The Advantages of Matrix Math Computer Languages
If you're a computer geek like me, then you may well need to do some
mathematical calculations now and then. While some straightforward math can be
accomplished with spreadsheets, if you need filtering, regression, and more
advanced mathematical support then I suggest that you consider Linux as your
operating system.
Linux Offers a Gold Mine of Matrix Languages
Why? For one  Linux, and all it contains, is free (listen up college
students). For another, Linux and most of the support for it is open source.
But in this case, mostly you should consider Linux because when it comes to
mathematics languages  Linux is loaded.
There are more mathematical, engineering, and scientific computing languages
in Linux than you can imagine. In my Debian Linux distribution, octave, R,
scilab, yorick, euler, maxima, jacal, pdl, and tela are all included. Each is a
matrix type math language of the ilk of the commercially available MATLAB. And
all I need do to have any or all of them available to me is to issue the easy
to use Debian aptget command. It's literally an embarrassment of riches.
Amazingly, this isn't an all inclusive list of Linux available math
languages. It's only the matrix math languages included in Debian. I suggest
you check out Mathematics Software For
Linux for additional suggestions.
What Do Matrix Languages Offer?
That brings us to scripting matrix oriented languages. They can grow on you.
For years I avoided them, largely because I knew that they were really memory
hogs, and only a few years ago PCs didn't have all that much memory. Not
enough, for example, to hold an entire time series of some space craft data,
plus all the intermediate matrices that might accumulate when you tried
processing said space craft data. But as PCs grew, I moved into the matrix
realm to try it out. For most of my math needs, I've never moved back.
I started with octave, then went to scilab. Then I discovered R, followed by
yorick, and finally pdl. Unable to decide on just one, I worked with the
primitive I/O commands in each language to create a simple binary file format
that could be created by each of the languages, and read by each of them. This
allowed me the luxury of being able to begin processing on one language because
of some specific capability it had, and move to another for some other
desirable property. I still work that way. It was tough at first, but now saves
me enormous time.
To make moving across languages easier, I created libraries for each that
had the same names, like fileio, math, and plots. Within
those libraries, I created routines to do what I wanted, and named them the
same in each language also, like binall for the binary input routine,
or plot3d for plotting a 3D image. In some cases, I was just basically
renaming the specific language's routine that already did what I wanted, but
often I was using the language's functions in combinations or with specific
parameters to accomplish what I wanted. Thus armed with similar libraries,
with similar or identical functions for familiar results, I am able to
move from language to language with little discomfort.
Each of the languages I'm familiar with has capabilities similar to those
of the popular commercial MATLAB language. Some are a little less populated
with math functions than MATLAB, but some are very robust in support. All are
matrix oriented in design. Each language, while bearing much in common to the
others, has some emphasis or particular capabilities that make it shine for
some kind of problem. I suggest that when it comes to these languages, you
don't let syntax familiarity overwhelm your considerations. Sometimes a
particular capability or formulation trumps syntax in the tackling of a
difficult problem.
If you're not overly familiar with matrix languages, you may be wondering
"What's the big deal." Why not just use BASIC, or Fortran, or C, or some other
general purpose language?
One Big Advantage is the Scripting
The main attraction of these languages is the interactive and scripting
capabilities they offer. They satisfy the desire to be able to quickly
prototype code without the time consuming compiling and linking that goes with
a more fundamental language. It's also handy to be able to interactively try
out code examples to see how they work before committing them to a code file.
For example, you can interactively test how a particular filter or graphics
routine works.
The problem with scripting languages is that an engine that processes the
script has to read, interpret and act upon each command. This takes time 
much more time than a compiled program takes.
However, if you have compiled code in your engine that works on large blocks
of numbers (matrices), then one interpreted command can result in compiled code
speed for performing a processing command on a large data block. So if one can
learn to express the solution to his or her problem with matrix notation, the
number of script entries versus the number of calculations made is trivial,
giving performance near that of compiled programs.
That's why matrix oriented scripting languages are so popular. That plus the
fact that most of them have easy to use input and output commands for saving
entire programs or matrices, and easy to use plotting commands for graphing the
contents of matrices in many ways. The end result is much less code to develop,
some interactive capabilities, yet quite reasonable execution speed.
The Trick is to Vectorize Your Solution
The trick is to be able to express math solutions in matrix nomenclature.
That means  avoid do loops whenever possible. One very good guideline,
documented with the yorick language distribution, is that if your solution
operates on a record at a time and could be operated on forward or backward
with the same result, it can likely be expressed in matrix nomenclature.
Another way to express this is that if each record of computation is only
dependent on data in that record (no dependence on previous records as in
integration), then it can be expressed in matrix nomenclature. That means it
can be processed in a matrix scripting language with tolerable speed. Note
that for prior record dependencies, most languages have a few built in
compiled routines to handle common problems of that type, like cumulative
summing, or integration.
As an example, assume you have a file with a number of x and y values, and
you want to read them in and compute a least squares line (y =mx + b).
In a conventional language this would entail some program preamble statements,
file opening statements, file reading statements, loops to compute the
necessary sums, and finally computation of the result.
In a matrix language, like octave, this could be done interactively as
simply as:
load binary "data" x y
m = cov(x,y) / var(x)
b = mean(y)  m*mean(x)
Get the idea? The matrix operators cov (covariance), var (variance) and mean
inherently contain loops to do the computations, returning the results. Once
you catch on the the matrix formulation of problem solution, you'll likely find
that you can save lots of time using a matrix scripting language.
Another hint is to think about matrix dimensionality. Remember the property
of matrix multiplication that an l x m matrix times an m x n
matrix results in an l x n matrix. The first size indicates the
number of rows in a matrix, and the second the number of columns. Sometimes
just studying what matrix sizes you have and what size you want to end up
with helps identify the steps you need to do to accomplish what you want.
You may need to construct a special multiplier matrix to create the result
you want, but when you see how to do it you'll be on the way to vectorizing
your solutions so you can solve problems with matrix operations and no loops.
A Speed Comparison of Some Matrix Languages
To do a quick speed test, I generated a million records of a function and
its component functions in each of 7 different matrix languages. I then had
each language perform a least squares solution to recover the scale factors of
the components of the generated function. The time for each language to loop
through the data generation and least squares solution 4 times was logged.
Full vectorization was used in each language for both data generation and
least squares solution, so no loops were used in the scripting code.
The model function was:
y(i) = 10 + 2*sin(r(i)) + 5*cos(r(i)) 3 * sin(2*r(i))  4*cos(2*r(i))

With i running from 1 to 1000000
This exercise tested both the languages' ability to compute large numbers of
transcendental functions as well as the matrix operations used to perform the
least squares solution. Each test was performed for the same number of data
points and iterations, and on the same computer. The following code snippet
shows the matrix operator method of creating the large data matrix:
Octave Code Example: Creating Data Matrix With Matrix Operations:
function x=testdat(n)
a = (1:1:n)';
r = a*pi/180;
y = 10+2*sin(r)+5*cos(r)3*sin(2*r)4*cos(2*r);
x = [a,y,a*0+1,sin(r),cos(r),sin(2*r),cos(2*r)];
endfunction

The next code snippet shows the use of a loop to create the large
data matrix:
Octave Code Example: Creating Data Matrix With A Loop:
function x=testdat2(n)
x = zeros(n,7);
for i=1:n
a = i*1.0;
r = a*pi/180;
x(i,1) = a;
x(i,3) = 1.0;
x(i,4) = sin(r);
x(i,5) = cos(r);
x(i,6) = sin(2*r);
x(i,7) = cos(2*r);
x(i,2) = 10.0+2*sin(r)+5*cos(r)3*sin(2*r)r*cos(2*r);
endfor
endfunction

In the following table, the Matrix Time column shows the program run
time in each scripting language when only matrix operators were used to create
the data and perform the least squares solution. The Loop Time column
shows the program run time that resulted when the large data matrix was
repeatedly formed using a loop procedure. The Time Ratio is the Loop
Time divided by the Matrix Time, showing the ratio of times just from looping
instead of using matrix operations.
Language 
Matrix Time 
Loop Time 
Time Ratio 
Yorick 
26 
54 
2.0 
Tela 
29 
36 
1.3 
Scilab 
30 
495 
16.5 
R 
30 
390 
13 
Octave 
33 
934 
29 
Perl Pdl 
37 
59 
1.6 
Euler 
49 
228 
4.75 
The table shows that not only did the languages do an enormous amount of
work in little time when optimized to use the built in matrix functions, but
also that the languages weren't really separated by that much in
performance.
The table also illustrates that if the scripts didn't make optimal use of
matrix operations and resorted to loops, the cost for some was heavy. The Perl
PDL, Yorick, and Tela languages seemed most forgiving in this regard. Euler
showed a moderate slowing when loops were used, and R, Octave, and Scilab were
much slower when loops were heavily used.
Certainly, except in special conditions, speed alone isn't the only
consideration for choosing a language. It has a lot to do with what overall
use the language will be applied, resource use, and even the programmer's
background and preferences. For that reason, I prepared another chart that
considers some other factors of several scripting languages.
A Qualitative Comparison of Some Matrix Languages
The graphic you see above presents a comparison of a number of math
languages I have worked with in Linux. Some of these languages are also
available in Windows, but my primary experience with them is in Linux.
Any such graph or table is subjective, and I admit that these are my own
personal appraisals. If anything, I've probably underrated some of the
languages in that I'm not an expert in all of them. But I have used all of them
and exercised many of their features.
I did not concentrate on syntax and overall features in making this
comparison. All of these languages are matrix oriented programs very capable of
solving a wide range of mathematical problems. They overlap considerably in
that regard, with some having a bit more extension into statistics (like R) or
engineering (like Octave and Scilab).
Instead I concentrated on features that seemed to separate the languages.
The features I chose are ones that I think can help provide a direction of
choice for many prospective users. I subjectively rate the languages on their
speed, I/O flexibility, support of higher level data structures (beyond
matrices), and ability to work with strings.
You might think speed is an attribute that could be tested and reported
quite objectively. The problem is the languages express their speed differently
in different kinds of tests. Some, like PDL, are extremely fast at solving
common matrix problems. In fact, in one speed test I ran that was full of large
matrix operations, PDL was the fastest.
But others, like Tela and Yorick, maintained speed even in scripted loops.
Most other languages handle loops at much decreased speed. I also consider
speed to be a factor that should include programmer time. Some languages use
features seemingly easy to learn and allow coding to be quickly done. Others,
like PDL, take a bit of getting used to, and use more verbose expressions that
take more coding time. My estimates take all of these factors into account, but
weights computational speed the most.
I/O flexibility may be of little consequence to those looking for a single
environment for doing most of their work. But if one intends to use one of
these tools in an environment where data is often operated on with other
products, then clearly math languages with more extensive and controllable I/O
features will be useful. Some of these languages stick to small handful of file
structures (like HDF or MATLAB file types) and only offer high level routines
for this purpose. Others give the user more fundamental I/O control so that
functions can be designed to handle most any kind of file structure.
All of these languages offer matrices as a data structure. Several go beyond
2 dimensions for that purpose. But for processing many kinds of data, it is
useful to be able to manage multiple data types in some kind of record
structure, associating perhaps a few different sized matrices, some scalars,
and even some strings into single entities. Some of these languages offer such
structures, others do not. The PDL offers a number of such elevated data
structures.
And finally, I consider the handling of strings. Granted, these languages
are primarily for working with numbers. But often these numbers come embedded
in files that also contain nonnumeric information, such as data labels. And in
any professional environment, it's likely that the results obtained with any of
these languages must be output in some format useful for reports. Some of these
languages offer very little of this kind of support, and some  like PDL with
it's underlying perl  are not only great for numerical processing, but string
processing as well.
I hope you use this information as only a starting point, and keep in mind
the subjective nature of the evaluations. But if you know what kind of
computing environment you'll be in, perhaps this graphic will give you a
direction to begin your exploration.
I invite you to read my more detailed reviews of these languages. Links to
the reviews are on the sidepanel menu.
