Matlab is a computer program for people doing numerical computation. It began as a "MATrix LABoratory" program, intended to provide interactive access to the libraries Linpack and Eispack. These are carefully tested, high-quality programming packages for solving linear equations and eigenvalue problems. The goal of Matlab was to enable scientists to use matrix-based techniques to solve problems, without having to write programs in traditional languages like C and Fortran. More capabilities have been added as time has passed (especially graphics support for X windows).
Matlab has been called "the lingua franca for the exchange of software and algorithms," rapidly displacing Fortran from that position due to its interactive interface, reliable algorithmic foundation, fully extensible environment, and computational speed. Matlab is available for many different computer systems. A student edition, for Windows and Macintosh, is available from local bookstores at a very economical price.
This document is intended to help you get started with Matlab, running an X windows system and the Unix operating system. It is intended to be used, while sitting at a computer terminal running Matlab. We will not include output, and many of the commands will fail to convey the intended lesson without it.
No familarity with Linear Algebra is required, although some of what we do cannot be fully appreciated, without it.
Matlab is available in the Math Department at WSU on UNIX workstations.
Our first command will make a record of the session, in a file named "session". [The ">>" is Matlab's prompt, you don't need to type it]. Enter:
>> diary sessionArithmetic uses some fairly standard notation. More than one command may be entered on a single line, if they are seperated by commas.
>> 2+3 >> 3*4, 4^2Powers are performed before division and multiplication, which are done before subtraction and addition.
>> 2+3*4^2The arrow keys allow "command-line editing," which cuts down on the amount of typing required, and allows easy error correction. Press the "up" arrow, and add "/2." What will this produce?
>> 2+3*4^2/2Parentheses may be used to group terms, or to make them more readable.
>> (2 + 3*4^2)/2The equality sign is used to assign values to variables.
>> x = 3 >> y = x^2 >> y/xIf no other name is given, an answer is saved in a variable named "ans."
>> z=2*ans, ansHere z was defined in terms of ans. The result was called z, so ans was unchanged.
To get a list of your variables, use one of
>> who, whosIn Matlab, like C or Fortran, variables must have a value [which might be numerical, or a string of characters, for example]. Complex numbers are automatically available [by default, both i and j are initially aliased to sqrt(-1)]. All arithmetic is done to double precision [about 16 decimal digits], even though results are normally displayed in a shorter form.
>> a=sqrt(2) >> format long, b=sqrt(2) >> a-b >> format shortTo save the value of the variable "x" to a plain text file named "x.value" use
>> save x.value x -asciiTo save all variables in a file named mysession.mat, in reloadable format, use
>> save mysessionTo restore the session, use
>> load mysessionTo find out about this kind of thing, consult the help system. There's even an HTML version!
>> help >> help general >> docFinally, to stop Matlab and logout, use
>> quitWithin Matlab, when using a Unix computer, current saved files (ascii format) from your session can be viewed. For example, to see the file "x.value", type
>> !more x.valueUse of the space bar allows each page of the given file to be viewed in sequence. Use of the "b" key allows the previous page to be viewed.
The names of all of your files from the current (and previous) session(s) will be displayed if you type
>> !lsIf you are in one of the Math Department Labs at WSU, the Unix command "lpr somefile" should send the file with name "somefile" to a printer in that lab. To print this file from within Matlab, type
>> !lpr somefileDo Not Use this command unless you are working in the lab. If you are working remotely (via telnet from some other computer, for example) you can use the Unix mailer to email your file to yourself with a Matlab command of the form
>> !mail firstname.lastname@example.org < somefileFor this command to work, you need to substitute your remote email address for "email@example.com". You can use this command to send yourself a copy of your "session" file if you have used the "diary" command, but you will need to use the command "diary off" before you email to yourself.
There is also a "web" command which can be used in the lab to access the web
>> web http://www.some.web.addressUse "help web" for further details about the "web" command.
Matlab is designed to make matrix manipulation as simple as possible. Every Matlab variable refers to a matrix [a number is a 1 by 1 matrix]. Start Matlab again, and enter the following command.
>> a = [1,2,3; 4 5 6]Note that:
>> a(1,2), a(2,3)It's very easy to modify matrices:
>> a(2,3) = 10The transpose of a matrix is the result of interchanging rows and columns. Matlab denotes the [conjugate] transpose by following the matrix with the single-quote [apostrophe].
>> a' >> b=[1+i 2 + 2*i 3 - 3*i]'New matrices may be formed out of old ones, in many ways. Enter the following commands; before pressing the enter key, try to predict their results!
>> c = [a; 7 8 9] >> [a; a; a] >> [a, a, a] >> [a', b] >> [ [a; a; a], [b; b] ]There are many built-in matrix constructions. Here are a few:
>> rand(1,3), rand(2) >> zeros(3) >> ones(3,2) >> eye(3), eye(2,3) >> magic(3) >> hilb(5)This last command creates the 5 by 5 "Hilbert matrix," a favorite example.
Use a semicolon to suppress output:
>> s = zeros(20,25);This is valuable, when working with large matrices. If you forget it, and start printing screenfuls of unwanted data, Control-C is Matlab's "break" key.
To get more information on these, look at the help pages for elementary and special matrices.
>> help elmat >> help specmatA central part of Matlab syntax is the "colon operator," which produces a list.
>> -3:3The default increment is by 1, but that can be changed.
>> x = -3 : .3 : 3This can be read: "x is the name of the list, which begins at -3, and whose entries increase by .3, until 3 is surpassed." You may think of x as a list, a vector, or a matrix, whichever you like.
You may wish use this construction to extract "subvectors," as follows.
>> x(2:12) >> x(9:-2:1)See if you can predict the result of the following.
>> x=10:100; >> x(40:5:60)The colon notation can also be combined with the earlier method of constructing matrices.
>> a = [1:6 ; 2:7 ; 4:9]A very common use of the colon notation is to extract rows, or columns, as a sort of "wild-card" operator which produces a default list. The following command produces the matrix a, followed by its first row [with all of its columns], and then its second column [with all of its rows].
>> a, a(1,:), a(:,2) >> s = rand(20,5); s(6:7, 2:9)Matrices may also be constructed by programming. Here is an example, creating a "program loop."
>> for i=1:10, >> for j=1:10, >> t(i,j) = i/j; >> end >> endThere are actually two loops here, with one nested inside the other; they define t(1,1), t(1,2), t(1,3) ... t(1,10), t(2,1), t(2,2) ... , t(2,10), ... t(10,10) [in that order].
If necessary, re-enter the matrices
>> a = [1 2 3 ; 4 5 6 ; 7 8 10], b = [1 1 1]'Scalars multiply matrices as expected, and matrices may be added in the usual way; both are done "element by element."
>> 2*a, a/4 >> a + [b,b,b]Scalars added to matrices produce a "strange" result, but one that is sometimes useful; the scalar is added to every element.
>> a+1, b+2Matrix multiplication requires that the sizes match. If they don't, an error message is generated.
>> a*b, b*a >> b'*a >> a*a', a'*a >> b'*b, b*b'To perform an operation on a matrix element-by-element, precede it by a period.
>> a^2, a.^2 >> a.*a, b.*b >> 1 ./ a >> 1./a.^2One of the main uses of matrices is in representing systems of linear equations. If a is a matrix containing the coefficients of a system of linear equations, x is a column vector containing the "unknowns," and b is the column vector of "right-hand sides," the constant terms, then the matrix equation
>> x = a \ bThis can be read "x equals a-inverse times b." To verify this assertion, look at
>> a*x, a*x - bChange b, and do the problem again.
>> b = [1 1 0]' >> x = a\b >> a*x, a*x - bIf there is no solution, a "least-squares" solution is provided [a*x - b is as small as possible]. Enter
>> a(3,3) = 9[which makes the matrix singular] and do those again. [Use the up-arrow, to recall the commands without retyping them].
There is a related problem, to solve x a = b (given a and b), which is done with
>> x = b / aThis can be read "B times A-inverse." Again, if there is no solution, a least-squares solution is found.
There are a number of builtin matrix functions, for example the determinant, rank, nullspace, and condition number.
>> det(a) >> rank(a) >> norm(a) >> null(a)Enter
>> a(3,3) = 10[which makes the matrix nonsingular] and do those again.
A very useful function for linear algebra students is rref. This function preforms a reduction of a matrix to reduced echelon form. Enter
>> rref(a)for an example. Try help rref for further details. Other valuable functions find the inverse, eigenvalues and eigenvectors of a matrix.
>> h=hilb(5) >> cond(a) >> inv(h) >> eig(h)The "eig" function has two forms of output. The last command produced a vector of eigenvalues. The next command produces two matrices, the first containing the eigenvectors as its columns, and the second containing the eigenvalues, along its diagonal.
>> [v,d]=eig(h)The matrix, h, times the first eigenvector, v(:,1), should equal the first eigenvalue, d(1,1), times that same eigenvector.
>> h*v(:,1) >> d(1,1)*v(:,1)"Round-off error" is a primary concern in numerical computing. Matlab does numerical computation, which is to say, it works with limited precision; all decimal expansions are truncated at the sixteenth place [roughly speaking]. Even if this is acceptable for any single calculation, its effects may accumulate with unacceptable consequences. The machine's round-off, the smallest distinguishable difference between two numbers as represented in Matlab, is denoted "eps".
>> help eps >> epsWe can check the assertion just made about eigenvectors and eigenvalues, as follows.
>> h*v(:,1) - d(1,1)*v(:,1)This is "the zero vector, modulo round-off error."
Matlab has outstanding graphics capabilities [you must be using a terminal which supports graphics, to use them]. Start with
>> x = -10:.1:10; >> plot( x.^2 ) >> plot( x, x.^2 ) >> plot( x.^2, x )Note that x must be assigned values, before the plot command is issued [although you could use plot( (-10 : .1 : 10) .^ 2 ) if you really really wanted to].
>> plot( x, x.*sin(x) ) >> plot( x.*cos(x), x.*sin(x) ) >> comet( x.*cos(x), x.*sin(x) )Functions of two variables may be plotted, as well, but some "setup" is required!
>> [x y] = meshgrid(-3:.1:3, -3:.1:3); >> z = x.^2 - y.^2; >> mesh(x,y,z) >> surf(x,y,z) >> contour(z) >> help sliceThere's a very interesting example, in the help page for slice; use the mouse to cut and paste it to the matlab prompt.
The following commands bring up lists of useful graphics commands [each has a help page of its own].
>> help plotxy >> help plotxyz >> help graphicsTo print Matlab graphics, just enter "print" at the Matlab prompt; the current figure window will be sent to the default printer. Note: If you are working in the Newton Lab at WSU, use the command
>> print -Plp4to send the current graphics figure to the printer.
Matlab statements can be prepared with any editor, and stored in a file for later use. The file is referred to as a script, or an "m-file" (since they must have names of the form foo.m). Writing m-files will make you much more productive.
Using your favorite editor, create the following file, named sketch.m:
[x y] = meshgrid(-3:.1:3, -3:.1:3); z = x.^2 - y.^2; mesh(x,y,z);Then start Matlab from the directory containing this file, and enter
>> sketchThe result is the same as if you had entered the three lines of the file, at the prompt.
You can also enter data this way: if a file named mymatrix.m in the current working directory contains the lines
A = [2 3 4; 5 6 7; 8 9 0] inv(A) quitthen the command
>> mymatrixreads that file, generates A and the inverse of A, and quits matlab [quitting is optional]. You may prefer to do this, if you use the same data repeatedly, or have an editor that you like to use. You can use Control-Z to suspend Matlab, then edit the file, and then use "fg" to bring matlab back to the foreground, to run it.
Matlab may be ran in "batch mode," in a very similar way. If a file named "test.in" contains the [non-graphics] commands you want processed, at the Unix prompt type:
% matlab < mymatrix.m > homework.outThis is read, "Run matlab, with input from test.in, and output to test.out." The input file does not need to be named "something-dot-m," but it must end with "quit"!
Functions are like scripts, but are compiled the first time they are used in a given session, for speed. Create a file, named sqroot.m, containing the following lines.
function sqroot(x) % Compute square root by Newton's method % Initial guess xstart = 1; for i = 1:100 xnew = ( xstart + x/xstart)/2; disp(xnew); if abs(xnew - xstart)/xnew < eps, break, end; xstart = xnew; endSave this file, start Matlab, and enter the commands
>> format long >> sqroot(19)A good exercise would be to create the STAT function described in the help file for function. Note that
>> [m,sd] = stat(x)produce different results.
The "m-files" which came with Matlab provide lots of examples! To find their location, use
>> pathThis will also lead you to some really nifty demos.
Some books which may be useful are
This document, as well as other information, is also available from the UCS Stat/Math Center's WWW home page; connect to
[from the IUB home page, select University Computing Services, and then Center for Statistical and Mathematical Computing].
Prepared by David Hart, Center for Statistical and Mathematical Computing, Indiana University, Bloomington, Copyright 1994, 1995. Last modification: 1/20/95. Minor modifications for WSU Math 220 classes by Alan Genz, Math Department, WSU, with last modification on 8/99.