global declarations are made function y mymean x mymean Average or mean value For vectors mymean x returns the mean value For matrices mymean x is a row vector containing the mean value of each column ae oP o m n size x if m 1 m n end y sum x m Figure 1 A User Defined Function The MATLAB function eval accepts a single string argument which it interprets as a MATLAB expression or state ment The following example demonstrates how you can use eval to pass one function to second function as an argument and have the second function call the first function within its body Suppose the following function is in the M file goo m function val goo x val x 2 and the following function is in the M file foo m function val foo fname a calls another function whose name is passed as a string to argument fname with the parameter a val eval fname a ede ee ee Constructs a string interpreted as an expression or statement by eval oe ol o ol Using these functions the following command gt gt foo goo 3 evaluates to ans 9 12 Graphics and Related Issues MATLAB provides users with a powerful array of functions for creating sophisticated graphics In CPSC 303 we will probably only use the 2 D plotting features but you should familiarize yourself with the breadth of MATLAB s capabilities For our work the two plotting fu
9. ement of the matrix is nonzero The MATLAB function any is useful when you want a different interpretation of the matrix resulting from the evaluation of a condition The simplest form of an if statement is if condition statements end For example gt gt A 6 3 9 gt gt n 3 gt gt if n 0 A A n end gt gt A results in the output A 2 1 3 Each semicolon in the previous example suppresses the output of a statement s result As in other programming languages if statements can also be used to choose between 2 or more alternatives For example the statement gt gt if n lt 0 x x abs n elseif rem n 2 0 amp n 0 x x n 2 else x x ntl end adds one of three possible elements to the end of the existing row vector x depending on the value of scalar n Note that elseif is one word Writing else and if separately would create a second if statement requiring its own end When working with matrices and vectors the ability to repeat statements with for and while statements is essential The general form of the for statement given below permits the statements in its body to be repeated a specific number of times for variable expression statements end The columns of expression are assigned to variable one by one and then the statements are executed As an example the following statements construct a vector containing the squares of the integers 2 through 5 gt g
10. from the columns of the matrix use the colon notation as previously discussed 13 Additional Features This brief guide has not touched upon many interesting MATLAB features MATLAB provides many additional functions that you can explore in the MATLAB manuals or using MATLAB s help facility If you intend to create complex MATLAB functions check out the description of MATLAB s debugger in the User s Manual 11
11. int Due to difficulties with the license server MATLAB may not always start the first time you try it even though licenses are available Before you conclude all licenses are in use try to start MATLAB several gt 5 times At the prompt users are free to enter sequences of MATLAB expressions commands and assignment statements See Section 3 MATLAB places all ASCII output in the xterm window from which MATLAB was invoked but automatically creates additional windows for graphical output To recall or edit previous command lines MATLAB provides command line editing using Emacs style keystrokes Using the f and keys you can cycle through previous commands The and keys move the cursor left and right along the current command line In its default mode MATLAB inserts characters before the cursor Cnt r1 T toggles the command line editor between the insert and over write modes The delete key eliminates the character lying under the cursor Once all necessary corrections if any have been made to the command line it can be entered by hitting lt rtn gt with the cursor in any position For more information consult the cedit help page by entering gt gt help cedit To terminate a MATLAB session enter the command gt gt exit or gt gt quit 3 Variables Expressions and Statements MATLAB statements typically take one of two forms variable expression or expression All variable and function names consist of a letter fo
12. llowed by any number of numbers letters and underscores MATLAB is case sensitive and only the first 19 characters of any name are significant Expressions are composed from operators function calls and variable names A carriage return normally signifies the end of a statement causing MATLAB to interpret the command and print its result If the last character of a statement is a semicolon however display of the result is suppressed This feature may be especially useful when the result of a computation is a large matrix Finally several statements separated by commas may be placed on a single line When an expression is not explicitly assigned to a variable with the assignment operator MATLAB automatically stores the result in the special variable ans During a MATLAB session you may forget the names of variables stored in your workspace The command who lists the name of all your variables If you want to know their size as well use the command whos By default MATLAB stores all variables until the session is terminated To remove a variable from the workspace use the command clear var_name WARNING clear with no arguments removes all variables from the workspace At any time you can interrupt the computation of a MATLAB statement with Cnt r1 c 4 Matrices and MATLAB Essentially the only data objects in MATLAB are rectangular numerical matrices There are no restrictions placed on the dimensions of a matrix except by system resource
13. nctions plot and fplot will be especially useful plot permits the plotting of vectors and matrices while fplot simplifies the plotting of functions Let s start by plotting sin between 0 and pi using plot First we must create two vectors containing the x and the y coordinates to the points we will use to create our plot gt gt x 0 pi 100 2 pi gt gt y sin x To create our plot of sin we enter the following statement gt gt plot x y The minus sign in single quotes tells MATLAB we want a solid line Other line types and colors can be specified in this fashion Enter help plot for more information Figure 2 A Plotting Example MATLAB permits the addition of another plot to the same graph by entering the command gt gt hold on Let s also plot the value of sin at 0 pi 4 pi 2 2 pi on our graph but this time instead of a solid line we will just mark each point with a plus sign gt gt plot 0 pi 4 2 pi sin 0 pi 4 2 pi Perhaps we also want to add a title and axis labels to the graph gt gt title Our sine plot from 0 to pi gt gt xlabel x gt gt ylabel sin x Text can also be added to a plot at user specified coordinates with the commands text and gtext Your final graph should look something like Figure 2 When you are finished with the graph enter the command gt gt hold off so that subsequent graphics do not include the current plot The MATLAB command fp1lot is similar
14. rom it For example suppose A is an 8 x 8 matrix A 1 4 5 8 is the 4 x 3 submatrix extracted from the first 4 rows and last 3 columns of the matrix When the colon operator is used by itself it denotes all of the rows or columns of a matrix Using the result of the table above gt gt AA 1 produces the first column of matrix AA ans 0 8540e 01 5708e 00 3562e 00 1416e 00 WN rR I 10 For Loops While Loops and Conditional Statements MATLAB also provides a programming language that includes looping statements for and while conditional state ments if and relational and logical operators for constructing conditions The execution of an if or while depends upon the evaluation of a condition To construct conditions MATLAB provides six relational operators lt less than lt less than or equal equal gt greaterthan gt greater than or equal not equal and three logical operators amp and or not Note that the relational operator compares two arguments while is used to assign a value to a variable When a condition compares scalar quantities it evaluates to 1 if true and 0 if false Relational operators can also be applied to matrices of the same dimension In this case a condition evaluates to a matrix of Os and 1s representing the result of applying the operator to the individual elements of the matrix operands A condition evaluating to a matrix is interpreted by if or while to be true if each el
15. s but special meaning is sometimes attached to 1 x 1 matrices scalars and matrices with only one row or column vectors The memory required for the storage of each matrix is automatically allocated by MATLAB The easiest way to enter a matrix into MATLAB is to provide an explicit list of elements enclosed in square brackets MATLAB uses the following conventions e A matrix element can be any valid MATLAB expression e Matrix elements are separated by blanks or commas e A semicolon or carriage return is used to indicate the end of a row For example entering the assignment statement gt gt A 1 2 4 5 8 2 0 6 5 results in the output A 1 0000 2 0000 4 5000 4 0000 6 0000 5 0000 The 2 x 3 matrix is saved in variable A for future reference If you want to see the contents of this or any other variable simply enter its name as a command To reference individual elements enclose their subscripts in parentheses after the variable name in the usual fashion gt gt A 2 3 ans 5 It is important to realize that MATLAB distinguishes between row and column vectors 1 2 3 is a row vector while 1 2 3 isa column vector Column vectors can also be created by applying MATLAB s transpose operator prime to a row vector For example gt gt 1 2 3 1 produces ans 1 2 3 The transpose operator may be applied to matrices of any dimension 5 Numbers and Arithmetic Expressions MATLAB uses conventional
16. t x J gt gt low 2 gt gt hi 5 gt gt for i Of course you could also write this for statement on a single line by separating its components by commas gt gt for i low hi x x i i end but using separate lines with appropriate indentation usually improves readability The following statement produces a vector with the same elements as x but they are arranged in the reverse order ee y Lis gt gt for i hi 1l low y y i i end You can also nest for statements The following statements create an m x n Hilbert matrix H Finally MATLAB also has its own version of the while loop which repeatedly executes a group of statements while a condition remains true while condition statements end Given a positive number val the following statements compute and display the even powers of 2 less than or equal to val gt gt n 0 gt gt while 2 n lt val if rem n 2 0 2 n n n t 1 else w g end end The previous example increments n by 1 each iteration instead of by 2 and exhibits other elements of poor programming style so that we can illustrate the nesting of an if statement inside a while The commands discussed in this section together with the discussion of scripts and functions in the following section permit students to do most of their 303 work within MATLAB 11 M Files Creating Your Own Scripts and Functions Users are able to tailor MATLAB by creating their own func
17. tic operators presented in Section 5 also operate on matrices In each case the operator behaves in a manner consistent with standard linear algebra practises The operators and permit the addition and subtraction of matrices and are defined whenever the matrices have the same dimension For example the following is a valid expression Seibel 2 Br Arb OR UPS 2 dee a ee The exception to this rule is the addition subtraction of a scalar to from a matrix In this case the scalar is added to or subtracted from each element of the matrix individually gt gt 1 2 3 1 ans 2 3 4 The multiplication of two matrices denoted by A B is defined whenever the inner dimensions of the operands A and B are equal For example if C 1 2 3 4 5 6 D 1 11 2 2 2 x 1 1 1 then C x x x an inner product x x an outer product and C D are defined but C D is not Give these examples a try and be sure you understand how MATLAB interprets them In the special case when one of the operands is a scalar each element of the matrix is multiplied by the scalar It is now time to explain the reason for MATLAB s two division operators If A is a square nonsingular matrix then A B and B A formally correspond to the left and right multiplication of B by A Note MATLAB does not actually compute the inverse of A when evaluating these expressions These expressions are used to solve the following types of systems of equations
18. tions and scripts of MATLAB commands and functions Both scripts and functions are ordinary ASCII text files external to MATLAB The name of each file must end in m and must be found on MATLAB s search path It is easiest to start MATLAB from the directory containing your M files See also the command path By default while an M file is executing its statements are not displayed This default behavior can be changed using the echo command A script may contain any sequence of MATLAB statements including references to other M files It is invoked like any other command without arguments and acts on the variables of the workspace globally Each command in the file is executed as though you had typed it into MATLAB Most of the demos provided by MATLAB are simply scripts The first line of a function M file starts with the word function and declares the name of the function and its input and output parameters All input parameters and variables defined within the file are local to the function Figure 1 provides a simple example from the MATLAB User s Guide If we type this function into a file called mymean m then we can call mymean like any other MATLAB function gt gt mymean 1 99 ans 50 The parameter supplied to mymean is passed by value not reference to the local variable x The variables m n y are also local to mymean and do not exist after it returns An M file function does not alter the value of variables in the workspac
19. to plot but accepts the name of a function in single quotes and an abscissa range It adaptively samples the function at enough points to give a representative graph and then plots the results As an example let s plot sin again gt gt fplot sin 0 2 pi To create a hardcopy of the current figure we use the MATLAB command print For example to create an PostScript version of our figure enter the following command gt gt print file_name ps dps The resulting file can be printed on any PostScript printer using the Unix command lpr or previewed using ghostview To create encapsulated PostScript suitable for inclusion into documents substitute deps for dps and file_name eps for file_name ps WARNING PostScript files can be quite large and eat up a lot of your disk quota In CPSC 303 we may often create data outside MATLAB and want to import it for plotting This is accomplished by placing the data pairs or triples into an ASCII file with each pair or triple of data points on a separate line separated by blanks In other words each vector of data to be imported is placed in a separate column of the file Be sure that numbers 10 in scientific notation are compatible with MATLAB s format See Section 5 To import the data enter the following command gt gt load file_name Variable f i 1e_name is now a matrix containing your data To check if it was properly imported simply enter fi le_name To extract the vectors of data
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