%% RLaB Primer
%
%% This file documents RLaB, a Vector and Matrix Programming Language
%% Copyright @copyright{} 1992, 1993, 1994 Ian R. Searle
%%
%% Permission is granted to make and distribute verbatim copies of
%% this manual provided the copyright notice and this permission notice
%% are preserved on all copies.
%%
%% Permission is granted to process this file through TeX and print the
%% results, provided the printed document carries a copying permission
%% notice identical to this one except for the removal of this paragraph
%% (this paragraph not being relevant to the printed manual).
%%
%% Permission is granted to copy and distribute modified versions of this
%% manual under the conditions for verbatim copying, provided also that the
%% sections entitled ``Distribution'' and ``General Public License'' are
%% included exactly as in the original, and provided that the entire
%% resulting derived work is distributed under the terms of a permission
%% notice identical to this one.
%%
%% Permission is granted to copy and distribute translations of this manual
%% into another language, under the above conditions for modified versions,
%% except that the sections entitled ``Distribution'' and ``General Public
%% License'' may be included in a translation approved by the author instead
%% of in the original English.
%
% This document copyright
%               Ian R. Searle
%		ians@eskimo.com
% and
%               Phillip Musumeci, Electrical Engineering Department,
%               University College, Australian Defence Force Academy,
%               Canberra A.C.T. 2600, AUSTRALIA.
%
%% For permissions relating to reproduction of this document,
%% see PERMISSIONS below.
%%
%% For corrections and suggestions relating to this document,
%% see SUGGESTIONS below.
%
 
\documentstyle[11pt]{article}

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%\typeout{RLaB primer v0.1 phillip@ee.adfa.oz.au 22/1/93}

\addtolength{\topmargin}{-1.9cm}        % shrink LaTeX's huge margins
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%%
%% Define RLaB logos. They are all the same
%%
\def\RLAB{{\rm R\kern-.16em\lower.5ex\hbox{L}\kern-.25em\raise.3ex
    \hbox{a}\kern-.1emB}\,}
\def\RLaB{{\rm R\kern-.16em\lower.5ex\hbox{L}\kern-.25em\raise.3ex
    \hbox{a}\kern-.1emB}\,}
\def\Rlab{{\rm R\kern-.16em\lower.5ex\hbox{L}\kern-.25em\raise.3ex
    \hbox{a}\kern-.1emB}\,}
\def\rlab{{\rm R\kern-.16em\lower.5ex\hbox{L}\kern-.25em\raise.3ex
    \hbox{a}\kern-.1emB}\,}
 
%%\def\RLAB{{rlab}}
%%\def\RLaB{{rlab}}
%%\def\Rlab{{rlab}}
%%\def\rlab{{rlab}}
 
\begin{document}
 
\begin{titlepage}

\begin{center}
{\Huge \rlab\ Primer \\}
{\Large Version 1.0}

\vskip 1cm
{\Large Ian R. Searle \& Phillip Musumeci}
\end{center}

\vskip 3cm

\begin{figure}[h]
  \begin{picture}(350,300)(0,50) % (x-size, y-size) (x-shift, y-shift)
    \special{psfile=logo.ps
             angle=90 hscale=120 vscale=120 voffset=-100 hoffset=700}
    \end{picture}
\end{figure}

\vskip 1cm

\begin{center}
  The \RLaB\ program is \copyright copyright 1993, 94 Ian R. Searle.\\ This
  document is \copyright copyright 1993, 94 Ian R. Searle \& Phillip Musumeci.
\end{center}

\end{titlepage}
\newpage
\tableofcontents
\newpage
\listoffigures
\newpage

\setcounter{section}{-1}
%\setcounter{page}{35}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Primer priming}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\rlab is freely available}

\RLaB\ stands for Our-Lab, since it is intended to be a freely available
program that anyone can use, and contribute to. To protect this freedom,
copying of the program is protected by the GNU General Public License.

The main ftp site is \verb+evans.ee.adfa.oz.au+. The directory
\verb+pub/RLaB+ contains the sources and binary versions for some
machines. On the North American continent \verb+csi.jpl.nasa.gov+
acts as an archive site for \rlab, look in \verb+pub/matlab/RLaB+.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Acknowledgments}

The availability of ``free'' software, such as GNU Emacs, GNU gcc,
gdb, gnuplot, Plplot, and the Netlib archives has made this project
possible.  The \RLaB\ author thanks both the authors and sponsors of
the GNU, LAPACK, RANLIB, FFTPACK, and Plplot projects.

Many individuals have contributed to \rlab\ in various ways. A list of
contributors can be found in the source distribution file
\verb+ACKNOWLEDGMENT+. A special thanks to Phillip Musumeci and
Matthew Wette who have provided FTP sites so that \rlab\ is available
to all.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Document reproduction and errors}

%% PERMISSIONS

The \RLaB\ Primer is freely available. Permission is granted to reproduce
the document in any way providing that it is distributed for free, except
for any reasonable charges for printing, distribution, staff time, etc.
Direct commercial exploitation is not permitted.  Extracts may be made from
this document providing an acknowledgment of the original \LaTeX\ source is
maintained.

%% SUGGESTIONS

We welcome reports of errors and suggestions for improvement in this
document and also in \Rlab.  Please mail these to {\tt
rlab-list@eskimo.com}. Unfortunately (for you), free software does not earn
quite enough to pay a bribe for each error-free error report received but
do feel free to email them.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Requirements}

\RLaB\ is written in C. The maths libraries used are written in
Fortran but the use of a publicly available Fortran$\rightarrow$C
converter reduces compiler requirements to C (the conversion tool f2c
is written in C). The library used for data display, PLPLOT, is
publicly available in C source code form for a wide variety of
platforms. This makes the whole \RLaB\ package a good candidate for
porting onto platforms with C, especially GNU C.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{How to Read This Primer}

This primer has intentionally been kept short, so you should be able
to read all of it without too much effort. Probably the best way to
read this primer is to do so sitting at a computer, trying the
examples as you encounter them.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} \label{sect-intro}

\RLaB\ brings the power of stable matrix maths tools plus a stable data
plotting facility together in a form that is freely available and ready to
be compiled and used on a variety of common computer systems. \RLaB\ allows
you to experiment with complex matrix maths in an interactive environment.
Because you enter commands at a high (mathematical) level, you can
concentrate on figuring out your solution and hopefully avoid becoming
bogged down in low level implementation details. By minimising the effort
required to implement algorithms, it is hoped that you will be more willing
to discard old programs when confronted by better algorithms that warrant
use.

\RLaB\ uses a structured language\footnote{These days, computer
languages do look very similar but we will try to point out a few {\em
useful} similarities!} which will be familiar to users of C and also
the Wirth-inspired languages such as Pascal and Modula.  An \RLaB\
program is a file containing a sequence of commands or instructions
that you could also enter from your terminal---these instructions
might perform a calculation and assign the result to a variable, or
call a function which returns a result which you display on your
terminal, and so on. Functions can be either built-in or user-defined.
In fact, the only form of ``subprogram'' in \RLaB\ is the function
and, just like in C, a function returns a single item as its answer.
Data storage declared in the main routine of your program is stored on
a global symbol table, and is available to all of your subprogram
functions. By default, data used within functions is local to the
function. Such local function storage exists only for the duration of
the function call, in a way similar to variables declared locally
within Pascal procedures.  Comments can be appended to any line in
your program by using a special symbol at the start of the
comment---this is similar to Fortran and C++, and avoids the possible
pitfall of ``run away'' comments which might be familiar to Pascal
users.  Overall, the language syntax is perhaps closest to C but if
you have ever programmed in C or Pascal, you will soon be at ease with
\RLaB.

\RLaB\ features strongly typed objects but with the emphasis on
usefulness, not on pedantics. In \RLaB\, we talk about the {\em class}
of an object and the available classes include numeric, string,
function, and list.  The first class of object, {\em numeric},
encompasses numeric scalars, vectors, and matrices, and should be
familiar to the matrix maths user. The remaining classes borrow
concepts, and implementation details from other languages such as C.

It is worth noting that a function can be thought of as just another
object---this means that when you come to write your own functions
that use input parameters, you will enjoy the flexibility of being
able to pass in other functions as well as data as input to your
function. Another feature of functions as implemented in \RLaB\ is
that they can call themselves---anyone who has written a program to
calculate factorials will appreciate the elegance that recursion can
bring to some programming solutions.

Having whetted your appetite, this primer aims to get you started with
\RLaB\ as both an interactive tool and as a programming language. The
ideal approach is for you to read (or re-read) this document with an
\RLaB\ session staring up at you. After showing you how to run \RLaB\
and get on-line help, we describe data types before moving back to a
``hands on'' description of basic operations.  Program structure is
then described and you will see how to write your own functions. As
\RLaB\ comes with quite a few handy functions already built-in, we
give examples of their use including the plot function at which point
we hope you will be able to start using \RLaB\ to develop your own
programs.

% and you will find the quick reference tables in the appendices most
% useful.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Starting to use \RLaB}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{How to run it}

A properly installed \RLaB\ can be started on your terminal by entering
 
\begin{verbatim}
$ rlab
\end{verbatim}

where typewriter-style dark text is meant to represent the text you
would see sitting in front of a display terminal. The first character
on the input line is always the prompt, in this case a Bourne-shell
prompt. The text following is what the user enters. Text echoed by a
program is not preceded by any prompt.

\RLaB\ will start with a message similar to:

\begin{verbatim}
Welcome to RLaB. New users type `help INTRO'
RLaB version 1.0 Copyright (C) 1992, 93, 94 Ian Searle
RLaB comes with ABSOLUTELY NO WARRANTY; for details type `help WARRANTY'
This is free software, and you are welcome to redistribute it under
certain conditions; type `help CONDITIONS' for details
>
\end{verbatim}

The \verb+>+ symbol on the last line next to the cursor is the \RLaB\ 
command prompt.  At this point, users should take the advice offered and
be usefully distracted from this primer by {\em actually reading} the
information available from {\tt help INTRO} - do not worry if you cannot
follow it yet. After you have read each screenful, press {\tt SPACE} (i.e.
the space bar) to see further screens of information.

   At this point it is only fair to tell you how to stop it. To stop a
   \rlab session you can type \verb+quit+ at the \rlab prompt. On Unix
   systems an \verb+EOF+ or \verb+^d+ (control-d) will also stop
   \rlab. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Help}

To get a taste of the functions for which help is available, enter

\begin{verbatim}
> help
\end{verbatim}

The first group of topics lists functions and special help topics
that are built into \RLaB. The special topics have names in upper case and
are of a general nature. Lawyers recommend that you {\em now} read the help
on topics {\tt CONDITIONS} and {\tt WARRANTY} by entering

\begin{verbatim}
> help CONDITIONS
\end{verbatim}
\begin{verbatim}
> help WARRANTY
\end{verbatim}

The subsequent topics refer to commands that have been written in
\RLaB\ script files, which we refer to as ``R-files''. These R-files
are stored in directories, which the \verb+help+ command searches. The
help files in the {\em \ldots /rlib} directory come as a standard part of
\RLaB, and the remainder refer to local R-files that have been setup
for you by whoever installed your \RLaB. 

In general, the functions listed in the first group are the most efficient
as they are compiled into the core of \RLaB. In contrast, \RLaB's R-files
have the extra overhead of reading and interpretation before they are
executed. This lower efficiency associated with R-file interpretation is
traded for the benefit of being able to write your own features into \RLaB.
If an R-file feature is really useful, it can be added to the core \RLaB\
program since you have the source code\footnote{Dynamic linking is
part of the plan for a future release}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simple calculations} \label{simple-calc}

\RLaB\ is designed for mathematical calculations so let's do some. The four
basic arithmetic operators have symbols {\verb|+, -, *, /|} representing
addition, subtraction, multiplication, and division respectively. Now enter
some one line expressions as shown here:

\begin{verbatim}
> 2*4
        8
> 1/2
      0.5
> 1+11
       12
> 1-11
      -10
> 1*2/3+4-5
   -0.333
> 1/0
      inf
> 0/(1/0)
        0
> 0/0
      NaN
\end{verbatim}

The $5^{th}$ expression illustrates compliance with the usual operator
hierarchy and then we observe that \RLaB\ can handle
exceptions\footnote{The ability to properly handle floating point
exceptions is dependent upon the capabilities of the hardware and
operating system, as well as the interests of the person installing
\RLaB\ .} such as when $\infty$ (\verb+inf+) is a result or an input to
further calculation; and also ``not-a-number'' (\verb+NaN+).  \RLaB\ can use
complex numbers as well as real numbers so now try

\begin{verbatim}
> 1/1i
        0 - 1i
> 1/1i + 1/1j
        0 - 2i
> 1/1i * 1/1j
       -1
> 1/1i/1j
       -1
\end{verbatim}

where we see that {\tt i} or {\tt j} can represent the complex
number $\sqrt{-1}$. No four function calculator is complete without a
memory so now we look at how to store results in a variable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Variable assignment and display}

In \RLaB, variables can have names of any length containing most printable
characters including $\{ a \ldots z , A \ldots Z , \_ \}$. You will observe
that we have to exclude special characters such as {\verb|+, -, *, /|} and
the {\tt SPACE} character. The actual assignment operator symbol is ``=''
and an assignment statement looks like

\begin{verbatim}
                variable_name = expression_to_evaluate
\end{verbatim}

and an example is

\begin{verbatim}
> radius=2
 radius =
        2
> circumference=2 * pi * radius
 circumference =
     12.6
\end{verbatim}

where a variable {\tt radius} is created and initialised with the
real value 2, and then a variable {\tt circumference} is created and filled
with the result of evaluating the right hand side of the equation. To see
the value of either of these variables, just enter their name and
\RLaB\ will print their value.  For a description of variable names, please read
the help on {\tt VARIABLES}.

As you have probably noticed by now, the result of each expression is
automatically printed to the screen. This feature can be controlled by
using the `\verb+;+' character. Terminating an expression with a
`\verb+;+' will suppress printing of the result. Likewise, terminating
an expression with the `\verb+?+' is an explicit way to force printing.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{User Interface: command recall \& editing}

Command line recall and editing is very useful for correcting command
errors or to allow your commands to evolve.  \RLaB\ provides a command
recall and edit facility modeled on (and sometimes actually using) the GNU
readline facility. If you are familiar with GNU emacs or the GNU bash
shell, then try entering {\tt C-p} to scroll back through previous commands
({\tt C-p} means hold down the {\tt control} key and press {\tt p}).  If
this is successful, test the standard character and word editing commands
to modify previous entries - if it works, skip to section
\ref{sec:objects}.

However, if the word GNU just means ``any of several African antelopes
constituting the genus {\em Connochaetes} ...''\footnote{Australian
Macquarie dictionary, ISBN: 0-949757-23-3.} but your keyboard does have the
arrow keys \{\ $\leftarrow\ \uparrow\ \downarrow\ \rightarrow\ $\}, then
you might still be able to take advantage of command line recall and
editing.  Try typing the $\uparrow$ key to see if any previous \RLaB\
commands are displayed - if they are, then confirm that $\downarrow$ also
displays more recent commands and then try horizontal cursor movement with
the \{\ $\leftarrow\ \rightarrow\ $\} and try some editing with the delete
key.  Typing {\tt C-d} ought to delete the character beneath the cursor.
When a new command has been created from an old, enter it in the usual way
by pressing {\tt RETURN}.  If this has worked for you, skip the remainder
of this section (and count yourself lucky that we weren't describing a
graphical user interface in one paragraph).

If your keyboard is missing the arrow keys but {\tt C-p} did cause previous
commands to pop up on the \RLaB\ command line, you will find that \{\
$\leftarrow\ \uparrow\ \downarrow\ \rightarrow\ $\} are the same as \{\
{\tt C-b C-p C-n C-f}\ \} - think of b for backwards, p for previous, n for
next, and f for forward.

Irrespective of what keystrokes you use for editing, the C-y keystroke will
restore text previously deleted. If you were unable to scroll back through
any previous commands (that you had just entered), then your \RLaB\ may have
been built without command line editing - this is unlucky.  As command
line editing is such a useful feature, you should consider getting a
better version of \RLaB\ if possible. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Objects---Basic Data Structures} \label{sec:objects}

In the most general form, an object in \RLaB\ can be data or a
function. It is even possible to construct an object that contains
both data {\em and} functions - a fact that no doubt excites the
hormones in the modern day object oriented programmer. We are going to
discuss basic data types before looking at how data can be ``grouped''
together for some useful purpose. We will also work through some
simple examples that manipulate data but first, what does \Rlab\
regard as data?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Data Types}

There are three {\em fundamental} types of data that you manipulate in
\RLaB: the string; the real number; and the complex number.  As we have
seen in section \ref{simple-calc}, it is straight-forward to
manipulate numerical quantities. Characters are available in the form
of strings which can contain 0 or more characters. In line with a
philosophy to ``keep it simple'', \RLaB\, which is primarily concerned
with numerical computation, has no special way to handle a single
character\footnote{And also one less data type that you need to learn
about.}. To enter a string, enclose the characters inside quotes like
\verb+"this"+ e.g.

\begin{verbatim}
> "Hello world"
Hello world
\end{verbatim}

Just as a number was previously stored in a variable, the same can be done
with a string of characters. To place a string into a variable, you
could enter a statement such as

\begin{verbatim}
> hw = "Hello world"
Hello world
\end{verbatim}

and the value of variable \verb+hw+ may be printed out by entering

\begin{verbatim}
> hw
Hello world
\end{verbatim}

The observant reader might be wondering what has happened to the
boolean data type? In \RLaB, {\em true} and {\em false} are
represented by the integers 1 and 0. Just as the data type char can be
handled as a rather small string (length=1), so the data type boolean
(or logical) can be handled by small numbers (value=0,1). We have now
met the 3 fundamental types of data processed in \RLaB\ and it is now
possible to understand a little more about how data structures and
functions are organised within \RLaB.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Object Hierarchy}

Scan your eyes down over Figure \ref{fig:objects} which shows the
hierarchical structure of objects in \RLaB\ - we shall now describe
this figure from the bottom up (ignoring lists until a little
later). Not all objects are created in the same way and what you can do with
or to them depends on their {\em class}. Items of class {\em function}
contain program instructions which is one form of data or
information. Items of class {\em numeric}, and {\em string} contain
data that \RLaB\ instructions can manipulate.

\begin{figure}[htbp]
  \begin{picture}(350,300)(0,25) % (x-size, y-size) (x-shift, y-shift)
    \special{psfile=object.ps
             angle=0 hscale=65 vscale=65 voffset=-55 hoffset=40}
    \end{picture}
\caption{\RLaB objects} \label{fig:objects}
\end{figure}

A numeric class item can store a real or complex number.  An item of
class string contains a null-terminated string of character(s). When
we want to access or create an array of items, we use an array syntax
that is the same for both string and numeric classes.

It is often helpful to a programmer to group together unlike data into
a single object - this is the purpose of the class list.  We are not
going to describe it in great detail here except to point out that it
serves a similar role to a record in Pascal or a structure in C, but
with a somewhat more flexible access mechanism. Note that lists can
contain any of the aforementioned objects, even another list.

One thing that you can always do with any item is ask \RLaB\ what its
class is. For example, \RLaB\ has a built-in command to calculate the
sin of an angular quantity - asking \RLaB\ about it gives the
following response

\begin{verbatim}
> class( sin )
function
\end{verbatim}

From the size of the list of topics that help is available on, you
probably realise that there are many built-in functions in \RLaB\ -
expect gratuitous use of these functions as further examples are
given.  Remember that you can find out about any function by typing
{\tt help} {\em function-name}. We are particularly interested in
exploring the use of \RLaB\ as a computation tool so now we describe
further numeric operations. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Numerics}

   The \RLaB\ numeric class includes objects of type real and
   complex. The numeric object also encompasses objects of scalar,
   vector, or matrix dimension. If you want to, you can think of all
   numeric objects as matrices. Thus, a vector is simply a 1-by-N
   matrix, and a scalar is a 1-by-1 matrix. Since the numeric object
   is most commonly used, it will get the most coverage.

\subsubsection{Matrix Creation }

   The simplest way to create a matrix is to type it in at the command
   line: 

\begin{verbatim}
> m = [ 1, 2, 3; 4, 5, 6; 7, 8, 9 ]
 m =
        1          2          3  
        4          5          6  
        7          8          9  
\end{verbatim}

   In this context the `\verb+[ ]+' signal \Rlab\ that a matrix should
   be created. The inputs (or arguments) for matrix creation are
   whatever is inside the `\verb+[ ]+'. The rows of the matrix are
   delimited with `\verb+;+' and the elements of each row are
   delimited with `\verb+,+'.

   Users can use most any expression when creating matrix elements.
   Other matrices, function evaluations, and arithmetic operations are
   allowed when creating matrix elements. In the next example, we will
   create a direction cosine matrix using the built-in trigonometric
   functions within the `\verb+[ ]+'.

\begin{verbatim}
> a = 45*(2*pi)/360
 a =
    0.785
> A = [ cos(a), sin(a); -sin(a), cos(a) ]
 A =
    0.707      0.707  
   -0.707      0.707  
\end{verbatim}

   Matrices can also be read from disk-files. The functions
   \verb+read+, \verb+readb+ and \verb+readm+ can read matrix values
   from a file. The \verb+read+ function uses a special ASCII text
   file format, and is capable of reading not only matrices, but
   strings, and lists as well. Since the file can contain many data
   objects, and their variable names, \verb+read+ is used like:

\begin{verbatim}
> read ( "file.dat" );
\end{verbatim}

   The variables are read from \verb+file.dat+ and installed in the
   global-symbol-table.

   The \verb+readb+ function works like \verb+read+, except it reads
   binary files for greater efficiency. The binary files created with
   \verb+writeb+ are portable across computers that use IEEE floating
   point format.

   The \verb+readm+ function reads a text file that contains
   white-space separated columns of numbers. \verb+readm+ is most
   often used to read in data created by other programs. Since
   \verb+readm+ is only capable of reading in one matrix per file, and
   no variable name information is available, \verb+readm+ is used
   like:

\begin{verbatim}
> a = read ( "a.dat" );
\end{verbatim}

\subsubsection{Vector Creation}

   Although there is no distinct vector type in \RLaB\ , you can
   pretend that there is. If your algorithm, or program does not need
   two dimensional arrays, then you can use matrices as singly
   dimensioned arrays.

   When using vectors, or single dimension arrays, row matrices are
   created. The simplest way to create a vector is with the `\verb+:+'
   operator(s), that is `\verb+start:end:inc+'. The leftmost operand,
   \verb+start+, specifies the starting value, the second operand,
   \verb+end+, specifies the last value. The default increment, or
   spacing, is 1.  A third optional operand, \verb+inc+, can be used
   to specify any increment.

\begin{verbatim}
> v = 1:4
 v =
        1          2          3          4  
\end{verbatim}

\subsubsection{Matrix Attributes}

   Matrix attributes, such as number of rows, number of columns, total
   number of elements, are accessible in several ways. All attributes
   are accessible through function calls, for example:

\begin{verbatim}
> a = rand(3,5);
> show (a)
   name:      a     
   class:     num   
   type:      real  
     nr:      3     
     nc:      5     
> size (a)
        3          5  
> class (a)
num
> type (a)
real
\end{verbatim}

   Matrix attributes are also accessible via a shorthand notation:

\begin{verbatim}
> a.nr
        3
> a.nc
        5
> a.n
       15
> a.class
num
> a.type
real
\end{verbatim}
  
   Note that these matrix attributes are ``read-only''. In other
   words: assignment to \verb+a.nr+ is pointless. In fact it will
   destroy the contents of \verb+a+ and create a list with element
   named \verb+nr+. If you wish to change a matrix attribute, you must
   do so by changing the data in \verb+a+. For example: if you want to
   make \verb+a+ complex:

\begin{verbatim}
> a = a + zeros (size (a))*1i;
> show(a)
   name:      a
   class:     num
   type:      complex
     nr:      3
     nc:      5
\end{verbatim}

   If you want to change the number of rows, or columns of \verb+a+:

\begin{verbatim}
> a = reshape (a, 1, 15);
> show(a)
   name:      a
   class:     num
   type:      complex
     nr:      1
     nc:      15
\end{verbatim}

\subsubsection{Element Referencing}

   Any expression that evaluates to a matrix can have its elements
   referenced. The simplest case occurs when a matrix has been created
   and assigned to a variable. One can reference single elements, or
   one can reference full or partial rows and/or columns of a matrix.
   Element referencing is performed via the `{\verb+[ ]+}' operators,
   using the `\verb+;+' to delimit row and column specifications, and
   the `\verb+,+' to delimit individual row or column specifications.

   To reference a single element:

\begin{verbatim}
> a = [1,2,3; 4,5,6; 7,8,9];
> a [ 2 ; 3 ]
        6
\end{verbatim}

   To reference an entire row, or column:

\begin{verbatim}
> a [ 2 ; ]
        4          5          6  
> a [ ; 3 ]
        3  
        6  
        9  
\end{verbatim}

   To reference a sub-matrix:

\begin{verbatim}
> a [ 2,3 ; 1,2 ]
        4          5  
        7          8  
\end{verbatim}

   As stated previously, any expression that evaluates to a matrix can
   have its elements referenced. A very common example is getting the
   row or column dimension of a matrix:

\begin{verbatim}
> size (a)[1]
        3
\end{verbatim}

   In the previous example the function \verb+size+ returns a
   two-element matrix, from which we extract the 1st element (the
   value of the row dimension). Note that we referenced the return
   value (a matrix) as if it were a vector. Referencing matrices in
   ``vector-fashion'' is allowed with all matrices. When
   vector-indexing is used, the matrix elements are referenced in
   column order. As with matrix indexing, a combination of vector
   elements can be referenced: 

\begin{verbatim}
> a[3]
        7
> a[3,4,9]
        7          2          9  
\end{verbatim}

\subsubsection{Assignment}

   Matrices can be assigned to in whole or in part. We have shown
   complete matrix assignment in the examples of the last few pages. In
   the same way that matrix elements can be referenced singly, or in
   groups, matrices can have single elements re-assigned, or groups of
   elements re-assigned. The result of an assignment expression is the
   left-hand-side (LHS). This is more convenient when working
   interactively, and when creating intermediate function arguments.

\begin{verbatim}
> a[2;2] = 200
 a =
        1          2          3  
        4        200          6  
        7          8          9  
> a[2,3;2,3] = [200,300;300,400]
 a =
        1          2          3  
        4        200        300  
        7        300        400  
\end{verbatim}

   The row and column dimensions of the matrices on the RHS, and the
   matrix description within the `\verb+[ ]+' must have the same
   dimensions.


\subsubsection{Matrix Operations}

   The usual mathematical operators {\verb|+,-,*,/|} operate on
   matrices as well as scalars. For {\verb+A binop B+}:

   \begin{description}
     \item[{\tt +}] Does element-by-element addition of two matrices.
              The row and column dimensions of both \verb+A+ and
              \verb+B+ must be the same. An exception to the
              aforementioned rule occurs when either \verb+A+ or
              \verb+B+ is a 1-by-1 matrix; in this case a
              scalar-matrix addition operation is performed.

     \item[{\tt -}] Does element-by-element subtraction of two matrices.
              The row and column dimensions of both \verb+A+ and
              \verb+B+ must be the same. An exception to the
              aforementioned rule occurs when either \verb+A+ or
              \verb+B+ is a 1-by-1 matrix; in this case a
              scalar-matrix addition operation is performed.

     \item[{\tt *}] Performs matrix multiplication on the two
              operands.  The column dimension of {\verb+A+} must match
              the row dimension of {\verb+B+}. An exception to the
              aforementioned rule occurs when either \verb+A+ or
              \verb+B+ is a 1-by-1 matrix; in this case a
              scalar-matrix multiplication is performed.

     \item[{\tt /}] Performs matrix ``right-division'' on its operands.
              The matrix right-division (\verb+B/A+) can be thought of
              as {\verb+B*inv (A)+}. The column dimensions of \verb+A+
              and \verb+B+ must be the same. Internally right division
              is the same as ``left-division'' with the arguments
              transposed. 

		\[ B / A = ( A^T \setminus B^T )^T \]

              The exception to the aforementioned dimension rule
              occurs when \verb+A+ is a 1-by-1 matrix; in this case a
              matrix-scalar divide occurs.

   \end{description}

   Additionally, \RLaB\ has several other operators that function on
   matrix operand(s). 


   \begin{description}
     \item[{\tt .+}] Performs element-by-element addition on
              its operands. The operands must have the same row and
              column dimensions. {\em Unless}:

	      \begin{itemize}

		\item \verb+A+ or \verb+B+ is a 1x1. In this case the
		operation is performed element-by-element over the
		entire matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a 1xN. and the other is
		MxN. In this instance the operation is performed
		element-by-element fashion for each row in the
		matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a Nx1. and the other is
		NxM. In this instance the operation is performed
		element-by-element fashion for each column in the
		matrix. The result is a NxM matrix.

              \end{itemize}

     \item[{\tt .-}] Performs element-by-element subtraction on
              its operands. The operands must have the same row and
              column dimensions. {\em Unless}:

	      \begin{itemize}

		\item \verb+A+ or \verb+B+ is a 1x1. In this case the
		operation is performed element-by-element over the
		entire matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a 1xN. and the other is
		MxN. In this instance the operation is performed
		element-by-element fashion for each row in the
		matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a Nx1. and the other is
		NxM. In this instance the operation is performed
		element-by-element fashion for each column in the
		matrix. The result is a NxM matrix.

              \end{itemize}

     \item[{\tt .*}] Performs element-by-element multiplication on
              its operands. The operands must have the same row and
              column dimensions. {\em Unless}:

	      \begin{itemize}

		\item \verb+A+ or \verb+B+ is a 1x1. In this case the
		operation is performed element-by-element over the
		entire matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a 1xN. and the other is
		MxN. In this instance the operation is performed
		element-by-element fashion for each row in the
		matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a Nx1. and the other is
		NxM. In this instance the operation is performed
		element-by-element fashion for each column in the
		matrix. The result is a NxM matrix.

              \end{itemize}

     \item[{\tt ./}] Performs element-by-element division on its
              operands. The operands must have the same row and column
              dimensions. {\em Unless}:

	      \begin{itemize}

		\item \verb+A+ or \verb+B+ is a 1x1. In this case the
		operation is performed element-by-element over the
		entire matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a 1xN. and the other is
		MxN. In this instance the operation is performed
		element-by-element fashion for each row in the
		matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a Nx1. and the other is
		NxM. In this instance the operation is performed
		element-by-element fashion for each column in the
		matrix. The result is a NxM matrix.

              \end{itemize}

     \item[$\setminus$] Performs matrix ``left-division''. Given
              operands {\verb+A\B+} matrix left division is the
              solution to the set of equations $Ax = B$. If $B$ has
              several columns, then each column of $x$ is a solution
              to {\verb+A*x[;i] = B[;i]+}. The row dimensions of
              \verb+A+ and \verb+B+ must agree.

     \item[$.\setminus$] Performs element-by-element left-division.
              Element-by-element left-division is provided for
              symmetry, and is equivalent to \verb+B./A+. The row and
              column dimensions of \verb+A+ and \verb+B+ must agree,
              {\em unless}:

	      \begin{itemize}

		\item \verb+A+ or \verb+B+ is a 1x1. In this case the
		operation is performed element-by-element over the
		entire matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a 1xN. and the other is
		MxN. In this instance the operation is performed
		element-by-element fashion for each row in the
		matrix. The result is a MxN matrix.

		\item \verb+A+ or \verb+B+ is a Nx1. and the other is
		NxM. In this instance the operation is performed
		element-by-element fashion for each column in the
		matrix. The result is a NxM matrix.

              \end{itemize}

     \item[${}^\wedge{}$] {\verb+A^B+} raises \verb+A+ to the \verb+B+
              power. When \verb+A+ is a matrix, and \verb+B+ is an
              integer scalar, the operation is performed by successive
              multiplications. When \verb+B+ is not an integer, then
              the operation is performed via \verb+A+'s eigenvalues
              and eigenvectors. The operation is not allowed if
              \verb+B+ is a matrix.

     \item[${.}^\wedge{}$] {\verb+A.^B+} raises \verb+A+ to the
              \verb+B+ power in an element-by-element fashion. Either
              \verb+A+ or \verb+B+ can be matrix or scalar. If both
              \verb+A+ and \verb+B+ are matrix, then the row and
              column dimensions must agree.

     \item[{\tt '}] This unary operator performs the matrix transpose
              operation. \verb+A'+ swaps the rows and columns of A.
              For a matrix with complex elements a complex conjugate
              transpose is performed.

     \item[{\tt .'}] This unary operator performs the matrix
              transpose operation. \verb+A.'+ swaps the rows and
              columns of A.  The difference between \verb+'+ and
              \verb+.'+ is only apparent when \verb+A+ is a complex
              matrix; then \verb+A.'+ does not perform a complex
              conjugate transpose.

   \end{description}

   Several details are important to note:

   \begin{itemize}

     \item The two character operators are just that, two characters. White
           space, or any other character in between the two symbols is
           an error, or may be interpreted differently.

     \item The expression \verb+2./A+ is {\bf not} interpreted as
           \verb+2. /A+. \Rlab\ is smart enough to group the period with
           the `\verb+/+'.

   \end{itemize}

\subsubsection{Matrix Relational Operations}

   The way matrices can be used within if-statement tests is special.
   The result of a matrix relational test, such as \verb+A == B+, is a
   matrix the same size as \verb+A+ and \verb+B+ filled with ones and
   zeros according to the result of an element-by-element test. If
   either of the operands is scalar, or a 1-by-1 matrix, then the
   element-by-element test is performed as before, by using the scalar
   value repeatedly. For example.

\begin{verbatim}
> a = [1, 2, 3; 4, 5, 6; 7, 8, 9];
> b = a';
> a == b
        1          0          0  
        0          1          0  
        0          0          1  
> a >= 5
        0          0          0  
        0          1          1  
        1          1          1  
\end{verbatim}

   \RLaB\ if-tests do not accept matrices. The built-in functions
   \verb+any()+ and \verb+all()+ can be used in combination with
   relational and logical tests to conditionally execute statements
   based upon matrix properties. For example: perform a test that
   returns true or false (0 or 1) if \verb+a+ contains the value 4.

\begin{verbatim}
> any ( any (a == 4))
\end{verbatim}

   The function \verb+any()+ returns true if any of the element(s) of
   its argument are non-zero. The function \verb+all()+ returns true
   if all of the element(s) of its argument are non-zero. Note that
   \verb+any+ is used twice; this is because \verb+any+ is a
   vector-oriented function. This will be discussed later.

\subsubsection{Examples}

   Now it is time for a few illustrative examples \ldots

   Suppose you are learning about normal-equations, orthogonal
   transformations, QR decompositions, etc.\footnote{You've been
   reading Kahaner, Moler, and Nash ``Numerical Methods and Software''
   for example.} You have read the proper sections in your text(s), and
   you want to try your hand at it to see if you really understand it.

   First you create an over-determined coefficient matrix, 3
   parameters, and 5 equations (\verb+a+). Then you create an
   observation matrix (\verb+b+):

\begin{verbatim}
> a = [3,4,1;0,2,2;0,0,7;zeros(2,3)];
> b = [14;10;21;6;2];
\end{verbatim}

   You've just read that the \RLaB\ operator `\verb+\+' solves systems
   of equations, so you try it out: 

\begin{verbatim}
> x = a \ b
 x =
        1  
        2  
        3  
\end{verbatim}

  You check the answer (note that this is a contrived problem):

\begin{verbatim}
> b - a*x
-7.11e-15  
-1.78e-15  
-1.42e-14  
        6  
        2  
\end{verbatim}

   and it looks ``OK''. The residual in the first three rows is near
   the machine-epsilon\footnote{Machine-epsilon is the smallest number
   that your computer can distinguish. A common way to determine
   machine-epsilon is to divide a variable ({\tt eps})
   by two until {\tt 1.0 + eps  == 1.0}.}.
   Now you wish to follow the example in the
   text more closely, in an attempt to reinforce your reading. The
   text has stated that the ``normal equations'' are: 
   $(A^{T}A)x = (A^{T}b)$.

   Not having read the chapter on Gaussian elimination, and matrix
   inverses yet you try: 

\begin{verbatim}
> x = inv (a'*a) * (a'*b)
 x =
        1  
        2  
        3  
\end{verbatim}

   Fortunately, the problem we are working with is not
   ill-conditioned, otherwise we may have produced a terrible result
   with the above procedure. If you want to pursue the reasoning
   behind the previous statement I suggest you read the section
   ``Linear Systems of Equations''.

   Well, this is all too easy, now you want to get dirty, so you move
   on to orthogonal transformations. You have read about the
   construction of Householder vectors and reflections; now you would
   like to try it first-hand. You know that:

   \[ P = I - 2(vv^{T})/(v^{T}v) \]

   Where $v$ is the Householder vector used to form the reflection
   matrix. First you must construct the vector. Your
   text\footnote{Mine in this case is Golub and VanLoan, ``Matrix
   Computations''.} tells you that a good method for constructing the
   Householder vector is:

  \[ v[2:n] = x[2:n] / (x[1] + sign(x[1]) * \|x\|_{2}) \]
  \[ v[1] = 1 \]

\begin{verbatim}
> a = rand(5,2);               // Start out with a more difficult [A]
> a
 a =
    0.655      0.265  
    0.129        0.7  
     0.91       0.95  
    0.112     0.0918  
    0.299      0.902  
> ac1 = a[;1];                 // grab the 1st column of [a] to work with
> u = norm (ac1, "2");         // compute the 2-norm of [ac1]
> v[2:5] = ac1[2:5] / (ac1[1] + sign (ac1[1])*u)
 v =
        0     0.0705      0.498     0.0611      0.164  
> v[1] = 1;
> v = v';                      // make v a column vector
\end{verbatim}

   By using the matrix creation, and element referencing features you
   have generated the vector in 4 commands. We could have used
   a signal command 

\begin{verbatim}
> v = [ 1 , a[;1][2:5]' / (a[1;1] + sign(a[1;1])*norm(a[;1],"2")) ]'
\end{verbatim}

   But, this is less than clear. Note that in this case, since we are
   working with vectors, we only use a single index when subscripting
   the variables.

   Now that we have our Householder vector, we are ready to assemble
   the Householder reflection (matrix).

\begin{verbatim}
> P = eye (5,5) - 2*(v*v')/(v'*v)
 P =
   -0.558      -0.11     -0.776    -0.0952     -0.255  
    -0.11      0.992    -0.0547   -0.00671     -0.018  
   -0.776    -0.0547      0.614    -0.0474     -0.127  
  -0.0952   -0.00671    -0.0474      0.994    -0.0156  
   -0.255     -0.018     -0.127    -0.0156      0.958  
> P*a
    -1.17       -1.2  
-1.65e-17      0.596  
-1.54e-16       0.22  
-1.31e-17    0.00217  
-5.39e-17      0.662  
\end{verbatim}

   As you can see it worked out just like they said it would. All the
   elements of the first column of $A$, below the diagonal, have been
   zeroed out\footnote{Although the numbers are not identically zero,
   they are on the order of machine precision, and can be considered
   to be zero. Some programs use fixed-point precision when printing.
   Although this makes their output look more accurate, it is
   not.}. In this manner we can proceed to transform $A$ into an upper
   triangular matrix.

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Program Flow Control}

   We must now take a small diversion before proceeding on with the
   rest of the objects and discuss flow-control. The flow-control
   statements available in \RLaB\ are the {\em if-statement}, the {\em
   while-statement}, the {\em for-statement}, the {\em break-statement}
   and the {\em continue-statement}.

   The flow-control statements use a syntax that is {\em similar} to
   the C-language\footnote{Note we said similar, not identical.}.
   Braces `\verb+{+' and `\verb+}+' are {\em required} in all of the
   flow-control statements.

\subsection{If-Statement}

   The {\em if-statement} performs a test on the expression in
   parenthesis, and executes the statements enclosed within braces if
   the expression is true. The expression must evaluate to a
   scalar-expression. If the expression evaluates to a vector or
   matrix a run-time error will result.

   \begin{tabbing}
    123456 \= 1234 \=  \kill
    \> if \ (\  {\it expression} \ )  \nopagebreak \\
    \> \{               \\
    \> \> {\it statements}    \\
    \> \}               \\
   \end{tabbing}

\begin{verbatim}
> if ( 1 ) { "TRUE" }
TRUE
> if ( 0 ) { "TRUE" }
\end{verbatim}

   An optional `\verb+else+' keyword is allowed to delineate statements
   that will be executed if the expression tests false:

\begin{verbatim}
> if ( 0 ) { "TRUE" else "FALSE" }
FALSE
\end{verbatim}

   The \verb+any+ and \verb+all+ functions are useful with {\em
   if-statements}. If we want to execute some statements, conditional
   on the contents of a matrix:

\begin{verbatim}
> a=[1,2;3,0];
> if (!all (all (a))) { "a has a zero element" }
a has a zero element
\end{verbatim}

\subsection{While-Statement}

   The {\em while-statement} tests the expression in parenthesis, and
   executes the statements enclosed within braces until the expression
   is false. The expression must evaluate to a scalar-expression. If
   the expression evaluates to a vector or matrix a run-time error
   will result.


   \begin{tabbing}
    123456 \= 1234 \=  \kill

    \> while \ (\  {\it expression} \ ) \\
    \> \{               \\
    \> \> {\it statements}    \\
    \> \}               \\
   \end{tabbing}

\begin{verbatim}
> while ( 0 ) { "TRUE" }
> i = 0;
> while ( i < 2 ) { i = i + 1 }
 i =
        1
 i =
        2
\end{verbatim}

\subsection{For-Statement}

   The {\em for-statement} executes the statements enclosed in braces
   $N$ times, where $N$ is the number of values in {\it
   vector-expression}. Each time the statements are executed {\it
   variable} is set to the $k^{th}$ value of {\it vector-expression},
   where $k = 1 \ldots\ N$.

   \begin{tabbing}
    123456 \= 1234 \=  \kill

    \> for \ (\  {\it variable} in {\it vector-expression} \ ) \\
    \> \{               \\
    \> \> {\it statements}    \\
    \> \}               \\
   \end{tabbing}

\begin{verbatim}
> for ( i in 1:3 ) { i }
 i =
        1
 i =
        2
 i =
        3
\end{verbatim}

   {\it vector-expression} can be any type of vector: real, complex,
   and string vectors are all acceptable.

\subsection{Break and Continue Statements}

   The {\em break} and {\em continue} statements are simply keywords.
   Usage of {\em break} and {\em continue} is only allowed within {\em
   while-statements} or {\em for-statements}. {\em break} will cause
   execution of the current loop to terminate. {\em continue} will
   cause the next iteration of the current loop to begin.

\begin{verbatim}
> for ( i in 1:100 ) { if (i == 3) { break } } i
 i =
        3
> for ( i in 1:4 ) { if (i == 2) { continue } i }
 i =
        1
 i =
        3
 i =
        4
\end{verbatim}

   Although they will not be explicitly discussed - there are more
   examples of flow-control statement usage throughout the remainder
   of the primer.
   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Objects---Program Functions}	\label{sect-functions}

   Like matrices and strings, functions are stored as ordinary
   variables in the symbol table. Function's treatment as variables
   explains the somewhat peculiar syntax required to create and store
   a function.

\begin{verbatim}
> logsin = function ( x ) { return log (x) .* sin (x) }
	<user-function>
\end{verbatim}

   The above statement creates a function, and assigns it to the
   variable \verb+logsin+. The function can then be used like:

\begin{verbatim}
> logsin ( 2 )
     0.63
\end{verbatim}

   Like variables, function can be copied, re-assigned, and destroyed. 

\begin{verbatim}
> y = logsin
	<user-function>
> y (2)
     0.63
> 
> // Overwrite it with a matrix
> logsin = rand (2,2);
> 
> // Check that y is still a function
> y (3)
    0.155
\end{verbatim}

   If you try re-assigning a built-in function you will get a run-time
   error message. The built-in functions, those that are programmed in
   C, are a permanent part of the environment. So that users may
   always rely on their availability, they cannot be re-assigned, or
   copied. 

   Variables that represent user-functions can also be part of list
   objects. Sometimes it can be useful to group functions that serve a
   similar purpose, or perform different parts of a larger procedure.

\begin{verbatim}
list = << logsin = logsin >>
   logsin       
> list.logsin (2)
     0.63
> list.expsin = function ( x ) { return exp (x) .* sin (x) }
   expsin       logsin       
> list.expsin (2)
     6.72
\end{verbatim}

\subsection{Function Syntax}

   The function syntax is fairly simple. The basic form of a function
   is:

   \begin{tabbing}
    123456 \= 1234 \=  \kill

    \> function \ (\  {\it argument list} \ ) \\
    \> \{               \\
    \> \>               \\
    \> \> {\it statements}    \\
    \> \>               \\
    \> \}               \\
   \end{tabbing}

   If a syntax error is encountered while the function is being
   entered (read), definition of the function must begin again from
   the very beginning.

   There are several statements that only make sense within functions:

   \begin{tabbing}
    123456 \= 1234 \=  \kill
    \> global \ (\  {\it global-var-1, global-var-2 \ldots} \ ) \\

    \> local \ (\  {\it local-var-1, local-var-2 \ldots} \ ) \\

    \> return {\it expression} \\

   \end{tabbing}


   The \verb+global+ and \verb+local+ statements are optional. There
   are no restrictions on the number of \verb+local+ or \verb+global+
   statements or where they occur in the function. However, since
   these two statements only affect variables that are used {\em
   after} the declaration, it is recommended that you use \verb+local+
   and \verb+global+ at the {\em beginning} of each function.

   The \verb+return+ statement is also optional. There are no
   restrictions on the number of return statements, or their
   placement. A function can return from any point in its
   execution. The return statement must return a value. The value can
   be any \RLaB\ object.

\subsection{Function Scoping Rules}

   When you start a \rlab\ session, either interactively or in
   batch-mode, you create an environment. The environment or workspace
   consists of the built-in functions, and any other variables or
   functions you may have added. The workspace will also be referred
   to as the global-symbol-table or the global scope.

   There are two other types of environment available: a function's
   local environment and a file's static environment\footnote{We will
   use the term environment and scope interchangeably.}

   A function's local scope is temporary, it is created when the
   function is invoked, and is destroyed when the function returns. A
   file's static scope is created when the file is loaded, and remains
   intact until the \rlab\ session is terminated.

   The different scopes serve to protect data from operations that
   occur in the other scopes. There is some degree of overlap in order
   to allow flexibility. Functions can affect file-static and global
   scopes; statements within files can affect statements within other
   files and the global scope. More simply put, the "lower" scopes
   generally have access to the "higher" scopes. When a variable is
   used, \rlab\ uses certain rules to "bind" the variable. When we use
   the term bind or bound, we mean that the variable name is
   associated with an entry in one of the three types of symbol
   tables.


   \begin{description}

   \item[File-Scope:] Variables that are in a file (but not within a
	function) are bound to the global-symbol-table (global-scope
	or global-environment) unless a static declaration is
	used. When a variable is declared static it is bound to the
	file's symbol table. From that point on, the variable will
	remain bound to the file's scope. When a variable is declared
	static, it is not visible from the global environment or from
	any other files.

   \item[Function Local Scope:] In general, variables used within a
	function (other than the function's arguments) are bound to
	the function's local scope (there are ways to override this
	behavior). Variables bound to a function's local scope are not
	visible from a file's scope or from the global scope. They are
	created (undefined) when the function is invoked, and
	destroyed when the function returns.

        There are exceptions: variables used in a function context are
	bound to the global-symbol-table. For example:

\begin{verbatim}
			x = a * sin ( pi )
\end{verbatim}

	\verb+sin+ is used in a function context, and is bound to the
	global scope, while \verb+x+, \verb+a+, and \verb+pi+ are
	bound to the function's local environment.

	Function's that are defined within a file have full access to
	the file's static variables.  Function variables will be bound
	to the file's scope before local binding occurs.

\begin{verbatim}
	    ---- beginning of file.r ----

	    static (A, pi)
	    pi = atan(1)*4;

	    fun = function ( a ) { return A*sin(pi*A*a); }

	    ---- end of file.r ----
\end{verbatim}

	When `fun' is created it binds `A' and `pi' to file.r's static
	environment.

	There are two declarations: `global' and `local' that can be
	used to override the default behavior if necessary. Variables
	declared local will be bound to the function's local scope,
	and variable declared global will be bound to the global
	scope.

\end{description}

\subsection{Function Argument Passing}

   Arguments can be passed by reference, or value. The default
   behavior is to pass by reference. Pass by references indicates that
   the variables used in the function call are directly referenced
   from within the function. Pass by value means that only the value
   of the argument is passed to the function. In other words, the
   arguments are copied. In order to pass a function argument by
   value, the user needs to declare that function argument as
   local. This method allows users to selectively determine how to
   pass each function argument.

   A function can be called with fewer arguments than specified in the
   definition; this is called a ``short-list''. When this situation
   occurs, \rlab\ pads the argument list with extra undefined
   variables. Arguments can be ``skipped'' when calling a
   function. The ``skipped'' arguments are passed to the function as
   undefined variables. To ``skip'' an argument just leave it out of
   the argument list, but don't forget the commas:

\begin{verbatim}
> x = function ( a, b, c, d ) { a ? b ? c ? d ? }
	<user-function>
> x (1);	// short-list
        1
	UNDEFINED
	UNDEFINED
	UNDEFINED
> x(1,,2);	// skipped arguments
        1
	UNDEFINED
        2
	UNDEFINED
\end{verbatim}

   As far as \RLaB\ is concerned undefined variables do not exist. The
   function \verb+exist+ will return true (1) if its argument exists,
   and false (0) if its argument is undefined.

   A function cannot be called with more arguments than specified in
   the function definition. If you attempt to do so, a run-time error
   will result.

   Function argument classes and class-types are not specified during
   definition. When writing ``robust'' functions the author should
   take some care to check that the function argument(s) are of the
   correct class and type, if necessary.  The documentation (comments)
   should clearly define the requisite argument types and the function
   return object. If the function documentation does not clearly state
   that the arguments will be modified during function execution, care
   should be exercised to avoid changing any of the function
   arguments. If necessary, the function arguments can be passed by
   value so that changes will not affect the caller's variables.

   If you wish to write function(s) to serve as often used utilities
   or libraries, then care should be taken to declare all variables
   (other than function arguments or built-in functions) as
   local. Declaring all function variables as local will prevent
   accidental destruction of user's global variables.

   \RLaB\ has a special built-in function (\verb+fvscope+) that
   analyzes user-functions, and makes a report describing which
   variables are local, arguments, global, or
   file-static. \verb+fvscope+ can be very helpful when writing your
   first function(s) to help you understand how \RLaB\ resolves
   variable references.

\subsection{Function Recursion}

   Functions can call themselves recursively. Each time execution
   passes into the function the local variables are (re)created. There
   is a special keyword: \verb+$self+, which must be used to force a
   function to refer to itself. 

\begin{verbatim}
fac = function ( a ) 
{
  if(a <= 1) 
  {
      return 1;
  else
      return a*$self (a-1);
  }
};
\end{verbatim}

   In the previous example a factorial computation is performed using
   recursion\footnote{Not necessarily an efficient way to compute the
   factorial.}. In the second return statement, the function calls
   itself until $a \leq 1$. 

\subsection{Files}

   Simple ``one-liner'' functions can be typed in at the command line.
   However, they are destroyed when the \RLaB\ session is ended. Most
   users will want to create their functions in a text-editor as
   ordinary ASCII files.

   The function \verb+load+ will execute the \rlab\ statements in a
   file as if they were typed at the command line. The \rlab\ command
   \verb+rfile+ searches a specified path for files with a `.r'
   extension. When the \verb+rfile+ command finds a file that matches
   its argument, it executes the \rlab\ statements in the file as if
   they were typed at the command line.

   Statements in a file are executed in the same manner as they would
   be had they been typed in interactively. Files containing ordinary
   commands and multiple functions are acceptable. In fact, complete
   programs can be written and run interactively or in batch mode. To
   run a program in batch mode you can try:

\begin{verbatim}
% rlab program.r &
\end{verbatim}

   Or the program could contain \verb+#!/usr/local/bin/rlab+ on the
   first line. Then, if your operating system provides the proper
   support, \rlab\ can execute your program, interactively, or in the
   background by simply typing:

\begin{verbatim}
$ chmod +x program.r
$ ./program.r
\end{verbatim}

\subsection{Example} \label{sect-func-example}

   Many functions are included with the \RLaB\ source distribution.
   Functions can be found in the distribution subdirectories
   \verb+./rlib+, \verb+./toolbox+, and \verb+./examples+. These 
   directories are normally installed under \verb|/usr/local/lib/rlab|.
   
   We will continue with some simple examples demonstrating
   function creation and usage. We will carry on with the exercise of
   learning least-squares techniques.

   In some earlier examples we played with solving a set of normal
   equations, and tried a simple experiment with Householder
   reflections. Now we want to try out this technique, and decompose a
   matrix into two matrices: $Q$ and $R$; such that $A =
   QR$.\footnote{For an excellent discussion of the $QR$ decomposition
   please refer to ``Matrix Computations'' Golub and Van Loan.}

   We are going to decompose an entire matrix, so we will want to
   automate the procedures we used in previous examples. The first was
   creating a Householder vector. Instead of typing in our function at
   the command-line, we will use a text-editor to create the function
   in a file so that we can correct our mistakes without retyping the
   entire function.

\begin{verbatim}
//
// house_v(): Given an N-vector V, generate an n-vector V
// with V[1] = 1, such that (eye(n,n) - 2*(V*V')/(V'*V))*X
// is zero in all but the 1st component.
//

static (sign)

house_v = function(v)
{
  local(v)
  
  n = max( size(v) );
  u = norm(v, "2");
  if(u != 0)
  {
    b = v[1] + sign(v[1])*u;
    if(n > 1) 
    {
      v[2:n] = v[2:n]/b;
    }
  }
  v[1] = 1;
  return v;
};

sign = function ( X )
{
  if (X >= 0) { return 1; else return -1; }
}
\end{verbatim}

   Note that our new function is more complicated than our earlier
   ``one-liner''. This is due to the fact that the function is more
   efficient, and does some input-checking. Notice that the variables
   \verb+b+, \verb+n+, \verb+u+, and \verb+v+ are {\em local}; these
   local variables will never be seen by the user, and will not
   interfere with any pre-existing variables by the same name in the
   global-workspace.

   Also note that the function argument, \verb+v+ is copied to the
   function local variable \verb+v+. This prevents the function from
   changing the values in the input argument, and thus destroying the
   contents of the caller's variable.

   One other important feature of the new function is the usage of
   the \verb+sign+ function. \verb+house_v+ requires that the sign
   function return either $1$ or $-1$. The \RLaB\ built-in \verb+sign+
   function will return zero when its argument is zero; this
   behavior is unacceptable, so we have written our own sign
   function. Declaring our new sign function to be static means that
   it will only affect statements within the file \verb+house_v.r+.

   To use our new function type:

\begin{verbatim}
> a = rand (4,2);
> x = house_v (a[;1])
 x =
        1  
    0.434  
    0.042  
    0.412  
\end{verbatim}

   Now that we can generate a Householder vector, we need to
   automatically form the Householder reflection, and use it to reduce
   $A$ to upper triangular form.

\begin{verbatim}
// P.r
// P: Generate P matrix

P = function ( V )
{
  m = max( size(V) );
  return [ eye(m,m) - 2*(V*V')./(V'*V) ];
};
\end{verbatim}

   This function is a small one, and simply implements the formula we
   demonstrated earlier. We can test out our two new functions like so:

\begin{verbatim}
> ( p1 = P (house_v (a[;1])) ) * a
    -1.49      -1.11  
-4.09e-17     0.0174  
-2.87e-18      0.354  
-8.89e-17     -0.779  
> p1' * p1
        1   4.79e-17   2.75e-18    1.8e-17  
 4.79e-17          1  -4.78e-18  -2.51e-18  
 2.75e-18  -4.78e-18          1      9e-18  
  1.8e-17  -2.51e-18      9e-18          1  
\end{verbatim}

   Our new function seems to be working as expected. The computed
   Householder reflection, \verb+p1+, zeros out all of the elements
   below the first in column one of \verb+a+. Additionally \verb+p1+
   is an orthogonal transformation, as demonstrated by computing
   ${P_{1}}^{T}P_{1}$. It is usually more efficient to build up
   programs as a collection of simple functions, like we are doing
   here, testing each as it is written, and making the appropriate
   fixes.

   Function debugging can be easily accomplished by simply removing key
   `\verb+;+' to turn expression printing on. Additionally, one can
   comment out \verb+local+ statements so that a function's variables
   can be examined after function execution.

   Now there are two more pieces, a better implementation of \verb+P+
   called \verb+house_row+\footnote{See ``Matrix Computations'' by
   Golub and VanLoan.}, and the final function (\verb+house_qr+) that
   will apply the transformations in sequence to $A$ to produce the
   upper-triangular $R$, and the orthogonal $Q$.

\begin{verbatim}
//
// house_row(): Given an MxN matrix A and a non-zero M-vector V
// with V[1] = 1, the following algorithm overwrites A with 
// P*A, where P = eye(m,m) - 2*(V*V')/(V'*V)
//

house_row = function(A, v)
{
  local(A)
  
  b = -2/(v'*v);
  w = b*A'*v;
  A = A + v*w';
  return A;
};
\end{verbatim}


\begin{verbatim}
// house_qr.r
// Given A, with M >= N, the following function finds Householder
// matrices H1,...Hn, such that if Q = H1*...Hn, then Q'*A = R is
// upper triangular.

// House.qr returns a list containing `q', and `r'

rfile house_row
rfile house_v
rfile P

house_qr = function ( A )
{
  local (A)
  
  m = A.nr; n = A.nc;
  v = zeros(m,1);
  q = eye (m, m);

  for(j in 1:n)
  {
    // Generate the Householder vector
    v[j:m] = house_v( A[j:m;j] );

    // Apply the Householder reflector to A
    A[j:m;j:n] = house_row( A[j:m;j:n], v[j:m] );

    // Create Q
    if(j < m) 
    {
      q = P ([ zeros (j-1,1); 1; v[j+1:m] ]) * q;
    }
  }
  return << q = q'; r = A >>;
};
\end{verbatim}

   Notice the three \verb+rfile+ statements near the top of the file.
   These statements ensure that the user-function dependencies are
   resolved {\em before} we try and execute the function. Also note
   how \verb+house_qr+ returns two matrices in a list, with element
   names \verb+q+, and \verb+r+. We can use, and test, this new
   function like:

\begin{verbatim}
> x = house_qr ( a )
   q            r            
> x.q * x.r
      0.7       0.96  
     0.95      0.915  
   0.0918      0.441  
    0.902     0.0735  
> a
 a =
      0.7       0.96  
     0.95      0.915  
   0.0918      0.441  
    0.902     0.0735  
\end{verbatim}

   A visual comparison shows that our function does indeed work. 
   Now we wish to use this factorization to compute a solution to our
   original least-squares problem. Since we have decomposed $A$ into
   two matrices, one of which is upper triangular, we can reformulate
   the problem as a simple back-substitution. The \RLaB\ built-in
   function \verb+solve+ will do this for us, since \verb+sr+ is
   already upper-triangular, all \verb+solve+ will do is the
   back-substitution.

\begin{verbatim}
> a = [3,4,1;0,2,2;0,0,7;zeros(2,3)];
> b = [14;10;21;6;2];
> x = house_qr (a);
> sq = x.q[;1:3];
> sr = x.r[1:3;];
> z = b'*sq;
> sol = solve (sr, z')
 sol =
        1  
        2  
        3  
> b - a*sol
        0  
        0  
        0  
        6  
        2  
\end{verbatim}

   Now that we have built our own \verb+qr()+ I should tell you that
   \RLaB\ has  a built-in \verb+qr()+ that is much more robust, and
   significantly faster.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Objects---Other Data Structures}

   Although \RLaB\ is primarily a linear-algebra programming tool,
   other data structures are necessary to allow the user some
   flexibility, and a little extensibility. The two remaining data
   structures do just that. Strings allow users/programmers to write
   intelligible error messages, and properly annotate program inputs
   and outputs, as well as label quantities during program execution.
   The list object is a very flexible data structure that can hold
   numeric, string, function and other list objects. Since lists are
   indexed in an associative fashion, they are a very powerful tool.

\subsection{Strings}

   When performing numeric computations, strings are not normally used
   for much except error messages and such. However, there are many
   other tasks for which strings are quite useful. 

   String objects can be treated in a manner similar to numeric
   objects, with the main difference being the different string
   operators. The only numeric operator that works on strings is the
   `\verb-+-' which catenates the left and right string operands
   together.

\begin{verbatim}
> "first part of the string " + "and the second part of the string"
first part of the string and the second part of the string
\end{verbatim}

   The other numeric operators will only produce a runtime error
   message when used with strings.

\begin{verbatim}
> "string1" * "string2"
rlab: NULL, NULL, 1st arg invalid for multiply
\end{verbatim}

   The relational and logical operator do work with string objects.
   For instance, you can compare two strings:

\begin{verbatim}
> "string-a" == "string-b"
        0
\end{verbatim}

   Or two string matrices:

\begin{verbatim}
> ["el-1", "el-2"] == ["el-1", "el-x"]
        1          0  
\end{verbatim}

   Functions exist to aid with string to number conversions
   (\verb+num2str+), and reading strings from the standard input
   (\verb+input+).

\begin{verbatim}
> x = input ("input a string > ", "s")
input a string > teststring
teststring
\end{verbatim}

   The function \verb+strsplt+ will split an arbitrary string into a
   matrix of one-character strings. This allows the user to operate on
   individual characters in a string; the `\verb-+-' operator can be
   used to ``glue'' the desired pieces back together again.

\begin{verbatim}
> x = "this is a single string";
> strsplt (x)
t  h  i  s     i  s     a     s  i  n  g  l  e     s  t  r  i  n  g  
\end{verbatim}
   
   We will do a little more with strings in Section~\ref{sect-lists}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Lists}	\label{sect-lists}

   A list is a heterogeneous associative array. Simply, a list is an
   array whose elements can be from different classes. Thus a list can
   contain numeric, string, function, and other list objects. Lists
   have many uses, only some of which we will demonstrate herein.

   To create a list-object use the `\verb+<<+' and `\verb+>>'+'
   operators. The list will be created, and the objects inside the
   `\verb+<< >>+' will be copied into the new list. If the objects are
   not renamed during the list-creation, they will be given numerical
   index values.

\begin{verbatim}
> a = rand(3,4); b = sqrt (a); c = 2*a + b;
> ll = << a ; b ; c >>
   1            2            3            
> ll2 = << A = a; b = b ; x = c >>
   A            b            x            
> ll2.A == ll.[1]
        1          1          1          1  
        1          1          1          1  
        1          1          1          1  
\end{verbatim}

   Lists are not indexed in what is perceived as the ``traditional
   manner''. Instead the list index is converted to a string, and the
   string value is used to look-up the referenced
   object\footnote{\RLaB\ lists are similar to AWK associative
   arrays.}. There are two methods for referencing the elements of a
   list, the first, a shorthand notation looks like:

   \begin{quote}
      list\_name . element\_name
   \end{quote}

   In this case, the {\em list\_name} and {\em element\_name} must
   follow the same rules as ordinary variable names. The second
   method for indexing a list is:

   \begin{quote}
      list\_name . \verb+[+ numeric\_or\_string\_expression \verb+]+
   \end{quote}

   The second method allows string and numeric variables to be
   evaluated before doing the conversion to string type\footnote{If one
   is necessary.}. 

   As you have seen in an earlier example lists can be used within
   functions when it is necessary to return several values. Because of
   the lists flexible nature a function can return matrices of
   differing sizes and types, as well as strings, other lists, or
   user-functions. 

\subsubsection{Examples}

   The \verb+getline+ function reads from a designated file, or pipe,
   and separates the input into whitespace separated tokens. The
   tokens are either numbers or strings. In this example we will use
   the \verb+getline+ function, and \RLaB\ piping capability to read
   the output from the UNIX ps command, and sum the numbers in the 5th
   column (the resident process size). 

\begin{verbatim}
> psize = 0;
> while (length (ans = getline ("|/usr/ucb/ps -aux | grep ian")))
  {
    psize = psize + ans.[5];
  }
> close ("|/usr/ucb/ps -aux | grep ian");
> psize
 psize =
 2.26e+03
\end{verbatim}

   The following simple example makes extensive use of the list
   object's capabilities. This example consists of two user-functions
   that were written to allow someone to index a matrix with strings,
   or numeric quantities that are not necessarily equal to the integer
   row and column numbers.

   Two special qualities of lists are used in this example. The first
   is the ability to group together related objects; thus the matrix,
   and its row and column labels are copied into a list-object. The
   second special quality is the associative nature of the list
   indices. When constructing the row and column labels, we create two
   lists that are members of the top-level list. These sub-lists
   contain the row or column label as the index value, and the integer
   row or column number as data.

\begin{verbatim}
//
// Create a list-object, a matrix with row and 
// column labels. Then the matrix can be indexed
// with the labels by using lb().
//

mklb = function (mat, rl, cl)
{
  lrl = <<>>; lcl = <<>>;
  if (!exist (rl))
  {
    rl = 1:mat.nr;
  else
    if (length (rl) != mat.nr) { error ("mklb: rl wrong size"); }
  }

  for (i in 1:rl.n)
  {
    lrl.[rl[i]] = i;
  }

  if (!exist (cl))
  {
    lcl = 1:mat.nc;
  else
    if (length (cl) != mat.nc) { error ("mklb: cl wrong size"); }
  }

  for (i in 1:cl.n)
  {
    lcl.[cl[i]] = i;
  }

  return << m = mat; rl = lrl; cl = lcl >>
};
\end{verbatim}

   the function \verb+mklb+ creates a list containing matrix, row,
   and column labels. Looking closely at the for-loops you will notice
   that the labels values are actually the index values of the
   lists \verb+lrl+ and \verb+lcl+. This method is used since it
   provides extremely easy access to the matrix elements in the next
   function \verb+lb+.

\begin{verbatim}
lb = function ( mlb, rl, cl )
{
  // Create row indices

  if (!exist (rl)) 
  { 
    irl = 1:mlb.m.nr;		// use all the rows
  else
    for (i in 1:rl.n)
    {
      irl[i] = mlb.rl.[rl[i]];
    }
  }

  // Create column indices

  if (!exist (cl))
  {
    icl = 1:mlb.m.nc;		// use all the columns
  else
    for (i in 1:cl.n)
    {
      icl[i] = mlb.cl.[cl[i]];
    }
  }

  return mlb.m[irl; icl]
};
\end{verbatim}

   The for-loops in \verb+lb+ simply loop through the specified
   indices, using them as index values to the internal row and column
   lists. Once the numeric vectors \verb+irl+ and \verb+icl+ have been
   formed the specified matrix elements are easily returned.

   To use the new functions you might:

\begin{verbatim}
> m = [1,2,3;4,5,6;7,8,9];
> ml = mklb (m, ["r1","r2","r3"], ["c1","c2","c3"])
   cl           m            rl           
> lb (ml, ["r1","r3"])
        1          2          3  
        7          8          9  
> lb (ml, ["r1","r3"], "c1")
        1  
        7  
> lb (ml, , "c2")
        2  
        5  
        8  
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Builtin functions}

   \RLaB\ comes with many built-in and user-functions. The built-in
   functions are available any time \rlab\ is run, regardless of the
   command-line options. The user-functions that are delivered with
   \RLaB\ may, or may not be accessible, depending upon how \rlab\ has
   been configured on your computer and the command-line options used.

   Typing `\verb+rlab -ql+' will run \rlab\ without executing the
   \verb+.rlab+ file or loading any of the delivered user-functions.
   The built-in functions in \RLaB\ are unlike user-functions in that
   users cannot destroy or reassign them to other
   variables\footnote{The exception to this statement is the use of
   static variables, which can obscure built-in functions - see the
   Section~\ref{sect-func-example}.}. This protection is in place
   so that users will be able to write portable libraries using the
   built-in functions.

   \RLaB\ does {\em not} automatically search \verb+RLAB_SEARCH_PATH+
   for R-files when an unresolved reference is encountered. Instead,
   the user must explicitly \verb+load+, or use the \verb+rfile+
   command to get \RLaB\ to load and compile the function or
   statements. To see what functions are currently available type:
   `\verb+what()+', this will list all of the currently available
   functions. Typing `\verb+rfile+' from within \rlab\ will list all
   of the R-files in the directories listed in the environment
   variable \verb+RLAB_SEARCH_PATH+. The files displayed by the
   \verb+rfile+ command can be loaded by typing `\verb+rfile filename+',
   where \verb+filename+ is the name of the R-file to load,
   {\em without} the \verb+.r+ extension.

\subsection{Function Behavior}

   Many of the delivered functions fall into one of three categories:
   scalar-functions, vector-functions, or matrix-functions. The list
   below gives examples from each type of function, but is not
   all-inclusive. 

   \begin{description}

     \item[Scalar Functions:] These functions operate on scalars, and
           treat matrices in an element-by-element fashion. Some
           examples are:

           \begin{tabbing}
             function1 \= function2 \= function2 \= function4 \kill        
             abs \> exp \> floor \> round \\
             cos \> sin \> tan \> ceil \\
             sqrt \> real \> imag \> conj \\
             isnan \> int \> \> \\
           \end{tabbing}

     \item[Vector Functions:] These functions operate on vectors,
           either row-vectors (1-by-$n$) or column vectors ($m$-by-1),
           in the same fashion.  If the argument is a matrix with $m
           \geq 1$ then the operation is performed in a column-wise
           fashion. Some examples are:

           \begin{tabbing}
             function1 \= function2 \= function2 \= function4 \kill        
             sum \> cumsum \> prod \> cumprod \\
             mean \> max \> min \> fft \\
             sort \> any \> all \> ifft \\
           \end{tabbing}

           When using a vector oriented function, like \verb+max()+ it
           is possible to operate on matrix objects. For example the
           maximum value in a matrix can be obtained via 
           \verb+max (max (a))+. The first call to \verb+max()+
           returns a vector of the maximum values from each column,
           and the second call to \verb+max()+ returns the maximum
           value in the matrix.

     \item[Matrix Functions:] These functions operate on matrices as a
           single entity. Some examples are:

           \begin{tabbing}
             function1 \= function2 \= function2 \= function4 \kill        
             balance \> chol \> det \> eig \\
             hess \> inv \> lu \> norm \\
             pinv \> qr \> rank \> rcond \\
             reshape \> solve \> svd \> symm \\
             factor \> backsub \> issymm \\
           \end{tabbing}

     \item[Misc. Functions:] These functions are in this particular
           category simply because they don't fit anywhere else. Some
           examples are:

           \begin{tabbing}
             function1 \= function2 \= function2 \= function4 \kill        
             system \> getline \> show \> who \\
             what \> tic \> toc \> printf \\
             format \> write \> read \> readm \\
           \end{tabbing}

   \end{description}
           
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Input and Output}

   There are two types if I/O in \RLaB\ .

\begin{enumerate}
  \item  Raw data, such as matrices, lists, and strings. In this case
   the formatting is irrelevant.
  \item  Formatted I/O, such as data from  other programs, and
   formatted output intended for reports and such.
\end{enumerate}

   In either case file names, or file-handles, are the same. A
   file-handle is {\em always} a string, either a string constant, a
   string variable, or a string expression. For example:

\begin{verbatim}
> write ("file.rd", A);

> f = "file.rd"; write (f, A);

> write ("file" + ".rd", A);
\end{verbatim}

   are all equivalent. 

   There are three special file-handles in \RLaB\ ; they are:
   \verb+"stdin"+, \verb+"stdout"+, and \verb+"stderr"+.

   \verb+"stdin"+ is connected to \RLaB's standard input, usually the
   keyboard. \verb+"stdout"+ is connected to \RLaB's standard output,
   usually the screen. And \verb+"stderr"+ is connected to the
   standard error, again, usually the screen.

   Files are automatically opened by the functions that perform input
   or output. In most cases the files will automatically be closed
   after the function has completed its task. The following will
   force a file closure after their task is complete: \verb+rfile+,
   \verb+load+, \verb+read+, and \verb+readm+.

   However, \verb+write+, \verb+writem+, \verb+fprintf+, and
   \verb+getline+ will leave their files open after completion, so
   that they may be used subsequently without complicated file
   positioning operations.

   \RLaB\ has another special type of file-handle. When a string,
   starting with a `\verb+|+' is used as a file-handle, a process is
   created. Either the sub-process input, or output is connected to
   \RLaB\ through the file-handle. For example:

\begin{verbatim}
> output = "| sort";
> for (i in 5:1:-1) { fprintf (output, "%i\n", i); } close (output);
1
2
3
4
5
\end{verbatim}

   If the \verb+fprintf+ output were not sent to the Unix program
   \verb+sort+, you would expect the output to appear in descending
   order. In fact, \verb+fprintf+ does write the output in descending
   order, to \verb+sort+. Then, \verb+sort+ re-orders its input, and
   writes it to \verb+stdout+. The same trick works in reverse;
   functions that expect input can get it from a sub-process. A good
   example is \verb+getline+:

\begin{verbatim}
> input = "| ls -la *.[ch]";
> fsum = 0;
> while (length (ans = getline (input))) { fsum = fsum + ans.[5]; } fsum
 fsum =
   943083
> close (input);
\end{verbatim}

   In this instance we used the sub-process capability to determine
   the number of bytes of C source code, and header files exist in the
   current working directory. The sub-process performs the long
   listing, and getline parses this input into numbers and strings.
   The $5^{th}$ column of the \verb+ls -la+ output is the number of
   bytes in each file.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Plotting}

   There are two available methods for plotting data from within
   \rlab. Probably the most desirable is the usage of the builtin
   plotting library, Plplot. The second method involves the usage of
   \rlab's I/O abilities (piping) to send data and commands to another
   running process - Gnuplot, for example.  A fairly extensive
   example of this second method is provided in the \rlab source
   distribution in the file \verb+misc/gnu_plot.r+. We shall focus
   on the builtin plotting capability.

   There are two new terms you must learn to understand plotting -
   plot-window, and sub-plot. The plot window is manifested on the
   plot-device, such as a Tektronix screen, X-window display, or a
   printer. \rlab will allow you to have multiple plot-windows, if
   your display device will support it. X-windows supports multiple
   plot-windows, but a Tektronix display, or DOS display will not.

   The next term is sub-plot. Each plot-window can have 1 or more
   sub-plots. The layout of the sub-plots is determined when
   \verb+plstart+ is called.

   Regardless of the number of plot-windows and sub-plots, there can
   only be one {\em current} plot-window, and one {\em current}
   sub-plot within the current plot-window. Each plot related function
   will only work on the current sub-plot in the current plot-window.
   This way the majority of the plot functions are the same,
   regardless of the output device. The current sub-plot is always the
   {\em next} plot. As soon as a plot function is used that actually
   draws the plot on the plot-device, such as \verb+plot+,
   \verb+plot3+ or \verb+plhist+, the current sub-plot is incremented.
   Therefore, all functions that effect the appearance of a plot must
   be used before the plot is drawn.

\subsection{2-D Plotting}
   
   To plot some data with a minimum of effort do:

\begin{verbatim}
> data = (0:pi:.1)';
> data[;2] = cos (2*pi*data*3);
> plstart (,,"xwin");
> plot (data);
\end{verbatim}

   The \verb+plstart+ function was called with the default sub-plot
   definition (1 subplot), and the X-windows driver as the display
   device. The \verb+plot+ function accepts matrices, or lists as
   arguments and plots the columns of a matrix, or the columns of each
   matrix of a list.

   Now for a slightly more complex example.

\begin{verbatim}
> t = (0:10:.05)';
> x = exp (-0.5*t) .* (cos (2*pi*3*t) + sin (2*pi*7*t));
> X = fft (x);
> rfile faxis
> freq = faxis (X, .05, 3);
> mag = abs (X);
> rfile angle
> phase = angle (X);
> plstart (1,2,"xwin");
>
> pltitle ("Magnitude of FFT")
> xlabel ("Frequency (Hertz)");
> plot ([freq,mag]);
>
> pltitle ("Angle (atan2(imag/real)) of FFT")
> xlabel ("Frequency (Hertz)");
> ylabel ("Angle (radians)");
> plot ([freq,phase]);
> plprint ("p3.ps");
\end{verbatim}

\vskip 1cm

\begin{figure}[htbp]
  \begin{picture}(350,350)(0,50) % (x-size, y-size) (x-shift, y-shift)
    \special{psfile=p3.ps
             angle=90 hscale=60 vscale=60 voffset=40 hoffset=460}
    \end{picture}
    \caption{Example Plot \label{p1}}
\end{figure}

   In this example we create a plot-window with two subplots so that
   we can display related information on the same plot-window. The
   arguments to \verb+plstart+ specify that there will be 1 plot in the
   horizontal direction, and 2 plots in the vertical direction, thus
   creating 2 sub-plots. The first \verb+pltitle+ and \verb+xlabel+
   function calls effect the first sub-plot. The first call to
   \verb+plot+ creates the first sub-plot. The subsequent calls to
   \verb+pltitle+, \verb+xlabel+ and \verb+ylabel+ effect the second
   sub-plot, and the last call to \verb+plot+ creates the second
   sub-plot. 

   The last line is a call to \verb+plprint+. The \verb+plprint+
   function creates a file that contains a copy of the contents of the
   current plot-window. The default hardcopy device for \verb+plprint+
   is Postscript, but color Postscipt, Xfig, plmeta, and HP-LJII can
   also be specified. The output from \verb+plprint+ is presented in
   Figure~\ref{p1}.  

\subsection{Histograms}

   Histograms can be plotted quite easily. At present the histogram
   plotting function, \verb+plhist+, will plot histograms of the
   columns of a matrix. For example:

\begin{verbatim}
> // Create data
> rand("normal", 0, pi);
> r = rand(2000,1);
>
> // Create plot
> pltitle("Histogram, 30 Bins");
> plhist (r, 30);
\end{verbatim}

\begin{figure}[htbp]
  \begin{picture}(350,300)(0,50) % (x-size, y-size) (x-shift, y-shift)
    \special{psfile=p5.ps
             angle=90 hscale=60 vscale=60 voffset=40 hoffset=460}
    \end{picture}
    \caption{Example Histogram Plot \label{p5}}
\end{figure}

   Figure~\ref{p5} contains the Postscript output from the previous
   example. 

\subsection{3-D Plotting}

   Plotting 3-dimensional data is quite straightforward, and very
   similar to the steps used for plotting 2-dimensional data. We will
   begin with an example:

\begin{verbatim}
> // Create the data
> X = -2:2:.2;
> Y = X;
> Z = [];
> for (i in 1:X.n) { 
>   for (j in 1:Y.n) { 
>     Z[i;j] = X[i] * exp (-X[i]^2 - Y[j]^2);
>   }
> }
> // Make the plot
> plstart(,,"xwin");
> plwid(4);
> pltitle("Sample 3-D Plot");
> xlabel("X-Axis");
> ylabel("Y-Axis");
> zlabel("Z-Axis");
> plmesh( << x = X; y = Y; z = Z >> );
> plprint("3d.ps");
\end{verbatim}

\begin{figure}[htbp]
  \begin{picture}(300,350)(0,120) % (x-size, y-size) (x-shift, y-shift)
    \special{psfile=3d.ps
             angle=90 hscale=80 vscale=80 voffset=40 hoffset=530}
    \end{picture}
    \caption{Example 3D Plot \label{p3d}}
\end{figure}

   The output from the plprint command appears in
   Figure~\ref{p3d}. Now we must take a minute and explain the data
   passed as input to \verb+plot3+. Three dimensional plots have two
   independent variables (vectors), \verb+x+, and \verb+y+. The
   dependent variable \verb+z+ (a matrix) is a function of both
   \verb+x+ and \verb+y+ and has row dimension the length of \verb+x+
   and column dimension the length \verb+y+. This storage scheme for
   three-dimensional data is economical in terms of memory, and allows
   the user some extra flexibility. Thus, the argument to \verb+plot3+
   is a list with three elements: \verb+x+, \verb+y+, and
   \verb+z+. \verb+plot3+ can accept up to three distinct lists for
   plotting.

   The same plot-functions: \verb+pltitle+, \verb+xlabel+, and
   \verb+ylabel+ can be used with 3-dimensional plots. Additionally,
   \verb+zlabel+ can be used to add labels to the \verb+z+ axis.

   The next example (Figure~\ref{p6}) demonstrates more plotting
   capabilities - including: changing fonts and pen widths, multiple
   plots per page and annotating a plot. Note that the calls to
   \verb+plptex+ are made {\em after} the \verb+plot+ calls. since
   \verb+plptex+ uses the plot coordinates to place text on the graph,
   the plot must be created first.

\begin{verbatim}
//
// Demonstrate the effect of adding terms to a Fourier expansion
//

// We want to approximate a square wave...
// The Fourier Series for a square wave is a 
// sum of odd harmonics - we will demonstrate.
// Start up a plot window...

plstart(2,2, "xwin");

// Compute the 1st term in the Fourier series and plot.

t = (0:10:.1)';
y = sin (t);

pltitle ("Fundamental Frequency");
plwid(10);
plfont(1);
plot ( [t,y] );
plptex ("Pen width = 10", 4, 0.4);
plptex ("Font = 1 (Normal)", 4, 0.1);
pause ();

// Now add the third harmonic to the fundamental, and plot.

y = sin (t) + sin (3*t)/3;

pltitle ("1st and 3rd Harmonics");
plwid(7);
plfont(4);
plot ( [t,y] );
plptex ("Pen width = 7", 4, 0.4);
plptex ("Font = 4 (Script)", 4, 0.1);
pause ();

// Now use the first, third, fifth, seventh, and ninth harmonics.

y = sin (t) + sin (3*t)/3 + sin (5*t)/5 + sin (7*t)/7 + sin (9*t)/9;
pltitle ("1st, 3rd, 5th, 7th, 9th Harmonics");
plwid(4);
plfont(3);
plot ( [t,y] ); 
plptex ("Pen width = 3", 4, 0.4);
plptex ("Font = 3 (Italic)", 4, 0.1);
pause ();

//
// Now create a matrix with rows that represent adding
// more and more terms to the series.
// 

t = (0:3.14:.02);
y = zeros(10,max(size(t)));
x = zeros(size(t));

for (k in 1:19:2)
{
  x = x + sin(k*t)/k; 
  y[(k+1)/2;] = x;
}

//
// Now make a nice 3-D plot that shows the effect
// of adding more and more terms to the series.
//

plfont (2);
plwid(1);
pltitle ("Square Wave via Fourier Series");
ylabel ("No. Terms");
plaz (140);
plot3 (<< x = t; y=1:10; z=y' >>);
plptex ("Pen width = 1", -1, 4.5);
plptex ("Font = 2 (Roman)", -1, 3.8);
plprint ("p6.ps");	// Make hardcopy (Postscript default).
\end{verbatim}

\begin{figure}[htbp]
  \begin{picture}(500,450)(0,50) % (x-size, y-size) (x-shift, y-shift)
    \special{psfile=p6.ps
             angle=90 hscale=70 vscale=80 voffset=40 hoffset=500}
    \end{picture}
    \caption{Square Wave Plot Example \label{p6}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage
\newpage

\section{Summary}

   \Rlab\ can do quite a few things that we have not covered here. The
   reference manual or the online help should be consulted for more
   detailed descriptions of language and function behavior. We will
   close out this primer with a listing of the currently available
   function (both built-in and user-functions).

\begin{verbatim}
> what ()
abs           epsilon       log           plot          solve         
acos          error         log10         plot3         sort          
acosh         eval          logspace      plprint       sprintf       
all           exist         lu            plptex        sqrt          
any           exp           lyap          plstyle       srand         
asin          eye           matrix        plwid         std           
asinh         factor        max           printf        strsplt       
atan          fft           maxi          printmat      strtod        
atan2         filter        mean          prod          sum           
atanh         find          members       plstart       svd           
backsub       finite        min           pltitle       sylv          
balance       fix           mini          plwin         symm          
cd            floor         mod           qr            system        
ceil          format        nan           rand          tan           
chol          fprintf       norm          rank          tanh          
class         fvscope       num2str       rcond         tic           
clear         getb          ode           read          tmp_file      
clearall      getenv        ones          readb         toc           
close         getline       open          readm         trace         
compan        hess          pause         real          tril          
complement    hilb          plclose       redit         triu          
conj          ifft          plend         replot        type          
cos           imag          plalt         reshape       union         
cosh          inf           plaspect      round         what          
cross         input         plaxis        save          who           
cumprod       int           plaz          scalar        whos          
cumsum        int2str       plegend       schord        write         
det           intersection  plfont        schur         writeb        
diag          inv           plgrid        set           writem        
diary         isempty       plgrid3       show          xlabel        
diff          isinf         plhist        showplwin     ylabel        
disp          isnan         plhistx       sign          zeros         
dot           issymm        plhold        sin           zlabel        
eig           length        plhold_off    sinh                        
eign          linspace      plimits       size                        
eigs          load          plmesh        sizeof                      
> quit
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
