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 | 1-norm, 2-norm, F-norm, <infinity>-norm* |

cond | condition number in the 2-norm |

rank | rank |

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.

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 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`

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)`

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 entry-wise, 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 2-by-2
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 **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.

MATLAB can execute a sequence of statements stored on diskfiles. Such files are called "M-files" 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 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
- ];

then the MATLAB statement `data`

will cause the assignment
given in `data.m`

to be carried out.

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;`

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 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);`

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 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 `

.
Try entering *functionname*`type eig`

, `type vander`

, and ```
type
rank
```

.

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 M-File, it causes execution to exit the M-file.

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: ')`

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.

While using MATLAB one frequently wishes to create or edit an M-file 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 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`

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 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.

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 time---and flop count---with 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.

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.

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.

MATLAB can produce both planar plots and 3-D mesh surface plots. To preview
some of these capabilities in version 3.5, enter the command `plotdemo`

.
**Planar plots.** The `plot`

command creates linear x-y plots;
if **x** and **y** are vectors of the same length, the command `plot(x,y)`

opens a graphics window and draws an x-y 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 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)`

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 | x-axis label |

ylabel | y-axis label |

gtext | interactively-positioned 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 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)`

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 mark-types are

- Linetypes: solid (
`-`

), dashed (\tt`--`

). dotted (`:`

), dashdot (`-.`

) - Marktypes: point (
`.`

), plus (`+`

), star (`*`

), circle (`o`

), x-mark (`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: 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 `

stores the current graphics
screen in a file named *filename**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`

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 system---for
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.

**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);`

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^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)`

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 3-D graphics capabilities of MATLAB have been considerably
expanded. Consult the on-line help for `plot3, mesh`

, and `surf`

.