[Solved] MAT 275 Laboratory 1

Introduction to MATLABMATLAB is a computer software commonly used in both education and industry to solve a wide range of problems.This Laboratory provides a brief introduction to MATLAB, and the tools and functions that help you to work with MATLAB variables and files.The MATLAB Environment To start MATLAB double-click on the MATLAB shortcut icon. The MATLAB desktop will open. On the left side you will generally find the Current Folder window and on the right the Workspace and Command History windows. The Command Window is where the MATLAB commands are entered and executed. Note that windows within the MATLAB desktop can be resized by dragging the separator bar(s).If you have never used MATLAB before, we suggest you type demo at the MATLAB prompt. Click on Getting Started with MATLAB and run the file.Basics And HelpCommands are entered in the Command Window. Basic operations are +, -, *, and /. The sequence>> a=2; b=3; a+b, a*bans =5ans =6defines variables a and b and assigns values 2 and 3, respectively, then computes the sum a+b and product ab. Each command ends with , (output is visible) or ; (output is suppressed). The last command on a line does not require a ,. Standard functions can be invoked using their usual mathematical notations. For example>> theta=pi/5;>> cos(theta)^2+sin(theta)^2ans =1verifies the trigonometric identity sin2  +cos2  = 1 for  = 5 . A list of elementary math functions canbe obtained by typing>> help elfun To obtain a description of the use of a particular function type help followed by the name of the function. For example>> help coshgives help on the hyperbolic cosine function. To get a list of other groups of MATLAB programs already available enter help:>> helpc2011 Stefania Tracogna, SoMSS, ASUMATLAB sessions: Laboratory 1 Another way to obtain help is through the desktop Help menu, Help > Product Help. MATLAB is case-sensitive. For example>> theta=1e-3, Theta=2e-5, ratio=theta/Thetatheta =1.0000e-003Theta =2.0000e-005ratio =50 The quantities Inf () and NaN (Not a Number) also appear frequently. Compare>> c=1/0c =Infwith>> d=0/0d =NaNPlotting with MATLAB To plot a function you have to create two arrays (vectors): one containing the abscissae, the other the corresponding function values. Both arrays should have the same length. For example, consider plottingthe functiony = f(x) =x2 sin(x) + exx 1for 0 x 2. First choose a sample of x values in this interval:>> x=[0,.1,.2,.3,.4,.5,.6,.7,.8,.9,1, 1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2]x =Columns 1 through 70 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000Columns 8 through 140.7000 0.8000 0.9000 1.0000 1.1000 1.2000 1.3000Columns 15 through 211.4000 1.5000 1.6000 1.7000 1.8000 1.9000 2.0000Note that an ellipsis was used to continue a command too long to fit in a single line.Rather than manually entering each entry of the vector x we can simply use>> x=0:.1:2or>> x=linspace(0,2,21)Both commands above generate the same output vector x. The output for x can be suppressed (by adding ; at the end of the command) or condensed by entering>> format compactc2011 Stefania Tracogna, SoMSS, ASUMATLAB sessions: Laboratory 1(This format was used for all previous outputs). To evaluate the function f simultaneously at all the values contained in x, type>> y=(x.^2-sin(pi.*x)+exp(x))./(x-1)y =Columns 1 through 6-1.0000 -0.8957 -0.8420 -0.9012 -1.1679 -1.7974Columns 7 through 12-3.0777 -5.6491 -11.3888 -29.6059 Inf 45.2318Columns 13 through 1826.7395 20.5610 17.4156 15.4634 14.1068 13.1042Columns 19 through 2112.3468 11.7832 11.3891Note that the function becomes infinite at x = 1 (vertical asymptote). The array y inherits the dimensionof x, namely 1 (row) by 21 (columns). Note also the use of parentheses.IMPORTANT REMARKIn the above example *, / and ^ are preceded by a dot . in order for the expression to be evaluated foreach component (entry) of x. This is necessary to prevent MATLAB from interpreting these symbolsas standard linear algebra symbols operating on arrays. Because the standard + and operations onarrays already work componentwise, a dot is not necessary for + and -.The command>> plot(x,y)creates a Figure window and shows the function. The figure can be edited and manipulated using theFigure window menus and buttons. Alternately, properties of the figure can also be defined directly atthe command line:>> x=0:.01:2;>> y=(x.^2-sin(pi.*x)+exp(x))./(x-1);>> plot(x,y,r-,LineWidth,2);>> axis([0,2,-10,20]); grid on;>> title(f(x)=(x^2-sin(pi x)+e^x)/(x-1));>> xlabel(x); ylabel(y);Remarks: The number of x-values has been increased for a smoother curve (note that the stepsize is now :01rather than :1). The option r- plots the curve in red. LineWidth,2 sets the width of the line to 2 points (the default is 0:5). The range of x and y values has been reset using axis([0,2,-10,20]) (always a good idea in thepresence of vertical asymptotes). The command grid on adds a grid to the plot. A title and labels have been added.The resulting new plot is shown in Fig. L1a. For more options type help plot in the CommandWindow.c2011 Stefania Tracogna, SoMSS, ASUMATLAB sessions: Laboratory 1Figure L1a: A Figure windowScripts and Functions Files containing MATLAB commands are called m-files and have a .m extension. They are two types:1. A script is simply a collection of MATLAB commands gathered in a single file. The value of thedata created in a script is still available in the Command Window after execution. To create anew script select the MATLAB desktop File menu File > New > Script. In the MATLAB texteditor window enter the commands as you would in the Command window. To save the file usethe menu File > Save or File > Save As, or the shortcut SAVE button .Variable defined in a script are accessible from the command window.2. A function is similar to a script, but can accept and return arguments. Unless otherwise specifiedany variable inside a function is local to the function and not available in the command window.To create a new function select the MATLAB desktop File menu File > New > Function. AMATLAB text editor window will open with the following predefined commandsfunction [ output_args ] = Untitled3( input_args )%UNTITLED3 Summary of this function goes here% Detailed explanation goes hereendThe output args are the output arguments, while the input args are the input arguments. Thelines beginning with % are to be replaced with comments describing what the functions does. Thecommand(s) defining the function must be inserted after these comments and before end.To save the file proceed similarly to the Script M-file.Use a function when a group of commands needs to be evaluated multiple times. Examples of script/function:1. scriptmyplot.mx=0:.01:2; % x-valuesy=(x.^2-sin(pi.*x)+exp(x))./(x-1); % y-valuesc2011 Stefania Tracogna, SoMSS, ASUMATLAB sessions: Laboratory 1plot(x,y,r-,LineWidth,2); % plot in red with wider lineaxis([0,2,-10,20]); grid on; % set range and add gridtitle(f(x)=(x^2-sin(pi x)+e^x)/(x-1)); % add titlexlabel(x); ylabel(y); % add labels2. script+function (two separate files)myplot2.m (driver script)x=0:.01:2; % x-valuesy=myfunction(x); % evaluate myfunction at xplot(x,y,r-,LineWidth,2); % plot in redaxis([0,2,-10,20]); grid on; % set range and add gridtitle(f(x)=(x^2-sin(pi x)+e^x)/(x-1)); % add titlexlabel(x); ylabel(y); % add labelsmyfunction.m (function)function y=myfunction(x) % defines functiony=(x.^2-sin(pi.*x)+exp(x))./(x-1); % y-values3. function+function (one single file)myplot1.m (driver script converted to function + function)function myplot1x=0:.01:2; % x-valuesy=myfunction(x); % evaluate myfunction at xplot(x,y,r-,LineWidth,2); % plot in redaxis([0,2,-10,20]); grid on; % set range and add gridtitle(f(x)=(x^2-sin(pi x)+e^x)/(x-1)); % add titlexlabel(x); ylabel(y); % add labels%function y=myfunction(x) % defines functiony=(x.^2-sin(pi.*x)+exp(x))./(x-1); % y-valuesIn case 2 myfunction.m can be used in any other m-file (just as other predefined MATLAB functions).In case 3 myfunction.m can be used by any other function in the same m-file (myplot1.m) only. Use 3when dealing with a single project and 2 when a function is used by several projects. Note that the function myplot1 does not have explicit input or output arguments, however we cannotuse a script since the construct script+function in one single file is not allowed. It is convenient to add descriptive comments into the script file. Anything appearing after % on anygiven line is understood as a comment (in green in the MATLAB text editor). To execute a script simply enter its name (without the .m extension) in the Command Window (orclick on the SAVE & RUN button ).The function myfunction can also be used independently if implemented in a separate file myfunction.m:>> x=2; y=myfunction(x)y =11.3891c2011 Stefania Tracogna, SoMSS, ASUMATLAB sessions: Laboratory 1A script can be called from another script or function (in which case it is local to that function).If any modification is made, the script or function can be re-executed by simply retyping the scriptor function name in the Command Window (or use the up-arrow on the keyboard to browse throughpast commands).IMPORTANT REMARKBy default MATLAB saves files in the Current Folder. To change directory use the Current Directorybox on top of the MATLAB desktop. A function file can contain a lot more than a simple evaluation of a function f(x) or f(t; y). But insimple cases f(x) or f(t; y) can simply be defined using the inline syntax.For instance, if we want to define the function f(t; y) = t2 y, we can write the function file f.mcontainingfunction dydt = f(t,y)dydt = t^2-y;and, in the command window, we can evaluate the function at different values:>> f(2,1) % evaluate the function f at t = 2 and y = 1ans =3or we can define the function directly on the command line with the inline command:>> f = inline(t^2-y,t,y)f =Inline function:f(t,y) = t^2-y>> f(2,1) % evaluate the function f at t = 2 and y = 1ans =3However, an inline function is only available where it is used and not to other functions. It is not recommendedwhen the function implemented is too complicated or involves too many statements.Alternatively, the function can be entered as an anonymous function>> f = @(t,y)(t^2-y) CAUTION! The names of script or function M-files must begin with a letter. The rest of the characters mayinclude digits and the underscore character. You may not use periods in the name other than thelast one in .m and the name cannot contain blank spaces. Avoid name clashes with built-in functions. It is a good idea to first check if a function or a scriptfile of the proposed name already exists. You can do this with the command exist(name),which returns zero if nothing with name name exists. NEVER name a script le or function le the same as the name of the variable it computes. WhenMATLAB looks for a name, it first searches the list of variables in the workspace. If a variable ofthe same name as the script file exists, MATLAB will never be able to access the script file.c2011 Stefania Tracogna, SoMSS, ASUMATLAB sessions: Laboratory 1ExercisesInstructions:You will need to record the results of your MATLAB session to generate your lab report. Create adirectory (folder) on your computer to save your MATLAB work in. Then use the Current Directoryfield in the desktop toolbar to change the directory to this folder. Now typediary lab1 yourname.txtfollowed by the Enter key. Now each computation you make in MATLAB will be save in your directoryin a text file named lab1 yourname.txt. When you have finished your MATLAB session you can turnoff the recording by typing diary off at the MATLAB prompt. You can then edit this file using yourfavorite text editor (e.g. MS Word).Lab Write-up: Now that your diary file is open, enter the command format compact (so that whenyou print out your diary file it will not have unnecessary blank lines), and the comment line% MAT 275 MATLAB Assignment # 1Include labels to mark the beginning of your work on each part of each question, so that your edited labwrite-up has the format% Exercise 1..% Exercise 2Final Editing of Lab Write-up: After you have worked through all the parts of the lab assignmentyou will need to edit your diary file. Remove all typing errors. Unless otherwise specified, your write-up should contain the MATLAB input commands,the corresponding output, and the answers to the questions that you have written. If the exercise asks you to write an M-file, copy and paste the file into your diary file in theappropriate position (after the problem number and before the output generated by the file). If the exercise asks for a graph, copy the figure and paste it into your diary file in the appropriateposition. Crop and resize the figure so that it does not take too much space. Use ; to suppressthe output from the vectors used to generate the graph. Make sure you use enough points for yourgraphs so that the resulting curves are nice and smooth. Clearly separate all exercises. The exercises numbers should be in a larger format and in boldface.Preview the document before printing and remove unnecessary page breaks and blank spaces. Put your name and class time on each page.Important: An unedited diary le without comments submitted as a lab write-up is notacceptable.1. All points with coordinates x = r cos() and y = r sin(), where r is a constant, lie on a circlewith radius r, i.e. satisfy the equation x2 + y2 = r2. Create a row vector for  with the values0; 4 ; 2 ; 34 ; ; and 54 .Take r = 2 and compute the row vectors x and y. Now check that x and y indeed satisfy theequation of a circle, by computing the radius r =x2 + y2.Hint: To calculate r you will need the array operator .^ for squaring x and y. Of course, youcould also compute x2 by x.*x.c2011 Stefania Tracogna, SoMSS, ASUMATLAB sessions: Laboratory 12. Use the linspace command or the colon operator : to create a vector t with 91 elements:1; 1:1; 1:2; : : : ; 10 and define the function y =et=10 sin(t)t2 + 1(make sure you use ; to suppressthe output for both t and y).(a) Plot the function y in black and include a title with the expression for y.(b) Make the same plot as in part (a), but rather than displaying the graph as a curve, show theunconnected data points. To display the data points with small circles, use plot(t,y,o).Now combine the two plots with the command plot(t,y,o-) to show the line through thedata points as well as the distinct data points.3. Use the command plot3(x,y,z) to plot the circular helix x(t) = sin t; y(t) = cos t; z(t) = t0 t 20.NOTE: Use semicolon to suppress the output when you define the vectors t, x, y and z. Make sureyou use enough points for your graph so that the resulting curve is nice and smooth.4. Plot y = cos x in red with a solid line and z = 1 x22 + x424 in blue with a dashed line for 0 x on the same plot.Hint: Use plot(x,y,r,x,z,).Add a grid to the plot using the command grid on.NOTE: Use semicolon to suppress the output when you define the vectors x, y and z. Make sureyou use enough points for your graph so that the resulting curves are nice and smooth.5. The general solution to the differential equationdydx= x + 2 isy(x) =x22+ 2x + C with y(0) = C:The goal of this exercise is to write a function file to plot the solutions to the differential equationin the interval 0 x 4, with initial conditions y(0) = 1; 0; 1.The function file should have the structure function+function (similarly to the M-file myplot1.mExample 3, page 5). The function that defines y(x) must be included in the same file (note thatthe function defining y(x) will have two input arguments: x and C).Your M-file should have the following structure (fill in all the ?? with the appropriate commands):function ex5x = ?? ; % define the vector x in the interval [0,4]y1 = f(??); % compute the solution with C = -1y2 = f(??); % compute the solution with C = 0y3 = f(??); % compute the solution with C = 1plot(??) % plot the three solutions with different line-stylestitle(??) % add a titlelegend(??) % add a legendendfunction y = f(x,C)y = ?? % fill-in with the expression for the general solutionendPlot the graphs in the same window and use different line-styles for each graph. To plot the graphsin the same window you can use the command hold on or use the plot command similarly toExercise 4.Add the title Solutions to dy/dx = x + 2.Add a legend on the top left corner of the plot with the list of C values used for each graph.c2011 Stefania Tracogna, SoMSS, ASUMATLAB sessions: Laboratory 1(Type help plot for a list of the different line-styles, and help legend for help on how to add alegend.) Include both the M-file and the plot in your report.NOTE: the only output of the function file should be the graph of the three curves. Make sureyou use enough points so that the curves are nice and smooth.6. (a) Enter the function f(x; y) = x3 +yexx + 1as an inline or anonymous function (see page 6).Evaluate the function at x = 2 and y = 1.(b) Type clear f to clear the value of the function from part (a). Now write a function M-filefor the function f(x; y) = x3 +yexx + 1. Save the file as f.m (include the M-file in your report).Evaluate the function at x = 2 and y = 1 by entering f(2,-1) in the command window.c2011 Stefania Tracogna, SoMSS, ASU