Matlab Beginning Exercises
R. J. Braun Department of Mathematical Sciences University of Delaware Version of September 5, 2008
1
Introduction
This collection of exercises is intended to help you start learning Matlab. Matlab is a huge package with many capabilities, but it is easy to use on many levels. The text indicated by the numbers will be Matlab commands that you can input to the Matlab prompt. text will indicate quantities that you need to replace with in a Matlab command. Text to the right of a % symbol are comments and need not be typed at the prompt unless you wish to have the comment in a diary or other le. In nearly all cases, no output will be shown; thats for you to nd!
2
Getting Started
In the computer classroom, you can start Matlab by clicking on the icon. In Windows, you can double click on an icon. In Linux, you can open Matlab by opening a terminal window and then typing the appropriate command. Typing the command matlab followed by <enter> will start Matlab using its window manager. In this mode, you can nd helpful pull down menus across the top of the window (or desktop). The default conguration has one main command window, and two other windows that allow access to the workspace, command history, current directory and launch pad. Typing the command matlab -nojvm runs Matlab without the window manager. This can avoid some annoying bugs in the window manager that may appear (particularly in linux) on starting up. This will give only the command window; plots will appear in separate graphics windows in both modes. To exit, use the pull down menu under File or type the command exit in the command window.
3
Scalar arithmetic
1. 3*2^4 2. (3*2)^4 3. 3-2^4 4. 3^4-3 5. 8/2^4 6. 2^4\8 7. 8^4/2 % same as previous! two different divisions, \ and / % parentheses have highest priority
Matlab can be used as a calculator. Some of the problems indicate the precedence of arithmetic operations.
Precedence of arithmetic operations is: (1) parentheses, (2) exponentiation, (3) multiplication and division, and (4) addition and subtraction. Calculations proceed from left to right on the line and equal precedence is settled in this way.
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4
Vectors
It will typically be that our basic unit for computing will be a vector. We need to be able to create and manipulate them. Save yourself some typing: use the arrow keys to recall commands and to edit them!
4.1
Creating vectors
% create a row vector (spaces) % colon generates list; default stride 1 % start : stride : stop specifies list % same as last; semicolon suppresses output
1. x = [3 4 7 11] 2. x = 3:8 3. x = 8:-1:0
4. xx = [ 8 7 6 5 4 3 2 1 0]; 5. xx 6. x = linspace(0,1,11) 7. x = 0:0.1:1 8. y = linspace(0,1); 9. length(x) 10. length(y) 11. size(x) 12. size(y) 13. y(3) 14. y(1:12) 15. y([3 6 9 12]) 16. x 17. z = [ 1+2*i 4-3*i ] 18. z 19. z. 20. 3*[1 2 5]
% display contents % generate vector automatically % same thing % note semicolon!
% access single element % access first twelve elements % access values specified in a vector! % transpose
% note difference in transposes! % factor replicated, multiplies each element
Matlab labels the arrays (vectors and matrices) beginning with 1; this will be an important programming detail on more than one occasion.
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4.2
Help with vectors and other things
Matlab has extensive online help; try these commands to get help at the command line. If the window manager is running in Matlab, you can also get browser-based interactive help. Try both. 1. help help 2. help length 3. help size 4. help linspace 5. help logspace 6. help clc 7. help clear 8. help who 9. help whos Note that Matlab help is also available in html format by using the menu bar at the top of the desktop. The help is more comprehensive in the browser-based help, but it is sometimes harder to nd if you already know the command name. % try Matlabs extensive help
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5.1
Mathematical functions and plotting
Elementary functions in Matlab
1. help elfun 2. Use the following few commands (a script) to make a plot. The evaluation of v = cos(u) in Matlab creates a vector whose elements are v(k) = cos(u(k)) where k = 1, 2, ...n. n = 11; u = linspace(0,2*pi,n); v = cos(u); % all function evaluations done at once! plot(u,v) 3. Once you have typed these to the Matlab prompt, you can recall and edit them one at a time, or you can save time by not having to retype them in the following way. You can save these commands in a le and execute the le. Click on the blank sheet icon at the upper left in the Matlab desktop; using the Matlab editor, you can type the commands or copy them to the editor and then save it to lename .m, where lename is the name you supply. (Avoid reserved words in Matlab, e.g., plot; see below for more info. Suggestion: call it rstplot.m.) You can execute this newly created le by typing the lename without extension (that is, without the .m) to the Matlab prompt. This collections of commands is essentially a small Matlab program, and this kind of le is called a script le. 4. The last plot had a crude curve representing the cosine function; repeat the last example with n=101 to improve the curve. 5. Add these commands to clean up the plot by adding axis labels and a title. The text inside the single quotes is a string which we intend to be the labels, etc.
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xlabel(u); ylabel(v); title(v = cos(u)); Further improvements in plotting appearance will be discussed later; the command help plot will be very helpful.
5.2
Script and function les
After this script le has run, if you type u at the prompt, you will see the contents of this array. Script les leave all variables that were created in the workspace. This can be advantageous during debugging and writing of Matlab code. There is a dierent type of le, called a function le, that creates its own work space, does calculations, returns values, then wipes out the workspace. This is analogous to what occurs in many programming languages. Functions generally execute faster than scripts and we will typically use them in this course. If you insert the line function rstplot at the beginning of the script le you made, execute clear all and close all, then execute the le rstplot.m, you will see the plot again, but typing u at the prompt will not give you its contents, since there will be no u in the workspace. We shall see how to return results from functions many times during the semester. It may be advantageous to develop a function as a script rst, and then convert it to the desired function when it is working as expected.
5.3
Reserved words in Matlab
The following words are already used in Matlab and they must be avoided: for, end, if, while, function, return, elseif, case, otherwise, switch, continue, else, try, catch, global, persistent, break. (You can use similar words by capitalizing one or more letters if your system is case sensitive.) the following variable names are special in Matlab and you should avoid changing them unless you are very careful about what you are doing: ans, beep, pi, eps, inf, NaN, nan, i, j, nargin, nargout, realmin, realmax, bitmax, varargin, varargout. Try the following to see the distinction between the two cases just described. 1. end = 1 2. End = 1 3. pi = 3 4. pi 5. clear pi 6. pi % back to the present! % you should get an error % could get error, depending on system % forgetting thousands of years of progress!
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Diary les
We can record what was typed and the text-based output during a Matlab session in a le called a diary le. This recording of a session will be useful for homework and projects. It is not a program and the Matlab commands cant be run in the desktop like a script or function le can. 1. e = [ 1 2 3; 4 5 6; 7 8 0] 2. f = [9 8 7; 6 5 4; 3 2 1] 3. g = [1 2 3 4; 5 6 7 8; 9 10 11 12] 4. h = [1 1 1 1; 2 2 2 2; 3 3 3 3]
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5. diary ThisClassIsGreat 6. g*h 7. e*f 8. diary off
% puts diary in file called ThisClassIsGreat % more about arithmetic ops later!
% diary file closed
9. dispWill you see this? % not this line! 10. diary ThisClassIsGreat 11. f*e 12. diary % diary toggles with no additional specification % reopening it will append additional work
(Note that in the computer classroom, this le is by default written in ~/desktop.)
7
Output
We will have the need to produce legible output; lets see how to do that. First, we will do some very basic output commands; then, we need to see how to repeat commands in Matlab. Finally, we will do a little nicer output.
7.1
Basic output
% inside quotes: verbatim
1. disp(The disp command writes text.) 2. t = logspace(-3,6,10); 3. x = 0.1234567890123456*t; 4. format short 5. t 6. x 7. format long 8. x 9. format long e 10. x 11. format short e 12. x 13. format short g 14. x 15. format long g 16. x
17. disp(sprintf(The 10th element of x is x(10) = %6.3f,x(10)))
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18. disp(sprintf(The 10th element of x is x(10) = %10.3f,x(10))) 19. disp(sprintf(The 10th element of x is x(10) = %12.3f,x(10))) 20. disp(sprintf(The 10th element of x is x(10) = %8.3e,x(10))) 21. disp(sprintf(The 10th element of x is x(10) = %8.3e,x(10))) 22. disp(sprintf(The 10th element of x is x(10) = %8.12e,x(10))) 23. disp(sprintf(The 10th element of x is x(10) = %24.15e,x(10))) 24. disp(sprintf(The 10th element of x is x(10) = %1.4g,x(10))) 25. disp(sprintf(The 10th element of x is x(10) = %16.12g,x(10))) 26. disp(sprintf(The 10th element of x is x(10) = %24.15g,x(10)))
7.2
Repeating Commands I
The for loop repeats the commands that are inside of it. The basic structure is for <list-of-values> <statements-to-be-done> end Use this structure to repeat output, or do calculations 1. % for loop now for k = 1:5 disp(sprintf(%8.5e,x(k))) end 2. % for loop doing a calculation total = 0 for k = 1:10 total = total + x(k); disp(sprintf(Total for k=%2.0f is %15.8g,k,total)) end 3. sum(x) % built in matlab command sums components of vector
Here the command sum can replace the loop; it adds up the elements of a vector.
7.3
Creating a table
1. Try this script to make a neat table. disp( ) % a blank line disp( k t x ) disp(----------------------------) for k=1:10 disp(sprintf(%2.0f %7.3g %7.7e,k,t(k),x(k))) end 6
8
More about working with vectors and matrices
We need more tools to eciently manipulate vectors and matrices in Matlab.
8.1
Vector and matrix arithmetic
Well now explore making matrices and doing arithmetic with them. There are some new arithmetic operations that you did not see before in linear algebra, but they will be useful nonetheless. 1. g = [1; 2; 3; 4] 2. g = [1 2 3 4; 5 6 7 8] 3. g - 2 4. 2*g-1 5. g = [1 2 3 4; 5 6 7 8; 9 10 11 12] 6. h = [1 1 1 1; 2 2 2 2; 3 3 3 3] 7. g-h 8. g*h 9. h*g 10. g.*h 11. g./h 12. h.\g 13. g*h 14. g*h 15. e = [ 1 2 3; 4 5 6; 7 8 0] 16. f = [9 8 7; 6 5 4; 3 2 1] 17. e*f 18. f*e 19. q = rand(3) 20. A^2 21. A^2 - A*A 22. A.^2 23. A.^2 - A*A % Not same! % matrix with elements uniformly on [0,1] % raise matrix to a power % verify with definition % square each element % not the same! % NOT the usual matrix multiplication! Elementwise! % elementwise division; denominator below slash % Last two same! % What happened?
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8.2
Standard arrays
Some matrices occur so often that Matlab has standard ways to produce them; this makes programming and use easier. 1. ones(3) 2. zeros(2,5) 3. size(zeros(2,5)) 4. g = [1 2 3 4; 5 6 7 8; 9 10 11 12] 5. size(g) 6. ones(size(g)) 7. eye(4) 8. eye(2,4) 9. eye(4,2) 10. rand(3) 11. b = eye(3) 12. rand(size(b)) 13. a = 1:4 14. diag(a) 15. diag(a,1) 16. diag(a,-2) % Random elements uniformly distributed between 0 and 1 % Identity matrix % All ones % All zeros
8.3
More about creating arrays
% slowest % slow % faster % fastest
1. d = exp(1) 2. d*ones(3,4) 3. d+zeros(3,4) 4. d(ones(3,4)) 5. repmat(d,3,4)
8.4
Selecting and changing parts of matrices
% single element % all rows, second column % second row, all columns
1. g = [1 2 3 4; 5 6 7 8; 9 10 11 12] 2. g(3,2) 3. g(:,2) 4. g(2,:) 5. g(1:2,:) 6. g(2,:) = [1 1 1 1] 7. g(:,2) = [2; 2; 2] % replace row % replace column 8
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A start at matrix algebra
In linear algebra, there was a need to solve systems, nd determinants and inverses, and to construct special matrices; we now scratch the surface of these capabilities in Matlab. 1. A = rand(5) 2. b = rand(5,1) 3. det(A) 4. inv(A) 5. x = inv(A)*b % matrix of coefficients from Ax=b % right hand side % determinant of A % inverse of A % solves system, NOT recommended!
6. size(x), size(b), size(A) 7. x2 = A\b 8. spy(A) 9. D = diag(1:5) 10. spy(D) 11. L = tril(A) 12. U = triu(A) 13. size(U), spy(U) % better way to solve systems % sparsity plot: show nonzero elements
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10.1
More vectors and plotting
Vectorizing function evaluations
We try a dierent rational function than in your textbook. The ideas behind rational functions approximations are discussed in Numerical Analysis by Burden and Faires (Section 8.4), which is on reserve for this course. (Maple can compute the rational function approximation with the ratpoly command.) Consider the rational function 3 1 3 1 5 x + 20 x2 60 x3 , (1) y= 2 1 2 1 + 5 x + 20 x which approximates ex on the interval x [0, 1]. We want to implement function this for plotting and to compare with the exponential function. 1. We could plot the function as follows; I recommend opening the editor in matlab and typing this into the editor and saving the le. The resulting script le can be run in Matlab by typing the lename without the extension .m to the command window prompt. The rst approach is: n = 150; x = linspace(0,1,n); y = zeros(1,n); for k = 1:n y(k) = (1-(3/5)*x(k)+(3/20)*x(k)^2 -(x(k)/60)*x(k)^2)/... (1+(2/5)*x(k)+(1/20)*x(k)^2) end plot(x,y) xlabel(x), ylabel(y); title(Rational function approximation to e^{-x}) 9
This approach makes no use of vectorization; the for loop computes each element one at a time. Note that x2 is computed several times. 2. Now make use of vectorization and try to minimize the amount of calculation of powers of x; you may type this in by modifying your previous script le, or download this script from the course web page. In this case, we % Script file: RatFunPlot % Compare rational function approximation to exp(-x) on [0,1] % f(x) = (1-3x/3-3x^2/20-x^3/60)/(1+2x/5+x^2/20) % n = 150; x = linspace(0,1,n); xsqd = x.^2; num = 1-(3/5)*x+(3/20)*xsqd -(x/60).*xsqd den = 1+(2/5)*x+(1/20)*xsqd y = num./den; plot(x,y,x,exp(-x)) xlabel(x); title(Rational function approximation to e^{-x}) legend(Rational Function,e^{-x},0) In this case, the calculation of x2 over all entries is computed in a single line; this is a vectorized calculation. The numerator and denominator are computed similarly, using the result of the squaring operation xsqd. The quotient is also a vectorized calculation. The legend is located by the number of the corner of the plot when counting starting from the upper right and proceeding counterclockwise (for values 1 through 4); a value of 0 attempts to nd the least conict with data and -1 locates the legend to the right of the plot. More details about the legend can be found in Matlab help.
10.2
More about plotting
x = linspace(0,1); N = length(x); y = 1 - x + x.^2/2 - x.^3/6; exact = exp(-x); figure % create new figure window for plotting y plot(x,y,r-.,LineWidth,2) figure % create new figure window plot(x,exact,LineWidth,2) hold on % causes subsequent curves to be added to current figure plot(x,y,r-.,LineWidth,2) hold off % returns to normal plotting mode title(sprintf(Approximate and exact with %3.0f points,N)) xlabel(x); legend(Exact,Approximate,Location,NorthEast) We can use hold on and hold off to control when to add curves to gures. Note the little trick to write a variables value in the title of the plot using sprintf. You will want to use this during this semester.
1. We can control the number of gures and add to gures that are open. Try this script:
2. close all
% will close all graphics windows 10
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11.1
Conditional Execution and Repeating Commands II
Values and Variables in Matlab
Values that variables may take in Matlab include oating point (double precision, e.g.), character and logical. Floating point numbers are the standard numerical values in Matlab; character values are messages and labels containing letters and numbers. The logical values are True and False, which are indicated in Matlab with a 1 and a 0, respectively. Variables with these values are not arithmetic and are used dierently that numerical array to access parts of arrays; more about this in class.
11.2
Relational operators
Relational operators carry out comparisons or tests and the result is to return logical values True (1) or False (0). The relational operators that are available (and their Matlab form) are: less than (<), less than or equal to (<=), greater than (<), greater than or equal to (>=), equal to (==), and not equal to (=). Note that == is a test between two quantities, while = assigns a value to a variable. We can use these to compare scalars and arrays; try out the following. 1. a = 4; b = -1; c = 1; 2. a > b 3. a < b 4. a >= b 5. mytest = (abs(b) == c) % parentheses optional but clearer % note that the output is a logical value
Things get a little more complicated with arrays. Using the g and h from above, try out the following. 1. g = [1 2 3 4; 5 6 7 8; 9 10 11 12] 2. h = [3 3 4 4; 5 5 6 6; 7 7 8 8] 3. h >= g 4. g == h % note that the output is a logical value
5. bigger = (g >= h) 6. g(bigger) 7. g([0 0 0 1; 1 1 1 1; 1 1 1 1]) % array argument not logical!
We will use some of this kind of thing programming later, but I dont want to elaborate now.
11.3
Logical operators
We can string relational operators together and manipulate logical variables using logical operators. They are (Matlab in parentheses): logical AND (&), logical OR (|) and logical NOT (). The NOT operator just changes the value of the logical variable to the other value. A truth table shows what the AND and OR operators do; it will be given in class. The upshot is that the both parts from an AND operator must be True to return a True; only one of the inputs for the OR operator need be True for it to return True. Continuing with the last set of Matlab commands, try these out. 1. h >= g 2. p = ~(h >= g) 11
3. ~p 4. (g == h) & (g > h) 5. (g == h) | (g > h) 6. g >= h
11.4
The while loop
Sometimes, we want to repeat commands until a condition is satised. If we dont know how many repetitions are required in advance, a while loop is preferable over a for loop. The general structure is while <expression> <commands> end The commands inside the while loop will be repeated as long as the expression returns a value of True, provided that expression returns a scalar. If the expression evaluates to an array, then all of the values that result from it must be True for commands to be evaluated. An example follows; it could be typed into a script le or at the command prompt. In the example, we halve a number until it is so small that the computer cant detect it when added to 1; the last number before this occurs is called machine epsilon and well talk about it more in class. num = 0; myeps = 1; while (1+myeps) > 1 myeps = myeps/2; num = num + 1; end num myeps = 2*myeps eps % matlabs preset variable for machine epsilon The result is that myeps = eps = 252 .
11.5
The if-else-end construct
Sometimes we want to compute one set of commands or another depending on the result of a relational test. There are several forms that this sort of construct can take. 11.5.1 Simplest
The simplest is: if <expression> <commands evaluated if True> end An example follows; given a positive number of pens, a cost is computed and displayed. if (pens >= 0) cost = pens*1.99; disp(sprintf(Cost of %3.0f pens is $ %5.2f,pens,cost)) end
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11.5.2
Intermediate
The next most complicated is: if <expression> <commands evaluated if True> else <commands evaluated if False> end The example now prints a warning for negative numbers of pens. if (pens >= 0) cost = pens*1.99; disp(sprintf(Cost of %3.0f pens is $ %5.2f,pens,cost)) else disp(An invalid number of pens was specified.) end 11.5.3 Most general
The most general form is: if <expression1> <commands evaluated elseif <expression2> <commands evaluated elseif <expression3> <commands evaluated elseif <expression4> <commands evaluated else <commands evaluated end if expression1 is True> if expression2 is True> if expression3 is True> if expression4 is True> if all other expressions are False>
We arent limited to 4 expressions here; we could add more. We could use this to price pens according to quantity, as follows. sale = false; if (pens >= 0) & (pens < 20) cost = pens; sale = true; elseif (pens >= 20) & (pens < 40) cost = pens*0.95; sale = true; elseif (pens >= 40) & (pens <= 100) cost = pens*0.90; sale = true; else disp(not a valid number of pens) end if sale disp(sprintf(Cost of %3.0f pens ...