MATLAB - University Of Hawaiʻi

CHAPTER 1 The MATLAB Environment3 Command History Window Command WindowThe standard mix of menus appears on the top of the MATLAB desktop thatallows you to do things like file management and debugging of files you create. Youwill also notice a drop-down list on the upper right side of the desktop that allowsyou to select a directory to work in. The most important item of business right nowis the Command Window.Command Window and Basic ArithmeticThe Command Window is found on the right-hand side of the MATLAB desktop.Commands are entered at the prompt with looks like two successive “greater than”signs: Let’s start by entering a few really basic commands. If you want to find the valueof a numerical expression, simply type it in. Let’s say we want to know the value of433.12 multiplied by 15.7. We type 433.12 * 15.7 at the MATLAB prompt and hitthe enter key. The result looks like this: 433.12*15.7ans 6.8000e 003MATLAB spits out the answer to our query conveniently named ans. This is avariable or symbolic name that can be used to represent the value later. Chances arewe will wish to use our own variable names. So for example, we might want to calla variable x. Suppose we want to set it equal to five multiplied by six. To do this, wetype the input as x 5*6x 30Once a variable has been entered into the system, we can refer to it later. Supposethat we want to compute a new quantity that we’ll call y, which is equal to xmultiplied by 3.56. Then we type y x * 3.56y 106.8000

MATLAB Demystified4Now, you will notice that in this example we put spaces in between each term inour equation. This was only done to enhance the readability and professionalappearance of our output. MATLAB does not require you to include these spaces inyour input. We could just as well type y x * 3.56 as y x * 3.56; however, thelatter presentation is cleaner and easier to read. When your expressions getcomplicated, it will be more important to keep things neat so it’s advisable toinclude the spaces.Let’s summarize basic arithmetical input in MATLAB. To write the multiplicationab, in MATLAB we typea * bFor division, the quantityais typed asba / bThis type of division is referred to as right division. MATLAB also allowsanother way to enter division, called left division. We can enter the quantity ba bytyping the slash mark used for division in the opposite way, that is, we use a backslash instead of a forward slasha \ bExponentiation ab is entered in the following waya bFinally, addition and subtraction are entered in the usual waya ba – bThe precedence followed in mathematical operations by MATLAB is the sameused in standard mathematics, but with the following caveat for left and rightdivision. That is, exponentiation takes precedence over multiplication and division,which fall on equal footing. Right division takes precedence over left division.Finally, addition and subtraction have the lowest precedence in MATLAB. Tooverride precedence, enclose expression in parentheses.EXAMPLE 1-1Use MATLAB to evaluate9 3 3 95 and 4 3 4 5 4 ( 2 )3

CHAPTER 1 The MATLAB Environment5SOLUTION 1-1The command required to find the value of the first expression is 5*(3/4) 9/5ans 5.5500For the second expression, we use some parentheses along with the exponentiationoperator a b. Although this is a simple expression, let’s enter it in pieces to getused to using variables. We obtain r 4 3r 64 s 3/4 9/(2*3)s 2.2500 t r*st 144The Assignment OperatorThe equals sign “ ” is known as the assignment operator. While it does what youthink it does much of the time, that is, describes an equation, at other times inMATLAB it’s more appropriate to think of it as an instruction to assign a value to avariable the way you would in a computer program. The distinction between thetwo interpretations can be illustrated in the following way. If you typex 6 90in MATLAB, you get the following response? x 6 90Error: The expression to the left of the equals sign is nota valid target for an assignment.So while the expression is a completely valid equation you could write down ifdoing algebra on paper, MATLAB doesn’t know what to do with it. On the otherhand, MATLAB is completely happy if you assign the value 90 – 6 to the variablex by writingx 90 – 6

6MATLAB DemystifiedAnother way that the assignment operator works more like an assignment in acomputer program is in a recursive type assignment to a variable. That is, MATLABallows you to writex x 4if we have previously defined the variable x. For example, the following sequenceis completely valid x 34 2x 1156 x x 4x 1160To use a variable on the right-hand side of the assignment operator, we mustassign a value to it beforehand. So while the following command sequence willgenerate an error x 2x 2 t x a? Undefined function or variable 'a'.The following sequence does not x 2x 2 a 3.5a 3.5000 t x at 5.5000In many instances, it is not desirable to have MATLAB spit out the result of anassignment. To suppress MATLAB output for an expression, simply add a semicolon(;) after the expression. In the following command sequence, first we just type inthe assignment x 3. MATLAB duly reports this back to us. On the next line, weenter x 3; so that MATLAB does not waste space by telling us something wealready know. Instead it comes back with the command prompt waiting for our nextinput: x 3x 3

CHAPTER 1 The MATLAB Environment7 x 3; We can include multiple assignments on the same line. For example, the followingexpressions are valid x 2; y 4; z x*yz 8Notice the two semicolons, they tell MATLAB we don’t want to see the values ofx and y.When doing a lot of calculations, you may end up with a large number ofvariables. You can refresh your memory by typing who in the MATLAB commandwindow. Doing this will tell MATLAB to display all of the variable names you haveused up to this point. For instance, in our case we have whoYour variables are:V aansrstxyzBy typing whos, we get a bit more information. This will tell us the variablescurrently in memory, their type, how much memory is allocated to each variable,and whether or not they are complex (see below). In our case we have doubledoubledoubledoublearrayarrayarray (complex)arrayarrayarrayarrayarrayarrayGrand total is 9 elements using 80 bytesNow suppose we want to start all over. We can do this by issuing a clear command.Clear can be applied globally by simply typing clear and then hitting the enter key,or to specific variables by typing clear followed by a space delimited variable list.If we wanted to reset or clear the variables x, y, and z that we have been using, thenwe could typeclear x y z

MATLAB Demystified8MATLAB will simply return the command prompt and won’t say anything else,but if you try to use these variables again without assigning them values it will beas if they had not been seen before.Long assignments can be extended to another line by typing an ellipsis which isjust three periods in a row. For example FirstClassHolders 72; Coach 121; Crew 8; TotalPeopleOnPlane FirstClassHolders Coach. CrewTotalPeopleOnPlane 201The ellipsis follows Coach on the line used to define TotalPeopleOnPlane. Afteryou type the ellipsis, just hit the enter key. MATLAB will move to the followingline awaiting further input.OK, something you are probably wondering, as I was when I started usingMATLAB, was how in the world do you control the way numbers are displayed onthe screen? So far in our examples, MATLAB has been spitting out numbers withfour decimal places. This is known as short format in MATLAB. It’s the default inMATLAB and if that’s all the precision you require, then you don’t have to doanything. If you want more, then you can tell MATLAB to add more digits to theright of the decimal point by using the format command. If we want 16 digitsinstead of 4, we type format long. To see how this works, look at the followingcalculation, displayed in both formats format long x 3 11/16 2 1.2x 5.98489670999407 format short x 3 11/16 2 1.2x 5.9849Comparing the long and short formats, notice that the fourth decimal place wasrounded up to nine when format short was used. If you want to do financialcalculations, you can use the format bank command. As expected, this roundseverything off to two decimal places. format bank hourly 35.55

CHAPTER 1 The MATLAB Environment9hourly 35.55 weekly hourly*40weekly 1422.00MATLAB displays large numbers using exponential notation. That is it represents5.4387 103 as 5.4387e 003. If you want all numbers to be represented in thisfashion, you can do so. This type of notation can also be defined using the short orlong formats. For short (four decimal places plus the exponent) you can type formatshort e. To allow 15 decimal digits plus the exponent, type format long e. Here is anexample of the short exponent format format short e 7.2*3.1ans 2.2320e 001If you type format rat, then MATLAB will find the closest rational expression itcan that corresponds to the result of a calculation. Pretty neat tool huh? Let’s repeatthe previous calculation format rat 7.2*3.1ans 558/25Basic Mathematical DefinitionsMATLAB comes with many basic or familiar mathematical quantities and functionsbuilt in. Let’s show how to use π in an example.EXAMPLE 1-2Find the volume of a sphere of radius 2 m.SOLUTION 1-2The volume of a sphere is given byV 4π R33

10MATLAB DemystifiedOf course MATLAB comes with π predefined. To use it, we just type pi. So afterdefining a variable to hold the radius, we can find the volume by typing r 2; V (4/3) *pi*r 3V 33.5103Another famous number that shows up in many mathematical applications is theexponential function. That is, e 2.718. We can reference e in MATLAB by typingexp(a) which gives us the value of ea. Here are a few quick examples exp(1)ans 2.7183 exp(2)ans 7.3891To find the square root of a number, we type sqrt. For example x sqrt(9)x 3 y sqrt(11)y 3.3166To find the natural log of a number x, type log(x). log(3.2)ans 1.1632 x 5; log(5)ans 1.6094If you want the base ten logarithm, type log10(x) x 3; log10(x)ans 0.4771MATLAB comes equipped with the basic trig functions and their inverses, takingradian argument by default. These are typed in lower case using the standardnotation. For instance

CHAPTER 1 The MATLAB Environment11 cos(pi/4)ans 0.7071To use an inverse of a trig function, add on an a before the name of the trigfunction. For example, to compute the inverse tangent of a number we can use thefollowing format rat atan(pi/3)ans 1110/1373Complex NumbersWe can also enter complex numbers in MATLAB. To remind members of ouraudience who are Aggie graduates, the square root of –1 is defined asi 1A complex number is one that can be written in the form z x iy, where x is thereal part of z and y is the imaginary part of z. It is easy to enter complex numbers inMATLAB, by default it recognizes i as the square root of minus one. We can docalculations with complex numbers in MATLAB pretty easily. For examplea 2 3ib 1 i a b 3 2iLet’s verify this in MATLAB. It is not necessary to add spaces or include amultiplication symbol (*) when typing i in MATLAB. format short a 2 3i; b 1 - i; c a bc 3.0000 2.0000i

MATLAB Demystified12Fixing TyposIt’s going to be a fact that now and then you are going to type in an expression withan error. If you hit the enter key and then later realize what happened, it’s notnecessary to retype the line. Just use your arrow keys to move back up to theoffending line. Fix your error, and then hit enter again and MATLAB will correctthe output.Some File BasicsLet’s round out the chapter by considering some basic operations with files.MATLAB wouldn’t be that useful if you couldn’t save and retrieve your workright? Let’s say you want to save all the expressions and variables you have enteredin the command window for use at a later time. You can do this by executing thefollowing actions:1. Click on the File pull-down menu2. Select Save Workspace As 3. Type in a file name4. Click on the Save buttonThis method creates a MATLAB file which has a .MAT file extension in Windows.If you save a file this way, you can retrieve all the commands in it and work with itagain just like you can when working with files in any other computer program.Sometimes, especially when working on complicated projects, you won’t wantto sit there and type every expression in a command window. It might be moreappropriate to type a long sequence of operations and store them in a file that canbe executed with a single command in the command window. This is done bycreating a script file. This type of file is known as a MATLAB program and is savedin a file format with a .M extension. For this reason, we also call them M-files. Wecan also create M-files that are function files.From what we’ve done so far, you already know how to create a script file. All ascript file comes down to is a saved sequence of MATLAB commands. Let’s createa simple script file that will compute ex for a few values of x. First, open theMATLAB editor. Either Click New M-File under the File pull-down menu Or click on the New File icon on our toolbar at the top of the screen

CHAPTER 1 The MATLAB Environment13Now type in the following lines:% script file example1.m to compute exponential of a set of numbersx [1:2:3:4];y exp(x)Notice the first line begins with a % sign. This line is a comment. This is a lineof text that is there for our benefit, it’s a descriptive note that MATLAB ignores.The next line creates an array or set of numbers. An array is denoted using squarebraces [] and by delimiting the elements of the array with colons or commas. Thefinal line will tell MATLAB to calculate the exponential of each member of thearray, in other words the values e1, e2, e3, e4. Save the file by clicking the Save iconin the file editor or by selecting Save As from the File pull-down menu. Save the fileas example1.m in your MATLAB directory.Now return to the MATLAB desktop command window. Type in example1. Ifyou did everything right, then you will see the following output example1y 2.71837.389120.085554.5982We can also use M-files to create and store data. Following an example fromanother McGraw-Hill book on MATLAB, let’s create a set of temperatures that wewill store in a file. We do this by creating a list of temperatures in the file editortemps [32,50,65,70,85]Now we save this as a file that we’ll call TemperatureData.m. We store this filein the MATLAB directory. To access it in the command window, we just type thename of the file. MATLAB responds by spitting out the list of numbers: TemperatureDatatemps 3250657085Now we can use the data by referring to the array name used in the file. Let’screate another set of numbers called Celsius that converts these familiar temps intothe European style Celsius temperatures we are so familiar with. This can be donewith the following command CelsiusTemps (5/9) * (temps - 32)CelsiusTemps 010.000018.333321.111129.4444Later, we will investigate programming in MATLAB and we will show you howto create functions that can be called later in the command window.

14MATLAB DemystifiedEnding Your MATLAB SessionOK we have gotten started with a few very basic MATLAB commands. You mightwant to save your work and then shut down MATLAB. How do you get it off yourscreen? Well you can end your MATLAB session by selecting exit from the Filepull-down menu, just like you would with any other program. Optionally, you cantype quit in the command window and MATLAB will close.QuizUse MATLAB to calculate the following quantities:111482. 5 3731. 53. 91.254. True or False. If y has not been assigned a value, MATLAB will allow youto define the equation x y 2 to store in memory for later use.5. If the volume of a cylinder of height h and radius r is given by V πr2h,use MATLAB to find the volume enclosed by a cylinder that is 12 cm highwith a diameter of 4 cm.6. Use MATLAB to compute the sin of π/3 expressed as a rational number.7. Create a MATLAB m file to display the results of sin(π/4), sin(π/3),sin(π/2) as rational numbers.

CHAPTER 2Vectors andMatricesOne area where MATLAB is particularly useful is in the computer implementationof linear algebra problems. This is because MATLAB has exceptional capabilitiesfor handling arrays of numbers, making it a useful tool for many scientific andengineering applications.VectorsA vector is a one-dimensional array of numbers. MATLAB allows you to createcolumn vectors or row vectors. A column vector can be created in MATLAB byenclosing a set of semicolon delimited numbers in square brackets. Vectors canhave any number of elements. For example, to create a column vector with threeelements we write: a [2; 1; 4]a 214Copyright 2007 by The McGraw-Hill Companies. Click here for terms of use.

MATLAB Demystified16Basic operations on column vectors can be executed by referencing the variablename used to create them. If we multiply a column vector by a number, this is calledscalar multiplication. Suppose that we wanted to create a new vector such that itscomponents were three times the components in the vector a we just created above.We could start by defining a scalar variable (remember that a semicolon after acommand suppresses the output): c 3;Next, we just perform the operation treating a like another variable: b c*ab 6312To create a row vector, we enclose a set of numbers in square brackets but thistime use a space or comma to delimit the numbers. For example: v [2 0 4]v 204Or using commas: w [1,1,9]w 119Column vectors can be turned into row vectors and vice versa using the transposeoperation. Suppose that we have a column vector with n elements denoted by: v1 v v 2 M vn Then the transpose is given by:vT v1v2 L vn

CHAPTER 2 Vectors and Matrices17In MATLAB, we represent the transpose operation with a single quote or tickmark(‘). Taking the transpose of a column vector produces a row vector: a [2; 1; 4]; y a'y 214Now let’s take the transpose of a row vector to produce a column vector: Q [2 1 3]Q 21 R Q'R 2133It is also possible to add or subtra

He is the author of Linear Algebra Demystified, Quantum Mechanics Demystified, Relativity Demystified, Signals and Systems Demystified