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9. Matrix functions
Much of MATLAB's power comes from its matrix functions. The most useful
ones are
eig 
eigenvalues and eigenvectors 
chol 
cholesky factorization 
svd 
singular value decomposition 
inv 
inverse 
lu 
LU factorization 
qr 
QR factorization 
hess 
hessenberg form 
schur 
schur decomposition 
rref 
reduced row echelon form 
expm 
matrix exponential 
sqrtm 
matrix square root 
poly 
characteristic polynomial 
det 
determinant 
size 
size 
norm 
1norm, 2norm, Fnorm, <infinity>norm* 
cond 
condition number in the 2norm 
rank 
rank 
* Note: <infinity> stands for mathematical infinity
symbol, which cannot be rendered in HTML at this time. Ed.
MATLAB functions may have single or multiple output arguments. For example,

y = eig(A), or simply eig(A)
produces a column vector containing the eigenvalues of A while

[U,D] = eig(A)
produces a matrix U whose columns are the eigenvectors of A
and a diagonal matrix D with the eigenvalues of A on its
diagonal. Try it.
10. Command line editing and recall
The command line in MATLAB can be easily edited. The cursor can be positioned
with the left/right arrows and the Backspace (or Delete) key used to delete
the character to the left of the cursor. Other editing features are also
available. On a PC try the Home, End, and Delete keys; on other systems
see help cedit or type cedit.
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 Mfile 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
If one wanted to compare plots of the functions y = sin mx and y
= sin nx on the interval [0,2\pi] for various m and n,
one might do the same for the command line:

m=2; n=3; x=0:.01:2*pi; y=sin(m*x); z=cos(n*x); plot(x,y,x,z)
11. Submatrices and colon notation
Vectors and submatrices are often used in MATLAB to achieve fairly complex
data manipulation effects. "Colon notation" (which is used both to generate
vectors and reference submatrices) and subscripting by vectors are keys
to efficient manipulation of these objects. Creative use of these features
permits one to minimize the use of loops (which slows MATLAB) and to make
code simple and readable. Special effort should be made to become familiar
with them.
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
gives [0.2, 0.4, 0.6, 0.8, 1.0, 1.2], and

5:1:1
gives [5 4 3 2 1]. The following statements will, for example,
generate a table of sines. Try it.

x = [0.0:0.1:2.0]';

y = sin(x);

[x y]
Note that since sin operates entrywise, it produces a vector
y from the vector x.
The colon notation can be used to access submatrices of a matrix. For
example,

A(1:4,3)
is the column vector consisting of the first four entries of the third
column of A. A colon by itself denotes an entire row or column:

A(:,3)
is the third column of A, and A(1:4,:) is the first four
rows. Arbitrary integral vectors can be used as subscripts:

A(:,[2 4])
contains as columns, columns 2 and 4 of A. Such subscripting can
be used on both sides of an assignment statement:

A(:,[2 4 5]) = B(:,1:3)
replaces columns 2,4,5 of b with the first three columns of B.
Note that the entire altered matrix A is printed and assigned.
Try it.
Columns 2 and 4 of A can be multiplied on the right by the 2by2
matrix [1 2;3 4]:

A(:,[2,4]) = A(:,[2,4])*[1 2;3 4]
Once again, the entire altered matrix is printed and assigned.
If x is an nvector, 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. Mfiles
MATLAB can execute a sequence of statements stored on diskfiles. Such files
are called "Mfiles" because they must have the file type of ".m" as the
last part of their filename. Much of your work with MATLAB will be in creating
and refining Mfiles.
There are two types of Mfiles: 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

];
then the MATLAB statement data will cause the assignment given
in data.m to be carried out.
An Mfile can reference other Mfiles, 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 mbyn 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 mbyn 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((ba+1)*rand(m,n)) + a;
This should be placed in a diskfile with filename randint.m (corresponding
to the function name). The first line declares the function name, input
arguments, and output arguments; without this line the file would be a
script file. Then a MATLAB statement

z = randint(4,5),
for example, will cause the numbers 4 and 5 to be passed to the variables
m and n in the function file with the output result being
passed out to the variable z. Since variables in a function file
are local, their names are independent of those in the current MATLAB environment.
Note that use of nargin ("number of input arguments") permits
one to set a default value of an omitted input variablesuch 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);
Once this is placed in a diskfile stat.m, a MATLAB command [xm,
xd] = stat(x), for example, will assign the mean and standard deviation
of the entries in the vector x to xm and xd, respectively.
Single assignments can also be made with a function having multiple output
arguments. For example, xm = stat(x) (no brackets needed around
xm) will assign the mean of x to xm.
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 Mfile, are available to the online 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 matrixscalar 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 functionsuch 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 realvalued function of a single

% real variable; ordinarily functions are defined in Mfiles.

% 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 builtin while others are distributed as
Mfiles. The actual listing of any MfileMATLAB's or your owncan
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 are entered into MATLAB surrounded by single quotes. For example,

s = 'This is a test'
assigns the given text string to the variable s.
Text strings can be displayed with the function disp. For example:

disp('this message is hereby displayed')
Error messages are best displayed with the function error

error('Sorry, the matrix must be symmetric')
since when placed in an MFile, it causes execution to exit the Mfile.
In an Mfile 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: ')
is encountered, the prompt message is displayed and execution pauses while
the user keys in the input data. Upon pressing the return key, the data
is assigned to the variable iter and execution resumes.
14. Managing Mfiles
While using MATLAB one frequently wishes to create or edit an Mfile and
then return to MATLAB. One wishes to keep MATLAB active while editing a
file since otherwise all variables would be lost upon exiting.
This can be easily done using the !feature. If, while in MATLAB, you
precede it with an !, any system commandsuch as those for editing, printing,
or copying a filecan be executed without exiting MATLAB. If, for example,
the system command ed accesses your editor, the MATLAB command

>> !ed rotate.m
will let you edit the file named rotate.m using your local editor.
Upon leaving the editor, you will be returned to MATLAB just where you
left it.
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 Mfiles
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 !.
Mfiles must be accessible to MATLAB. On most mainframe or workstation
network installations, personal Mfiles 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
Two measures of the efficiency of an algorithm are the number of floating
point operations (flops) performed and the elapsed time.
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)
You may wish to compare this timeand flop countwith that for solving
the system using x = inv(A)*b;. Try it. Version 4.0 has the more
convenient tic and toc.
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
While all computations in MATLAB are performed in double precision, the
format of the displayed output can be controlled by the following commands.
format short 
fixed point with 4 decimal places (the default) 
format long 
fixed point with 14 decimal place 
format short e 
scientific notation with 4 decimal places 
format long e 
scientific notation with 15 decimal places 
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
Hardcopy is most easily obtained with the diary command. The command

diary filename
causes what appears subsequently on the screen (except graphics) to be
written to the named diskfile (if the filename is omitted it will be written
to a default file named diary) until one gives the command diary
off; the command diary on will cause writing to the file
to resume, etc. When finished, you can edit the file as desired and print
it out on the local system. The !feature (see section 14) will permit
you to edit and print the file without leaving MATLAB.
18. Graphics
MATLAB can produce both planar plots and 3D mesh surface plots. To preview
some of these capabilities in version 3.5, enter the command plotdemo.
Planar plots. The plot command creates linear xy plots;
if x and y are vectors of the same length, the command plot(x,y)
opens a graphics window and draws an xy plot of the elements of x
versus the elements of y. You can, for example, draw the graph of
the sine function over the interval 4 to 4 with the following commands:

x = 4:.01:4; y = sin(x); plot(x,y)
Try it. The vector x is a partition of the domain with meshsize
0.01 while y is a vector giving the values of sine at the nodes
of this partition (recall that sin operates entrywise).
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
exposedbut moved to the sideand 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)
Note that one must precede ^ by a period to ensure that it operates entrywise
(see section 3).
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.
title 
graph title 
xlabel 
xaxis label 
ylabel 
yaxis label 
gtext 
interactivelypositioned text 
text 
position text at specified coordinates 
For example, the command

title('Best Least Squares Fit')
gives a graph a title. The command gtext('The Spot') allows a
mouse or the arrow keys to position a crosshair on the graph, at which
the text will be placed when any key is pressed.
By default, the axes are autoscaled. This can be overridden by the
command axis. If c = [xmin,xmax,ymin,ymax] is a 4element
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 autoscaling. 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)
and by forming a matrix Y containing the functional values as
columns

x=0:.01:2*pi; Y=[sin(x)', sin(2*x)', sin(4*x)']; plot(x,Y)
Another way is with hold. The command hold freezes the
current graphics screen so that subsequent plots are superimposed on it.
Entering hold again releases the "hold." The commands hold
on and hold off are also available in version 4.0. One can
override the default linetypes and pointtypes. For example,

x=0:.01:2*pi; y1=sin(x); y2=sin(2*x); y3=sin(4*x); plot(x,y1,'',x,y2,':',x,y3,'+')
renders a dashed line and dotted line for the first two graphs while for
the third the symbol + is placed at each node. The line and marktypes
are

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

Marktypes: point (.), plus (+), star (*), circle
(o), xmark (x)
See help plot for line and mark colors.
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: ShiftPrtScr.] A hardcopy
of the graphics screen can be most easily obtained with the MATLAB command
print. It will send a highresolution 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 metafilewhich
may now contain several plotsmay be processed later with the graphics
postprocessor (GPP) program to produce highresolution hardcopy, two plots
per page.
The program GPP (graphics postprocessor) 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 deviceindependent
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
will produce files filename.ps and filename.jet suitable
for printing on, respectively, PostScript and HP LaserJet printers. They
can be printed using the usual printing command for the local systemfor
example, lpr filename.ps on a Unix system. Entering the system
command gpp with no arguments gives a list of all supported hardcopy
devices. On a PC, most of this can be automated by appropriately editing
the file gpp.bat distributed with MATLAB.
3D mesh plots. Three dimensional mesh surface plots are drawn
with the function mesh. The command mesh(z) creates a
threedimensional perspective plot of the elements of the matrix z.
The mesh surface is defined by the zcoordinates of points above a rectangular
grid in the xy 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);
One then computes a matrix z, obtained by evaluating f entrywise
over the matrices x and y, to which mesh can be
applied.
You can, for example, draw the graph of z = e^(x^2y^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)
One could, of course, replace the first three lines of the preceding with

[x,y] = meshdom(2:.1:2, 2:.1:2);
You are referred to the User's Guide for further details regarding mesh.
In version 4.0, the 3D graphics capabilities of MATLAB have been considerably
expanded. Consult the online help for plot3, mesh, and surf.
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