Considers various basic features of Matla

Maunder, Rob

English

EdShare

ELEC6021 An Introduction to MATLAB: Session 1Lecturer: Rob Maunder rm@ecs.soton.ac.ukShort  for  “Matrix  Laboratory”,  MATLAB  is  a  proprietary  numerical  computing environment.   To  perform  numerical  calculations  MATLAB  uses  a  scripting or interpreted programming language, meaning that the source code is not compiled such as in  the  C  programming  language  but  rather  is  interpreted  each  time  the  program  is executed.   MATLAB  is  commonly  used  in  industry  and  academia  for  its  ease  of programming and vast libraries of functions.  This session will guide you through the basics  of  the  programming  language  and  does  not  cover  everything.   On  Windows systems, MATLAB can be started through the Start button.  Once opened, the MATLAB environment should look similar to Figure 1. Figure 1 The MATLAB environment.1. Finding your way aroundTASK 1 Identify components of the environment: Command History Window  Command Window  Start Button and Launch Pad Help Browser Current Directory Browser Workspace Browser1TASK 2 Documentation on MATLAB is readily available  via the  Help Browser,  the Internet, or the help command.  At the Command Window type:help helpPress Enter to execute the command.  This command will display text in the Command Window explaining what  help does and how to use it.  The most common use of the help is to find out what a certain command or function does in MATLAB.  2. The BasicsTASK 3  Try some simple operations.  For scalar addition, subtraction, multiplication, and division type in the Command Window:6+29-546*0.58/5Notice  how  MATLAB  always  saves  the  result  in  a  variable  called  ans.   Good programming practice in MATLAB is to never use ans as a variable name.  If a value is assigned to a variable ans is not used.  Try:a = 2b = 5;QUESTION What does the semicolon do?MATLAB also has native support for complex numbers, both i and j have the value of the square root of –1.  So be careful when assigning values to either of these variables, as that will override the default value.  To enter a complex number just type:4.9+j*5.6a+j*b3. Matrix ManipulationMATLAB’s biggest  advantage  is  its  relative  ease  of  matrix  manipulations.   In  other programming languages matrices are dealt with by using loops.TASK 4 Create an array (also referred to as a vector); type in the Command Window:a = [16 4 67 48 9 90]2As shorthand for sequence of numbers you can type:a = 1:10This will give you a sequence of numbers from 1 to 10 with unit interval.  To change the interval type:a = 1:2:10a = 1.45:0.1:2.3a = 0:pi/4:piTASK 5 Now to enter matrices, enter Dürer’s matrix.  The semicolon marks the end of the row.A=[16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1]This matrix is a famous magic square where all numbers in a row, column, or diagonal sum to 34.  NB. MATLAB has a function magic for generating magic squares.  Use the sum function by typing:sum(A)This is a row vector of the sums of the columns of A.  To transpose A to get the sum of rows type:A’sum(A’)’A column vector containing the row sums is displayed.  Sum the diagonal elements with:sum(diag(A))Using the  help command, look up how the  fliplr function works to sum the anti-diagonal elements of A.  Now, type the following:A(1,4)Subscripts This is specific indexing, such that it returns the value of row 1 column 4 of the matrix A.  But it is essential that you master the colon, since this allows MATLAB to use its own internal routines which run much faster than loops using specific indexing and gives rise to much more readable code.A(1:n,k)displays the first n elements of the k-th column of A.  So:3sum(A(1:4,4))computes the sum of the 4th column.  The colon by itself refers to all the elements in a row or column.sum(A(:,end))computes the sum of the elements of the last column of A.  To reorder the columns in A type:A(:,[4 3 2 1])orA(:,4:-1:1)Both code statements say that for each of the rows of A reorder the elements in the order 4 3 2 1.4. Array and Matrix OperatorsIn other programming languages matrix maths operations are commonly performed using loops.  But in MATLAB operations are written intuitively.  The operators as you would expect are shown in Table 1  Table 1 Array and Matrix Arithmetic OperatorsOperator Matrix Array (element-by-element)Addition + +Subtraction - -Multiplication * .*Right Division / ./Left Division \ .\Power ^ .^Transpose ‘ .‘Brackets (are used to specify the order of evaluation)( ) ( )TASK 6 Type the following:x = (0:0.1:1)*piy = linspace(0,pi,11)This generates two identical vectors.  Try some scalar operations like:4z = x-2w = z-xVectors in MATLAB are either single row or single column matrices.  Create the vector c and find the outer and inner products by typing:c = 1:5c’*cc*c’Now, verify that:W = rand(4)b = rand(4,1)x = W\bsolves the set of matrix equations Wx = b.TASK 7 For array manipulations MATLAB also offers element-by-element operators to carry out common tasks shown in Table 1.  Again with vector c.c = 1:5c’*cc’.*cDon’t  panic!   The  error  is  a  common error  you  may come across  when using array operators.  Instead use the following code:c.*cNotice the difference between matrix and array operators?5. FunctionsFunctions perform some processing on a number of inputs, in order to provide a number of  outputs.  So  far,  we've  seen  a  few  examples  of  functions,  namely  sum,  diag, linspace and rand. Matlab has lots of built-in functions, which allow you to do all sorts of things.TASK 8 For example the rand function can take two inputs and returns one output. It outputs a matrix of random numbers, where the first input specifies how many rows the matrix should have and the second input specifies how many columns. You can try this out using the commandrand(3,4)5which should output a 3×4 matrix of random numbers.The  rand function can also take only one input.  In this case, it  will  output a square matrix of random numbers, having dimensions that are specified by the input. You can see this for yourself by inputting the commandrand(3)Also, the  rand function can take no inputs at all! In this case, it will output a single random number. Try this by inputting the commandrandTASK 9 The  eig function is an example of a function that can return two outputs. It outputs the eigenvalues and eigenvectors of the input matrix. You can try this out for yourself using the commandsX=rand(3)[V,D]=eig(X)You should find that  X×V = V×D. Note that  a function's  outputs can be stored in  a variable by using the equals operator. When there is more than one output, the variables should be listed inside square brackets, like in the example above.If you're ever wondering what one of Matlab's built-in functions does, you can find out using the help command. Try this for the rand function by typinghelp randTASK 10 Use Matlab's  help command to find out about each of the other functions we've used so far.6. Using FilesTASK 11 Outside of MATLAB, use a text editor (such as Notepad) to create a file called fred.dat containing a data array looking like:1 2 3 45 6 7 88 7 6 54 3 2 1Save the file in the MATLAB work directory and then enter in the MATLAB Command Window:6load fred.datLook at the variable  fred.  There are many commands and functions for saving and retrieving data such as fread, fprintf, save, etc.TASK 12 So far, you have been typing commands into the Command Window.  This is not a good way to realise larger programs.  Scripts and functions can be stored in files with a .m extension.  You can create these using the MATLAB editor, which is started from File-> New-> M-File or by clicking the New M-File button found on the top left-hand corner.  Code written as a script, when executed is run as if typing each line into the Command Window yourself.In the Editor Window type the following previous example code:c = 1:5 %Generates a vector of numbers 1 to 5c’*c %Inner product by matrix operatorsc’.*c %This throws an error due to dimension mismatchc.*c %Multiply element-by-elementNote that the % sign in MATLAB is for inserting comments (these are ignored by the interpreter).  It is good programming practice to write comments so that others and you are  aware  of  what  the  code  does.   Save  the  file  with  the  name  script1.m in  the MATLAB work directory.  To call the script, click the run button on the Editor Window or type the script's name in the Command Window, in this case script1.TASK 13 From now on type code into M-files as this makes life easier, unless otherwise instructed.  If Matlab does not have a built-in function to do what you want, you can write your own function! To create a function in MATLAB, you write an M-file having a first line like:function [y, z] = funcname(x, i)The word function is a MATLAB keyword that declares a function.  The variables in square brackets [ ] are the variables returned by the function.  Brackets can be omitted if there is only one return variable.  The variables in round brackets ( ) are input argument variables passed to the function and  funcname is the function’s name used to call it, which should be the same as that of the M-file. So, a function called function1 should be saved in the M-file function1.m.Differences between scripts and functions:1. Scripts  are  lines  of  code  that  you  would  type  into  the  Command Window, so variables are created in your workspace.2. Functions have local variables (unless otherwise declared global by global), so they are destroyed once the function is returned.7Create  a  function  called  logicarr that  uses  logical  arrays.  These  are  useful  for writing efficient codes.  You can gain understanding of what they are and how they work by  studying  the  following  sequence  of  commands  and  the  plots  generated.   Their meanings are later described in Table 2.function z = logicarr(x,y)figure(1);plot(x,y);z = (y >= 0).*y;z = z + 0.5*(y<0);z = (x <= 3*pi/2).*z;figure(2);plot(x,z);At the Command Window create the arrays:x = linspace(0,2*pi,30);y = sin(x);And call the function:logicarr(x,y);By not assigning the function to a variable the returned array is assigned to ans.7. Plotting GraphsA major benefit of the MATLAB environment is its useful graph plotting capabilities.TASK 14 Creating a plot.  Use the colon operator to make a vector having values in the range (0, 2π) and plot the sine of this vector.x = 0:pi/100:2*pi;y = sin(x);plot(x,y);Remember the semicolons stop results printing to screen which makes the program run faster.  Now label the axes and add a title.xlabel(’x = 0:2\pi’);ylabel(’y = sin(x)’);title(’Plot of the sine function’,’FontSize’,12);Find out how to print the figure.8TASK 15 Multiple data sets on one graph.  Create another two functions dependent on x and put them in the plot function:y2 = sin(x-0.25);y3 = sin(x-0.5);plot(x,y,x,y2,x,y3);To make it easier to identify the different lines use a legend:legend(’sin(x)’,’sin(x-0.25)’,’sin(x-0.5)’);TASK 16 Specify line colours and stylesThe plot function allows for many customisations such as line colours, line styles, etc. Use the help command to understand what the following example does.x1 = 0:pi/100:2*pi;x2 = 0:pi/10:2*pi;plot(x1,sin(x1),’r:’,x2,sin(x2),’b+’);TASK 17 Imaginary and complex dataWhen plot is used on complex data the imaginary part is ignored, except when plot is given  a  single  complex  argument.   For  this  case  the  real  part  is  plotted  versus  the imaginary part.  So plot(Z) is equivalent to plot(real(Z),imag(Z)).t = 0:pi/10:2*pi;plot(exp(j*t),’-o’);axis equalTASK 18 Adding multiple graphs to a plotBy using the following command you can add more graphs to a plot as shown in the proceeding example.hold onThis example code overlaps a contour plot with a psuedo colour plot:[x,y,z] = peaks;pcolor(x,y,z)shading interphold oncontour(x, y, z, 20,’k’)hold offTASK 19 Figure windows9To have more than one figure use the  figure function every time you wish to have another Figure Window before plotting it.  To create another figure but control its figure number, use:x1 = 0:pi/100:2*pix2 = 0:pi/10:2*pifigure(1)plot(x1,sin(x1),’r:’);figure(2)plot(x2, sin(x2),’b+’);TASK 20 Multiple plots in one figureUse the subplot function for this:t=0:pi/10:2*pi;[X,Y,Z] = cylinder(4*cos(t));subplot(2,2,1);mesh(X);subplot(2,2,2);mesh(Y);subplot(2,2,3);mesh(Z);subplot(2,2,4);mesh(X,Y,Z);grid lines can be set using:grid ongrid off8. Flow ControlTASK 21 User input and formatted output into your programs.  To prompt a user for an input:z = input(’enter a value for z’);The user can input (for instance) [1,2,3; 4,5,6] which is assigned to the variable z.  For formatted output look up how the  disp will  generate  output more readable than the default and fprintf will produce user-specified formatted output.TASK 22 The if statement in MATLAB follows the format of:if condition 110statement 1elseif condition 2statement 2elsestatement 3endThe  conditions  are  expressions  made  from  using  logical  and  relational  operators  as described in Table 2 and Table 3.  Try this example code:temperature = input(’enter a temperature ’)if temperature >= 90  disp(’It’s getting hot’);elseif temperature < 90 & temperature > 50  disp(’This is just right’);else  disp(’I think I will get my coat’);end Table 2 Relational OperatorsOperation OperatorEqual to ==Not equal to ~=Less than <Greater than >Less  than  or equal<=Greater  than  or equal>=Table 3 Logical OperatorsOperator Description&& Returns  logical  1  (true)  if  both  inputs evaluate  to  true,  and  logical  0  (false) otherwise.|| Returns logical 1 (true) if either input, or both, evaluate to true, and logical 0 (false) otherwise.To test for equality between scalars if A == B works, but for matrices tests where the two matrices are equal, resulting in a matrix of zeros or ones.  The proper way to test for equality between two variables is if isequal(A,B).  Functions that are helpful for reducing  the  results  of  matrix  comparisons  to  scalar  conditions  for  use with  if are: isequal and isempty.TASK 23 The switch statement follows the format below.  11switch switch_expr case case_expr1    statement 1  case case_expr2    statement 2  otherwise    statement 3endThe temperature comparator can also be described by:switch (1)  case temperature >= 90    disp(’It’’s getting hot’);  case temperature > 50    disp(’This is just right’);  otherwise    disp(’I think I’’ll get my coat’);endOnly the first matching case is executed.  There must always be an  end to match the switch.TASK 24 The for loop.  This example repeats a statement a predetermined number of times:function factx = fact(x)%this  function  accepts  a  scalar  input  and  return  its %factoriala=1;for k=1:x  a=a*k;endfactx=a;Note the matching end at the end.  N.B.  Do not use for on all the elements of an array and do not use loops when you can use the colon notation.  MATLAB is an interpreted language, not compiled like C and consequently long  for loops and complex logic is generally slow to execute. Check that this function operates correctly when you call it from the Command Window.TASK 25 The while loop repeats a group of statements an indefinite number of times under the control of a logical condition.  Here is a complete program that finds the roots of a polynomial by interval bisection.a=0; fa=-Inf;12b=3;fb=Inf;while b-a > eps*b  x=(a+b)/2;  fx=xˆ3-2*x-5;  if(sign(fx) == sign(fa))    a=x; fa=fx;  else    b=x;fb=fx;  endendxTwo statements that you should also be aware sometimes useful for writing loops are the continue and  break statements.  The  continue statement passes on to the next iteration of a for or while loop.  The break statement is an early exit from a for or while loop.9. Further ReadingThere  are  many books on MATLAB and how to use  MATLAB certain  fields.   The University’s library has quite a few.  There is also a vast amount of documentation online on MATLAB functions.A first place to look online would be:http://www.mathworks.com/access/helpdesk/help/techdoc/matlab.html or a popular starting point if the help command is not adequate is www.google.com. 13

Laboratory on Matlab

Considers function handles, matrix manipulation, 3D plots and programmin

ELEC6021 Research MethodsMatlab Session 2In the first session, we looked at how to write our own Matlab functions,among other things. This session poses a number of problems and asks youto write Matlab functions to solve them. However, before we attempt thoseexercises, let’s remind ourselves about writing and running Matlab functions.Writing and running functionsLet’s start by writing a Matlab function to perform the mathematical func-tion f(x) = x2. Use the Matlab text editor to create a new M-file and typethe following commands into it.function y=my_function(x)y=x^2;Save your M-file as my function.m. You can run your function by typingthe following command in the main command window.my_function(3)Matlab should display a value of 9, which is f(3) = 32.Function handlesThe name of your function can be assigned to a string variable using thecommandf=’my_function’;This string variable is referred to as a function handle. The built-in Matlabfunction feval can be used to evaluate the function specified by a functionhandle. You can try this using the command1feval(f,4)This time, Matlab should display a value of 16, which is f(4) = 42.The feval function is usually used inside functions that take functionhandles as arguments. An example of such a function is fplot. You can getthis to plot your function using the following command.fplot(f,[0 4])Here, the argument [0 4] tells fplot to plot the function for inputs in therange 0 to 4.Note that using a function handle and the fplot function has saved usa lot of effort. The commands required to generate the same plot withoutusing a function handle are as follows.x=0:0.08:4;for index=1:length(x)y(index)=my_function(x(index));endplot(x,y);For further information on feval and fplot, type the following commandsat the main command window.help fevalhelp fplotExercise 1 Random integer generatorsIn this exercise we shall consider some random integer generators. Differentrandom integer generators may output different integers with different prob-abilities. For example, a six-sided dice is a random integer generator thatoutputs the integers 1 to 6, each with a probability of 1/6. The probabilityof a six-sided dice outputting any integer outside the range 1 to 6 is zero.By contrast, a ten-sided dice is a different random integer generator thatoutputs the integers 1 to 10, each with a probability of 1/10.Some random integer generators may output different integers with differ-ent probabilities. For example, the act of rolling two six-sided dice, dividingthe results by two, rounding up to the nearest integers and summing thetwo resultant integers together can be considered to be a random integergenerator. This outputs the integers 2 and 6, each with a probability of 1/9,the integers 3 and 5, each with a probability of 2/9 and the integer 4 with aprobability of 3/9.2You can write a Matlab function that will output the probability P (s, k)of an s-sided dice outputting the integer k as follows% Function that determines the probability p of an s-sided dice% outputting the integer kfunction p=uniform_dist(s,k)% Check that the inputs are integers and in the correct rangeif round(k) ∼= kerror(’k should be an integer!’);endif round(s) ∼= serror(’s should be an integer!’);endif s < 1error(’s should be positive!’);end% Calculate the integer probabilityif k >= 1 && k <= sp=1/s;elsep=0;endNote that this function will generate an error if either s or k is not an integeror if s has a value less than 1. It is always a good idea to include someerror checking in your functions so that they only ever return meaningfulvalues. This function also includes some comments. In general, commentsare invaluable when you want to share your code with a friend; without anycomments, your friend would have no idea what your Matlab code does!The function listed above is called uniform dist, since it describes adiscrete uniform probability distribution, in whichP (s, k) ={1/s if 1 ≤ k ≤ s0 otherwise(1)where s ≥ 1 and k are integers.Your first task is to write similar functions for the discrete geometric andzeta probability distributions, remembering to include some error checking.In the discrete geometric distributionP (s, k) ={(s− 1)s−k if k ≥ 10 otherwise, (2)3where k is an integer and s > 1 is a real-valued parameter of the distribution,similar to in the discrete uniform distribution described above. Meanwhile,the discrete zeta distribution is defined byP (s, k) ={k−sζ(s)if k ≥ 10 otherwise, (3)where the zeta function ζ is implemented by Matlab’s built-in function zeta,k is an integer and s > 1 is real-valued.The following function determines the sum of the probabilities of theintegers in the vector k values, when using the probability distribution thatis specified using the function handle P and a particular value of s.% Function that determines the sum of the probabilities of the% integers in the vector k_values, when using the probability% distribution that is specified using the function handle P% and the parameter s.function sum = sum_P(P,s,k_values)% Initialise the sum to zerosum = 0;% Accumulate the probabilities of the various integersfor index = 1:length(k_values)sum = sum + feval(P,s,k_values(index));endFor each of your probability distributions and for a range of values for s > 1,use the sum P function to verify that no matter what vector of k values youuse, the sum of the probabilities is never greater than 1. This demonstratesthat our probability distributions are valid.The following function generates random integers using the probabilitydistribution that is specified using the function handle P and a particularvalue of s.% Function that generates random integers using the% probability distribution that is specified using the% function handle P and a the parameter s.function k = random(P,s)% Generate a random number in the range 0 to 1, taken from a% continuous uniform distributiona = rand;% Initialise the integerk=1;4% Find the integer value having the probability that matches% the random numberwhile a > feval(P,s,k)a = a - feval(P,s,k);k = k + 1;endYour next task is to figure out how this works. To help you out with this,Figure 1 provides a flow chart for the random function. Try generating abunch of random integers using each of your probability distribution func-tions. Note that to start with, s = 2 is a good choice for the parametervalues of the discrete geometric and zeta distributions. See what happens tothe random integers generated by these distributions as you use larger andlarger values for s.yesk ← k + 1a← a− P (s, k)k ← 1a← U(0, 1)P and ska > P (s, k)noFigure 1: Flow diagram for the random function. Here, U(0, 1) generates arandom real value from a continuous uniform distribution over the range 0to 1.It is easier to understand what is happening to the random integers ass changes by plotting P (s, k) versus k. Your final task in this exercise is towrite a Matlab function that has a similar functionality to fplot, but whichis specifically designed to plot P (s, k) versus k for a specified probability5distribution and value of s. The first line of your function should thereforebefunction plot_distribution(P,s,k_values)where P is a function handle and k values is a vector that specifies whichvalues of k to use. You may find that drawing a flow diagram makes it easierto work out what loops are required. To help you out with this exercise, theoutput you should get for plot distribution(’geometric dist’,2,1:10)is exemplified in Figure 2. Try to get your function to automatically generateplots that look like Figure 2, without having to manually insert titles andaxis labels in the plot window.1 2 3 4 5 6 7 8 9 1000.050.10.150.20.250.30.350.40.450.5geometric_dist s=2kP(k,s)Figure 2: Discrete geometric probability distribution for s = 2.Use your plot distribution function to see how the probability distri-butions change as you vary the value of s. You should find that this explainsthe observations you made about the random integers generated by the dis-crete geometric and zeta distributions, when you used larger and larger valuesof s.6Finally, you may like to write a Matlab script that repeatedly calls therandom function and collects statistics about how many times each integervalue is generated. You can then compare these statistics with the probabil-ities shown in the plots generated using your plot distribution function.In this way, you can verify that the random function generates each integervalue with the correct probability.Exercise 2 Sudoku checkerA sudoku puzzle consists of a 9×9 grid of numbers, most of which are initiallyunknown to the player. The objective is to fill in the missing numbers byexploiting the following three rules:• each of the integers in the range 1 to 9 must appear in each row of thegrid exactly once;• each of the integers in the range 1 to 9 must appear in each column ofthe grid exactly once;• each of the integers in the range 1 to 9 must appear in each 3× 3 boxin the grid exactly once.An example of a sudoku puzzle is shown in Figure 3.555337776 1 99 8 68 64 8 3 12 66 2 84 1 98 9555337776 1 99 8 68 64 8 3 12 66 2 84 1 98 94 6 8 9 1 27 2 3 4 81 3 4 2 5 75 9 7 1 4 22 6 5 7 91 3 9 4 8 59 1 5 3 7 42 8 7 6 33 4 5 2 6 1(b)(a)Figure 3: (a) A sudoku puzzle and (b) its solution.Your task in this exercise is to write a Matlab function that checkswhether or not a sudoku solution adheres to the three rules listed above.The first line of your function should befunction valid = check_sudoku(solution)7where solution is expected to be a 9 × 9 matrix of integers and valid isgiven a value of 1 when the solution adheres to the above-listed rules and avalue of 0 otherwise.Remember to include comments and some error checking in your function.This should check that solution is a 9 × 9 matrix of integers in the range1 to 9. You should also try to make your function as elegant and efficientas possible. You may find that Matlab’s built-in function unique is usefulin this exercise. You can find out how to use this by using Matlab’s helpfacility.Once you have written your function, test its functionality using the datafiles that can be downloaded fromhttp://www.ecs.soton.ac.uk/notes/elec6021/matlab/sudokuYou can perform the tests by saving the data files to your Matlab workingdirectory and using commands like the following ones.load solution1.datcheck_sudoku(solution1)Note that some of the data files will test the error handling of your function.Exercise 3 - Plotting elevation dataSouthampton is surrounded by many great places to visit. These includeBournemouth, the New Forest, Salisbury, Winchester, Portsmouth and theIsle of Wight. Some of these are shown in the satellite imagery of Figure 4. Inthis exercise, we will use Matlab to plot the elevation of the area surroundingSouthampton.The elevation data for the area surrounding Southampton is stored in thefile elevation.dat, which can be downloaded fromhttp://www.ecs.soton.ac.uk/notes/elec6021/matlab/elevationThis text file can be loaded into Matlab using the load command, like inExercise 2. The resultant matrix elevation contains the elevations (in me-tres above sea-level) of positions arranged in a grid at intervals of 75 m, overan area of 50 km by 50 km.We need two vectors to describe the co-ordinates of the data points: onefor the East-West co-ordinates and one for the North-South co-ordinates.The length of the East-West vector should be equal to the number of columnsin the elevation matrix. You can therefore obtain this vector using thecommand8Figure 4: Satellite imagery of Southampton, Portsmouth, the New Forest, theSolent and the Isle of Wight. The label ‘A’ indicates the Highfield Campusat the University of Southampton. c©Google 2009.x = (0:size(elevation,2)-1)*75;where 75 is used because the elevations are recorded for positions that arespaced 75 m apart. Use Matlab’s help command if you’re not sure whatthe extra argument for the size function is for. Similarly, the length of theNorth-South vector should be equal to the number of rows in the elevationmatrix. Adjust the argument of the size function in the command above toobtain the North-South vector y.You can generate a 3D surface plot of the elevation data using the com-mandsurf(x,y,elevation,’LineStyle’,’none’);Here, ’LineStyle’,’none’ is used to remove the grid that would otherwisebe drawn over the surface plot, obscuring the colours from view.You can give appropriate lengths to the three axes of the plot by usingthe commandsxlim([0,max(x)])ylim([0,max(y)])zlim([-1000,1000])axis square9You can use the ‘Rotate 3D’ tool in the plot window to view it from differentangles. Note that in order to make the plot more exciting, the commands Iprovided above make the hills look like mountains! You can obtain a morerealistic view by typing the commandaxis equalI think you’ll agree though that this makes the plot look too boring! Youcan undo this by typing the following command again.axis squareYou can find out more about the axis command by typinghelp axisRemember that it is always important to annotate the axes of your plotsand to include the units. You can do this using the commandsxlabel(’Easting (m)’)ylabel(’Northing (m)’)zlabel(’Elevation (m)’)If you like, you can also include a key to explain the various colours by usingthe colorbar command.Next, let’s draw a contour plot for the elevation data. You can do thisusing the commandcontourf(x,y,elevation)Remember to set the lengths of the axes and to annotate them, as describedabove.Next, we can merge the elevation data with the satellite imagery of Fig-ure 4 to generate a 3D representation of the area surrounding Southampton.Start by downloading the JPEG satellite.jpg from the URL providedabove. You can load this into Matlab using the commandsatellite = imread(’satellite.jpg’);You can display the image by typing the commandimshow(satellite)Notice that the image shows the exact same area as the elevation data; thistook quite a lot of trial and error to get right!You can merge the satellite imagery and the elevation data by using thecommand10warp(x,y,flipud(elevation),satellite)This command maps an image onto a surface plot. Here, the flipud functionis required because the rows of an image are numbered from top to bottom,not from bottom to top like the y axis of a plot. You can find out more aboutthe flipud command using Matlab’s help function. Again. remember toset the lengths of the axes and to annotate them, as described above.Try using the ‘Zoom In’ and ‘Rotate 3D’ tools in the plot window tosee what the Isle of Wight looks like when you’re stood at the top of thetallest building on the Highfield Campus at the University of Southampton.Alternatively, you can change the viewing angle using the commandview(az,el)where az is the azimuth in the range 0 to 360 degrees and el is the elevationin the range 0 to 90 degrees. Also, you can zoom in using the commandzoom(2)and zoom out using the commandzoom(0.5)Finally, you can draw the contours into the same plot by using the com-mandshold oncontour3(x,y,flipud(elevation))Again, you may like to include a key by typing the colorbar command.Pretty neat, huh?Exercise 4 - SortingIn this exercise, your task is to write a Matlab function that will sort the ele-ments of a vector into ascending order. There are lots of different algorithmsthat will achieve this and many are described on Wikipedia athttp://en.wikipedia.org/wiki/Sorting_algorithmFeel free to pick whichever algorithm you like, but notice that different algo-rithms have different speeds, memory requirements and algorithmic complex-ities. You should draw a flowchart for your algorithm to help you visualiseits operation.The first line of your function should be11function out = my_sort(in)where in is expected to be a vector and out should contain the same elementsas in, but sorted into ascending order. Try to make your function as elegantand efficient as possible and remember to include some useful comments.You can check that your algorithm works by using the Matlab commandissorted(my_sort(rand(1,10000)))which will use your algorithm to sort a vector of 10000 random numbers andreturn a 1 if it is successful or a 0 if not.You can test the efficiency of your algorithm using the commandtic; my_sort(rand(1,10000)); tocThis will tell you how long it takes for your algorithm to sort a vector of10000 random numbers. You may like to compare this to the efficiency ofMatlab’s built-in sort function by using the commandtic; sort(rand(1,10000)); tocYou can find out about the tic and toc commands by using the commandshelp tichelp tocRob Maunderrm@ecs.soton.ac.ukSeptember 200912

Laboratory on Matlab

Abstract

Similar works

Full text

Available Versions

EdShare

EdShare