MATLAB functions may have single or multiple output arguments. For example,

y = eig(A), or simply eig(A)

[U,D] = eig(A)

10. Command line editing and recall

A convenient feature is use of the up/down arrows to scroll through the stack of previous commands. One can, therefore, recall a previous command line, edit it, and execute the revised command line. For small routines, this is much more convenient that using an M-file which requires moving between MATLAB and the editor (see sections 12 and 14). For example, flopcounts (see section 15) for computing the inverse of matrices of various sizes could be compared by repeatedly recalling, editing, and executing

a = rand(8); flops(0), inv(a); flops

m=2; n=3; x=0:.01:2*pi; y=sin(m*x); z=cos(n*x); plot(x,y,x,z)

The expression 1:5 (met earlier in for statements) is actually the row vector [1 2 3 4 5]. The numbers need not be integers nor the increment one. For example,

0.2:0.2:1.2

5:-1:1

x = [0.0:0.1:2.0]';

y = sin(x);

[x y]

The colon notation can be used to access submatrices of a matrix. For example,

A(1:4,3)

A(:,3)

A(:,[2 4])

A(:,[2 4 5]) = B(:,1:3)

Columns 2 and 4 of A can be multiplied on the right by the 2-by-2 matrix [1 2;3 4]:

A(:,[2,4]) = A(:,[2,4])*[1 2;3 4]

If x is an n-vector, what is the effect of the statement x = x(n:-1:1)? Try it.

To appreciate the usefulness of these features, compare these MATLAB statements with a Pascal, FORTRAN, or C routine to effect the same.

12. M-files

There are two types of M-files: script files and function files.

Script files. A script file consists of a sequence of normal MATLAB statements. If the file has the filename, say, rotate.m, then the MATLAB command rotate will cause the statements in the file to be executed. Variables in a script file are global and will change the value of variables of the same name in the environment of the current MATLAB session.

Script files are often used to enter data into a large matrix; in such a file, entry errors can be easily edited out. If, for example, one enters in a diskfile data.m

A = [

1 2 3 4

5 6 7 8

];

An M-file can reference other M-files, including referencing itself recursively.

Function files. Function files provide extensibility to MATLAB. You can create new functions specific to your problem which will then have the same status as other MATLAB functions. Variables in a function file are by default local. However, version 4.0 permits a variable to be declared global.

We first illustrate with a simple example of a function file.

function a = randint(m,n)

%RANDINT Randomly generated integral matrix.

% randint(m,n) returns an m-by-n such matrix with entries

% between 0 and 9.

a = floor(10*rand(m,n));

A more general version of this function is the following:

function a = randint(m,n,a,b)

%RANDINT Randomly generated integral matrix.

% randint(m,n) returns an m-by-n such matrix with entries

% between 0 and 9.

% rand(m,n,a,b) return entries between integers a and b.

if nargin < 3, a = 0, b = 9; end

a = floor((b-a+1)*rand(m,n)) + a;

z = randint(4,5),

Note that use of nargin ("number of input arguments") permits one to set a default value of an omitted input variable---such as a and b in the example.

A function may also have multiple output arguments. For example:

function [mean, stdev] = stat(x)

% STAT Mean and standard deviation

% For a vector x, stat(x) returns the

% mean and standard deviation of x.

% For a matrix x, stat(x) returns two row vectors containing,

% respectively, the mean and standard deviation of each column.

[m n] = size(x);

if m == 1

m = n; % handle case of a row vector

end

mean = sum(x)/m;

stdev = sqrt(sum(x.^2)/m - mean.^2);

The % symbol indicates that the rest of the line is a comment; MATLAB will ignore the rest of the line. However, the first few comment lines, which document the M-file, are available to the on-line help facility and will be displayed if, for example, help stat is entered. Such documentation should always be included in a function file.

This function illustrates some of the MATLAB features that can be used to produce efficient code. Note, for example, that x.^2 is the matrix of squares of the entries of x, that sum is a vector function (section 8), that sqrt is a scalar function (section 7), and that the division in sum(x)/m is a matrix-scalar operation.

The following function, which gives the greatest common divisor of two integers via the Euclidean algorithm, illustrates the use of an error message (see the next section).

function a = gcd(a,b)

% GCD Greatest common divisor

% gcd(a,b) is the greatest common divisor of

% the integers a and b, not both zero.

a = round(abs(a)); b = round(abs(b));

if a == 0 & b == 0

error('The gcd is not defined when both numbers are zero')

else

while b ~= 0

r = rem(a,b)

a = b; b = r;

end

end

Some more advanced features are illustrated by the following function. As noted earlier, some of the input arguments of a function---such as tol in the example, may be made optional through use of nargin ("number of input arguments"). The variable nargout can be similarly used. Note that the fact that a relation is a number (1 when true; 0 when false) is used and that, when while or if evaluates a relation, "nonzero" means "true" and 0 means "false". Finally, the MATLAB function feval permits one to have as an input variable a string naming another function.

function [b, steps] = bisect(fun, x, tol)

%BISECT Zero of a function of one variable via the bisection method.

% bisect(fun,x) returns a zero of the function. fun is a string

% containing the name of a real-valued function of a single

% real variable; ordinarily functions are defined in M-files.

% x is a starting guess. The value returned is near a point

% where fun changes sign. For example,

% bisect('sin',3) is pi. Note the quotes around sin.

%

% An optional third input argument sets a tolerence for the

% relative accuracy of the result. The default is eps.

% An optional second output argument gives a matrix containing a

% trace of the steps; the rows are of form [c f(c)].

% Initialization

if nargin < 3, tol = eps; end

trace = (nargout == 2);

if x ~= 0, dx = x/20; else, dx = 1/20; end

a = x - dx; fa = feval(fun,a);

b = x + dx; fb = feval(fun,b);

% Find change of sign.

while (fa > 0) == (fb > 0)

dx = 2.0*dx;

a = x - dx; fa = feval(fun,a);

if (fa > 0) ~= (fb > 0), break, end

b = x + dx; fb = feval(fun,b);

end

if trace, steps = [a fa; b fb] end

% Main loop

while abs(b - a) > 2.0*tol*max(abs(b),1.0)

c = a + 0.5*(b - a); fc = feval(fun,c);

if trace, steps = [steps; [c fc]];

if (fb > 0) == (fc > 0)

b = c; fb = fc;

else

a = c; fa = fc;

end

end

Some of MATLAB's functions are built-in while others are distributed as M-files. The actual listing of any M-file---MATLAB's or your own---can be viewed with the MATLAB command type functionname. Try entering type eig, type vander, and type rank.

13. Text strings, error messages, input

Text strings can be displayed with the function disp. For example:

disp('this message is hereby displayed')

In an M-file the user can be prompted to interactively enter input data with the function input. When, for example, the statement

iter = input('Enter the number of iterations: ')

This can be easily done using the !-feature. If, while in MATLAB, you precede it with an !, any system command---such as those for editing, printing, or copying a file---can be executed without exiting MATLAB. If, for example, the system command ed accesses your editor, the MATLAB command

>> !ed rotate.m

As noted in section 1, on systems permitting multiple processes, such as one running Unix, it may be preferable to keep both MATLAB and your local editor active, keeping one process suspended while working in the other. If these processes can be run in multiple windows, as on a workstation, you will want to keep MATLAB active in one window and your editor active in another.

You may consult your instructor or your local computing center for details of the local installation.

Version 4.0 has many debbugging tools. See help dbtype and references given there.

When in MATLAB, the command dir will list the contents of the current directory while the command what will list only the M-files in the directory. The MATLAB commands delete and type can be used to delete a diskfile and print a file to the screen, respectively, and chdir can be used to change the working directory. While these commands may duplicate system commands, they avoid the use of an !.

M-files must be accessible to MATLAB. On most mainframe or workstation network installations, personal M-files which are stored in a subdirectory of one's home directory named matlab will be accessible to MATLAB from any directory in which one is working. See the discussion of MATLABPATH in the User's Guide for further information.

15. Comparing efficiency of algorithms: flops and etime

The MATLAB function flops keeps a running total of the flops performed. The command flops(0) (not flops = 0!) will reset flops to 0. Hence, entering flops(0) immediately before executing an algorithm and flops immediately after gives the flop count for the algorithm.

The MATLAB function clock gives the current time accurate to a hundreth of a second (see help clock). Given two such times t1 and t2, etime(t2,t1) gives the elapsed time from t1 to t2. One can, for example, measure the time required to solve a given linear system A x = b using Gaussian elimination as follows:

t = clock; x = A\b; time = etime(clock,t)

It should be noted that, on timesharing machines, etime may not be a reliable measure of the efficiency of an algorithm since the rate of execution depends on how busy the computer is at the time.

16. Output format

Once invoked, the chosen format remains in effect until changed.

The command format compact will suppress most blank lines allowing more information to be placed on the screen or page. It is independent of the other format commands.

17. Hardcopy

diary filename

When in the graphics screen, pressing any key will return you to the command screen while the command shg (show graph) will then return you to the current graphics screen. If your machine supports multiple windows with a separate graphics window, you will want to keep the graphics window exposed---but moved to the side---and the command window active.

As a second example, you can draw the graph of y = e^(-x^2) over the interval -1.5 to 1.5 as follows:

x = -1.5:.01:1.5; y = exp(-x.^2); plot(x,y)

Plots of parametrically defined curves can also be made. Try, for example,

t=0:.001:2*pi; x=cos(3*t); y=sin(2*t); plot(x,y)

The command grid will place grid lines on the current graph.

The graphs can be given titles, axes labeled, and text placed within the graph with the following commands which take a string as an argument.

For example, the command

title('Best Least Squares Fit')

By default, the axes are auto-scaled. This can be overridden by the command axis. If c = [xmin,xmax,ymin,ymax] is a 4-element vector, then axis(c) sets the axis scaling to the precribed limits. By itself, axis freezes the current scaling for subsequent graphs; entering axis again returns to auto-scaling. The command axis('square') ensures that the same scale is used on both axes. In version 4.0, axis has been significantly changed; see help axis.

Two ways to make multiple plots on a single graph are illustrated by

x=0:.01:2*pi;y1=sin(x);y2=sin(2*x);y3=sin(4*x);plot(x,y1,x,y2,x,y3)

Linetypes: solid (-), dashed (\tt--). dotted (:), dashdot (-.)

Marktypes: point (.), plus (+), star (*), circle (o), x-mark (x)

The command subplot can be used to partition the screen so that up to four plots can be viewed simultaneously. See help subplot.

Graphics hardcopy [The features described in this subsection are not available with the Student Edition of Matlab. Graphics hardcopy can be obtained there only with a screen dump: Shift-PrtScr.] A hardcopy of the graphics screen can be most easily obtained with the MATLAB command print. It will send a high-resolution copy of the current graphics screen to the printer, placing the graph on the top half of the page.

In version 4.0 the meta and gpp commands described below have been absorbed into the print command. See help print.

Producing unified hard copy of several plots requires more effort. The Matlab command meta filename stores the current graphics screen in a file named filename.met (a "metafile") in the current directory. Subsequent meta (no filename) commands append a new current graphics screen to the previously named metafile. This metafile---which may now contain several plots---may be processed later with the graphics post-processor (GPP) program to produce high-resolution hardcopy, two plots per page.

The program GPP (graphics post-processor) is a system command, not a MATLAB command. However, in practice it is usually involked from within MATLAB using the "!" feature (see section 14). It acts on a device-independent metafile to produce an output file appropriate for many different hardcopy devices.

The selection of the specific hardcopy device is made with the option key "/d". For example, the system commands

gpp filename /dps

gpp filename /djet

3-D mesh plots. Three dimensional mesh surface plots are drawn with the function mesh. The command mesh(z) creates a three-dimensional perspective plot of the elements of the matrix z. The mesh surface is defined by the z-coordinates of points above a rectangular grid in the x-y plane. Try mesh(eye(10)).

To draw the graph of a function z = f(x,y) over a rectangle, one first defines vectors xx and yy which give partitions of the sides of the rectangle. With the function meshdom (mesh domain; called meshgrid in version 4.0) one then creates a matrix x, each row of which equals xx and whose column length is the length of yy, and similarly a matrix y, each column of which equals yy, as follows:

[x,y] = meshdom(xx,yy);

You can, for example, draw the graph of z = e^(-x^2-y^2) over the square [-2,2]\times[-2,2] as follows (try it):

xx = -2:.1:2;

yy = xx;

[x,y] = meshdom(xx,yy);

z = exp(-x.^2 - y.^2);

mesh(z)

In version 4.0, the 3-D graphics capabilities of MATLAB have been considerably expanded. Consult the on-line help for plot3, mesh, and surf.

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