- Accessing MATLAB
- Entering matrices
- Matrix operations, array operations
- Statements, expressions, and variables; saving a session
- Matrix building functions
- For, while, if - and relations
- Scalar functions
- Vector functions
- Matrix functions
- Command line editing and recall
- Submatrices and colon notation
- M-files
- Text strings, error messages, input
- Managing M-files
- Comparing efficiency of algorithms: flops and etime
- Output format
- Hardcopy
- Graphics
- Reference (Outside this document)

On most systems, after logging in one can enter MATLAB with the system command
`matlab` and exit MATLAB with the command `exit` or `quit`.
On a PC, for example, if properly installed,
one may enter MATLAB with the command:

and exit it with the command:C> matlab

>> quit

On systems permitting multiple processes, such as a Unix system, you will find it convenient, for reasons discussed in section 14, to keep both MATLAB and your local editor active. If you are working on a workstation which runs processes in multiple windows, you will want to keep MATLAB active in one window and your local editor active in another. You should consult your instructor or your local computer center for details of the local installation.

MATLAB works with essentially only one kind of object-a rectangular numerical matrix with possibly complex entries; all variables represent matrices. In some situations, 1-by-1 matrices are interpreted as scalars and matrices with only one row or one column are interpreted as vectors.

Matrices can be introduced into MATLAB in several different ways:

- Entered by an explicit list of elements,
- Generated by built-in statements and functions,
- Created in M-files (see sections 12 and 14 below),
- Loaded from external data files (see User's Guide).

andA = [1 2 3; 4 5 6; 7 8 9]

creates the obvious 3-by-3 matrix and assigns it to a variableA = [ 1 2 3 4 5 6 7 8 9 ]

When listing a number in exponential form (e.g. 2.34e-9), blank spaces must be avoided. Listing entries of a large matrix is best done in an M-file, where errors can be easily edited away (see sections 12 and 14).

The built-in functions `rand`, `magic`, and `hilb`, for example,
provide an easy way to create matrices with which to experiment. The command
`rand(n)` will create an `n x n` matrix with randomly generated
entries distributed uniformly between 0 and 1, while `rand(m,n)`
will create an `m x n ` one. `magic(n)` will create an
integral `n x n` matrix which is a magic square (rows and columns
have common sum); `hilb(n)` will create the `n x n` Hilbert
matrix, the king of ill-conditioned matrices ( m and n denote, of
course, positive integers). Matrices can also be generated with a
for-loop (see section 6 below).

Individual matrix and vector entries can be referenced with indices inside
parentheses in the usual manner. For example, `A(2,3)` denotes the entry in
the second row, third column of matrix `A` and
`x(3) ` denotes the third
coordinate of vector `x`. Try it. A matrix or a vector will only
accept *positive* integers as indices.

The following matrix operations are available in MATLAB:

These matrix operations apply, of course, to scalars (1-by-1 matrices) as well. If the sizes of the matrices are incompatible for the matrix operation, an error message will result, except in the case of scalar-matrix operations (for addition, subtraction, and division as well as for multiplication) in which case each entry of the matrix is operated on by the scalar.+ addition - subtraction * multiplication ^ power ' transpose \ left division / right division

The ``matrix division'' operations deserve special comment. If `A` is an
invertible square matrix and `b ` is a compatible column, resp. row, vector,
then

is the solution ofx=A \ b

is the solution ofx=b/A

b/A=(A' \ b')'

can be made to operate entry-wise by preceding them by a period. For example, either* , ^ , \ , and /,

or[1,2,3,4].*[1,2,3,4]

will yield[1,2,3,4].^2

MATLAB is an *expression* language; the expressions you type are
interpreted and evaluated. MATLAB statements are usually of the form

or simplyvariable = expression,

expression

Expressions are usually composed from operators, functions, and variable names.
Evaluation of the expression produces a matrix, which is then displayed on the
screen and assigned to the variable for future use. If the variable name and
= sign are omitted, a variable `ans` (for answer) is automatically
created to which the result is assigned.

A statement is normally terminated with the carriage return. However, a statement can be continued to the next line with three or more periods followed by a carriage return. On the other hand, several statements can be placed on a single line if separated by commas or semicolons.

If the last character of a statement is a semicolon, the printing is suppressed, but the assignment is carried out. This is essential in suppressing unwanted printing of intermediate results.

MATLAB is case-sensitive in the names of commands, functions, and variables.
For example, `solveUT` is not the same as `solveut`.

The command `who` will list the variables currently in the workspace.
A variable can be cleared from the workspace with the command
`clear` *variablename*. The command `clear` alone will clear all
nonpermanent variables.

The permanent variable `eps` (epsilon) gives the machine
precision-about on most machines. It is useful in
determining tolerences for convergence of iterative processes.

A runaway display or computation can be stopped on most machines without leaving MATLAB with CTRL-C (CTRL-BREAK on a PC).

**Saving a session.** When one logs out or exits MATLAB all
variables are lost. However, invoking the command `save` before
exiting causes all variables to be written to a non-human-readable
diskfile named `matlab.mat`. When one later reenters MATLAB, the
command `load` will restore the workspace to its former state.

Convenient matrix building functions are

For example,eye(10) identity matrix zeros(10) matrix of zeros ones(10) matrix of ones diag(A) returns diagonal M.E. as vector triu(A) upper triangular part of a matrix tril(A) lower triangular part of a matrix rand(10) randomly generated matrix hilb(5) Hilbert matrix magic(5) magic square toeplitz see help toeplitz

If `x` is a vector, `diag(x)` is the diagonal matrix with
`x` down the
diagonal; if `A` is a square matrix, then `diag(A)` is a
vector consisting
of the diagonal of `A` . What is `diag(diag(A))`? Try it.

Matrices can be built from blocks. For example, if A is a 3-by-3 matrix, then

will build a certain 5-by-5 matrix. Try it.B = [A, zeros(3,2); zeros(2,3), eye(2)]

Convenient matrix building functions are

For example,eye(10) identity matrix zeros(10) matrix of zeros ones(10) matrix of ones diag(A) returns diagonal M.E. as vector triu(A) upper triangular part of a matrix tril(A) lower triangular part of a matrix rand(10) randomly generated matrix hilb(5) Hilbert matrix magic(5) magic square toeplitz see help toeplitz

If `x` is a vector, `diag(x)` is the diagonal matrix with
`x` down the
diagonal; if `A` is a square matrix, then `diag(A)` is a
vector consisting
of the diagonal of `A` . What is `diag(diag(A))`? Try it.

Matrices can be built from blocks. For example, if A is a 3-by-3 matrix, then

will build a certain 5-by-5 matrix. Try it.B = [A, zeros(3,2); zeros(2,3), eye(2)]

In their basic forms, these MATLAB flow control statements operate like those in most computer languages.

**For.** For example, for a given n, the statement

orx = []; for i = 1:n, x=[x,i^2 ], end

will produce a certain n-vector and the statementx = []; for i = 1:n x = [x,i^2 ] end

will produce the same vector in reverse order. Try them. Note that a matrix may be empty (such asx = []; for i = n:-1:1, x=[x,i^2 ], end

will produce and print to the screen the m-by-n hilbert matrix. The semicolon on the inner statement suppresses printing of unwanted intermediate results while the lastfor i = 1:m for j = 1:n H(i, j) = 1/(i+j-1); end end H

**While.** The general form of a `while` loop is

The statements will be repeatedly executed as long as the relation remains true. For example, for a given numberwhile relation statements end

n = 0; while 2^n < a n = n + 1; end n

The statements will be executed only if the relation is true. Multiple branching is also possible, as is illustrated byif relation statements end

In two-way branching the elseif portion would, of course, be omitted.if n < 0 parity = 0; elseif rem(n,2) == 0 parity = 2; else parity = 1; end

**Relations.** The relational operators in MATLAB are

Note that ``='' is used in an assignment statement while ``=='' is used in a relation. Relations may be connected or quantified by the logical operators< less than > greater than <= less than or equal >= greater than or equal == equal ~= not equal.

& and | or ~ not.

When applied to scalars, a relation is actually the scalar 1 or 0 depending on
whether the relation is true or false. Try `3 < 5, 3 > 5, 3 == 5,` and
`3 == 3`.
When applied to matrices of the same size, a relation is a matrix of
0's and 1's
giving the value of the relation between corresponding entries.
Try `a = rand(5), b = triu(a), a == b`.

A relation between matrices is interpreted by `while` and `if` to be
true if each entry of the relation matrix is nonzero. Hence, if
you wish to execute *statement* when matrices `A` and `B`
are equal you could type

but if you wish to executeif A == B statement end

or, more simply,if any(any(A ~ B)) ( statement ) end

Note that the seemingly obviousif A == B else { statement} end

will not give what is intended sinceif A ~= B, ( statement ), end

The `for` statement permits any matrix to be used instead of `1:n`.
See the User's Guide for details of how this feature expands the power of
the `for` statement.

Certain MATLAB functions operate essentially on scalars, but operate element-wise when applied to a matrix. The most common such functions are

sin asin exp abs round cos acos log (natural log) sqrt floor tan atan rem (remainder) sign ceil

Other MATLAB functions operate essentially on a vector (row or column), but act
on an m-by-n matrix () in a column-by-column fashion to produce a
row vector containing the results of their application to each column.
Row-by-row action can be obtained by using the transpose; for example,
`mean(A')'`. A few of these functions are

For example, the maximum entry in a matrixmax sum median any min prod mean all sort std

Much of MATLAB's power comes from its matrix functions. The most useful ones are

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

produces a column vector containing the eigenvalues ofy = eig(A) or simply eig(A)

produces a matrix[U,D] = eig(A)

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

If one wanted to compare plots of the functionsa = 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)

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,

gives0.2:0.2:1.2

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

Note that sincex = [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)` 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 `A`

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]:`

Once again, the entire altered matrix is printed and assigned.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.

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`

then the MATLAB statementA = [ 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:

This should be placed in a diskfile with filenamefunction 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;

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:

Once this is placed in a diskfilefunction [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

is the matrix of squares of the entries ofx.^2

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]]; end 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

Text strings are entered into MATLAB surrounded by single quotes. For example,

assigns the given text string to the variables = 'This is a test'

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

Error messages are best displayed with the functiondisp('this message is hereby displayed')

since when placed in an M-File, it causes execution to exit the M-file.error('Sorry, the matrix must be symmetric')

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

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

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

will let you edit the file named>> !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.

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
`Ax=b` using Gaussian elimination as follows:

You may wish to compare this time-and flop count-with that for solving the system usingt = 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.

While all computations in MATLAB are performed in double precision, the format of the displayed output can be controlled by the following commands.

Once invoked, the chosen format remains in effect until changed.format short fixed point with 4 decimal places (the default) format long fixed point with 14 decimal places format short e scientific notation with 4 decimal places format long e scientific notation with 15 decimal places

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

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 nameddiary {\it filename}

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:

Try it. The vectorx = -4:.01:4; y = sin(x); plot(x,y)

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 over the interval -1.5 to 1.5 as follows:

Note that one must precede the power-to sign by a period to ensure that it operates entrywise (see section 3).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 commandtitle graph title xlabel x-axis label ylabel y-axis label gtext interactively-positioned text text position text at specified coordinates

gives a graph a title. The commandtitle('Best Least Squares Fit')

By default, the axes are auto-scaled. This can be overridden by the command
`axis`. If 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

and by forming a matrixx=0:.01:2*pi;y1=sin(x);y2=sin(2*x); y3=sin(4*x);plot(x,y1,x,y2,x,y3)

Another way is withx=0:.01:2*pi; Y=[sin(x)', sin(2*x)', sin(4*x)']; plot(x,Y)

One can override the default linetypes and pointtypes. For example,

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 arex=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 (`-`). 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 Note that most of this section is no longer relevant
`

`
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

*Note that in newer versions of Matlab the above text is no longer relevant*

**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:

One then computes a matrix[x,y] = meshdom(xx,yy);

You can, for example, draw the graph of over the square
`[-2,2] x [-2,2] ` as follows (try it):

One could, of course, replace the first three lines of the preceding withxx = -2:.1:2; yy = xx; [x,y] = meshdom(xx,yy); z = exp(-x.^2 - y.$^2); mesh(z)

[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`.