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### 1.2 Simple Examples

The following chapters describe all of Octave's features in detail, but before doing that, it might be helpful to give a sampling of some of its capabilities.

If you are new to Octave, I recommend that you try these examples to begin learning Octave by using it. Lines marked with `octave:13>' are lines you type, ending each with a carriage return. Octave will respond with an answer, or by displaying a graph.

#### Creating a Matrix

To create a new matrix and store it in a variable so that it you can refer to it later, type the command

```     octave:1> a = [ 1, 1, 2; 3, 5, 8; 13, 21, 34 ]
```

Octave will respond by printing the matrix in neatly aligned columns. Ending a command with a semicolon tells Octave to not print the result of a command. For example

```     octave:2> b = rand (3, 2);
```

will create a 3 row, 2 column matrix with each element set to a random value between zero and one.

To display the value of any variable, simply type the name of the variable. For example, to display the value stored in the matrix `b`, type the command

```     octave:3> b
```

#### Matrix Arithmetic

Octave has a convenient operator notation for performing matrix arithmetic. For example, to multiply the matrix `a` by a scalar value, type the command

```     octave:4> 2 * a
```

To multiply the two matrices `a` and `b`, type the command

```     octave:5> a * b
```

To form the matrix product `transpose (a) * a`, type the command

```     octave:6> a' * a
```

#### Solving Linear Equations

To solve the set of linear equations `a`x` = b`, use the left division operator, `\':

```     octave:7> a \ b
```

This is conceptually equivalent to `inv (a) * b`, but avoids computing the inverse of a matrix directly.

If the coefficient matrix is singular, Octave will print a warning message and compute a minimum norm solution.

#### Integrating Differential Equations

Octave has built-in functions for solving nonlinear differential equations of the form

```     dx
-- = f (x, t)
dt
```

with the initial condition

```     x(t = t0) = x0
```

For Octave to integrate equations of this form, you must first provide a definition of the function `f(x,t)`. This is straightforward, and may be accomplished by entering the function body directly on the command line. For example, the following commands define the right hand side function for an interesting pair of nonlinear differential equations. Note that while you are entering a function, Octave responds with a different prompt, to indicate that it is waiting for you to complete your input.

```     octave:8> function xdot = f (x, t)
>
>  r = 0.25;
>  k = 1.4;
>  a = 1.5;
>  b = 0.16;
>  c = 0.9;
>  d = 0.8;
>
>  xdot(1) = r*x(1)*(1 - x(1)/k) - a*x(1)*x(2)/(1 + b*x(1));
>  xdot(2) = c*a*x(1)*x(2)/(1 + b*x(1)) - d*x(2);
>
> endfunction
```

Given the initial condition

```     x0 = [1; 2];
```

and the set of output times as a column vector (note that the first output time corresponds to the initial condition given above)

```     t = linspace (0, 50, 200)';
```

it is easy to integrate the set of differential equations:

```     x = lsode ("f", x0, t);
```

The function `lsode` uses the Livermore Solver for Ordinary Differential Equations, described in A. C. Hindmarsh, ODEPACK, a Systematized Collection of ODE Solvers, in: Scientific Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam, 1983, pages 55–64.

#### Producing Graphical Output

To display the solution of the previous example graphically, use the command

```     plot (t, x)
```

If you are using the X Window System, Octave will automatically create a separate window to display the plot. If you are using a terminal that supports some other graphics commands, you will need to tell Octave what kind of terminal you have. Type the command

```     gset term
```

to see a list of the supported terminal types. Octave uses `gnuplot` to display graphics, and can display graphics on any terminal that is supported by `gnuplot`.

To capture the output of the plot command in a file rather than sending the output directly to your terminal, you can use a set of commands like this

```     gset term postscript
gset output "foo.ps"
replot
```

This will work for other types of output devices as well. Octave's `gset` command is really just piped to the `gnuplot` subprocess, so that once you have a plot on the screen that you like, you should be able to do something like this to create an output file suitable for your graphics printer.

Or, you can eliminate the intermediate file by using commands like this

```     gset term postscript
gset output "|lpr -Pname_of_your_graphics_printer"
replot
```

#### Editing What You Have Typed

At the Octave prompt, you can recall, edit, and reissue previous commands using Emacs- or vi-style editing commands. The default keybindings use Emacs-style commands. For example, to recall the previous command, type Control-p (usually written C-p for short). C-p gets its name from the fact that you type it by holding down <CTRL> and then pressing <p>. Doing this will normally bring back the previous line of input. C-n will bring up the next line of input, C-b will move the cursor backward on the line, C-f will move the cursor forward on the line, etc.

A complete description of the command line editing capability is given in this manual in Command Line Editing.

#### Help and Documentation

Octave has an extensive help facility. The same documentation that is available in printed form is also available from the Octave prompt, because both forms of the documentation are created from the same input file.

In order to get good help you first need to know the name of the command that you want to use. This name of the function may not always be obvious, but a good place to start is to just type `help`. This will show you all the operators, reserved words, functions, built-in variables, and function files. You can then get more help on anything that is listed by simply including the name as an argument to help. For example,

```     help plot
```

will display the help text for the `plot` function.

Octave sends output that is too long to fit on one screen through a pager like `less` or `more`. Type a <RET> to advance one line, a <SPC> to advance one page, and <q> to exit the pager.

The part of Octave's help facility that allows you to read the complete text of the printed manual from within Octave normally uses a separate program called Info. When you invoke Info you will be put into a menu driven program that contains the entire Octave manual. Help for using Info is provided in this manual in Getting Help.