% Submitted, 14 Jan 98 to
%  Gazette of the Australian Mathematical Society 
% by keady@maths.uwa.edu.au
%
% by e-mail to
% austms.gazette@unisa.edu.au, kevin.white@unisa.edu.au
%
% Here is the review we discussed. I had problems with the
% gazette.sty file which I got from UNSW's gazette Web pages,
% but I hope that it isn't too much bother for you to edit this
% deliberately very simple and uncomplicated LaTeX2e to your
% needs.

% Alternatively, if you can e-mail me a sample paper which
% works with the gazette syle file, I will edit it over.
% (The sample papers at the UNSW did not work with the style
% file there.)
%  
% I very much hope that the Review can be got into the next issue,
% as  timeliness is important for comments on software.

% My fax is WA (08-) 9380 3372
% and I would be truly grateful if I could see how the printed
% version would look.
%
% Regards (and to Basil Benjamin too)
%
% Grant Keady


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\title{Maple V release 5}

\author{Reviewed by Grant Keady}


\date{ January 1998
}

\begin{document}

\maketitle

\section*{Introduction}

This article aims to be a bit more than a review.
As a review, though, the focus is narrowed by concentrating
on changes made in moving from release 4 to release 5.
The other items, not actually reviewing, are as follows.\\
(i) Readers of the {\em Gazette} are offered 
my suggestions concerning whether to upgrade, though
{\em you} have to decide the scale of investment you can afford.\\
(ii) There are some speculations concerning the future, and some
requests to Waterloo Maple.

Maple is the widely-used Computer Algebra System (CAS).
In Australian Mathematics Departments it is widely used
both in research and in teaching.
(For an Australian conference reporting on the latter,
see \cite{B1}.)
Maple's home company is Waterloo Maple in Canada, but the
development in recent years has involved several sites,
certainly ETH, Zurich as well as Canadian sites.
The first commercial releases of Maple were in the very early 1980s. 
I guess that numbering system has V as a version number,
and the release number is next. 
Many more details are at\\
\verb$http://www.maplesoft.com$\\
or via Australian distributors\\
\verb$http://www.ceanet.com.au$
% http://www.ikg.rt.bw.schule.de/maple/mvr5.htm  NICE link

Mathematica/SMP is of a CAS of a similar vintage to Maple.
(The transition from SMP to Mathematica proper - the commercial
release of Mathematica in late 1988 - was bigger than a version
change.) Both Maple and Mathematica are targetting a similar
set of customers, the ordinary users of applied calculus,
differential equations and the like. There are more non-mathematician
users of this than mathematician users and this no doubt influences
the development of the packages. Wolfram, the author of Mathematica,
is a physicist. Engineers are becoming the largest single groups of
users and, importantly, they have more money to spend on useful tools 
than do academic mathematicians. 
(CEANET, also the vendor for numerical software -  Matlab and NAG -
is an appropriate vendor.) There will be more on  
`interactive text books', `interactive engineering handbooks' later.
One reason for mentioning Mathematica early in the review is
to explain, to readers familiar with it, that Maple Vr5 has
added palettes for input of mathematics, rather like those in
Mathematica 3 (and Maple Vr4 added sectioning rather like
cell grouping in Mathematica Notebooks, hyperlinks as in
Mathematica 3 and so on).

Mathematica's assessment of the Applied CAS market place has 
Mathematica as the main vendor. As far as numbers of users are
concerned, Maple is certainly staying there as overwhelmingly the
main competitor to Mathematica. (Of course, numbers of users might
not be the only measure. Other criteria, well-implemented OOP style,
usefulness in other areas of mathematics, etc., leave a role for other
packages -
AXIOM/Aldor, MuPAD and Magma to name a few. However, this review will
concentrate on the ordinary mass-market applied-CAS genre defined
by Mathematica and Maple uses.) Some recommendations are made
%in bold face 
at various points in the review. Here are two:\\
{%\bf 
If you are already a Maple user at a site with several other Maple
users, 
stay with Maple. }\\
{%\bf 
If you are a Maple user and using a version of Maple earlier than
Vr4,
and you are using Maple, at least partly, for teaching related tasks,
{\bf upgrade now}. The presentation and interface matters make Vr3 seem 
primitive now.}\\
There are rumours around that there will be another release
within 18 months or 2 years: this might be appropriate
because of factors outside Waterloo Maple's control.

\section*{Maple Vr5}

In my first review of Maple a decade ago, I concentrated on the range
of ``ordinary applied calculus" the package could do on small computers.
(My reviews have, prior to now, been of Macintosh versions.
The review of Maple Vr4, a copy of which is up at the 
author's homepages, might be sufficiently recent to be of interest
to people familiar with Maple from before Vr4.
The Vr4 review was of the Student Version, and the summary is 
(i) that it works well, and
(ii) that
Student Versions on students' home computers are likely to be
the main way that the software gets used in large-class teaching
uses.)

The assessment of Maple for this review was done on a Pentium PC
running Windows 95, the sort of machine many of our students will
have at home. The software used was a Vr5 beta-version CD-ROM supplied
by CEANET.) 
The 1988 review waxed lyrical about how much could
be done on small machines, and I still have similar feelings,
though today's ``small" machines are different than those of
a decade ago.

\subsection*{User interface matters, etc.}

Times change; machines get more powerful.
Many engineers and other users of applied calculus now expect -
  and are getting - more
attention to presentation and user-interface matters.
Mathematical word-processing, certainly to the extent needed by
on-screen
display of mathematics, has become integrated within the packages.
(As regards printing documents, Maple has also had its Export to LaTeX 
facility since Vr3, and now, from Vr5, for people who happen not to
have Maple but need to read documents produced in it,
we have also Export to HTML. I am completely confident that
Maple will also support the HTML Math, MathML, when this is in use.)
Hyperlinks in documents - and they are truly easy to use - have been
available since Vr4. It is sensible to ask
`Why is all this happenning?'. 
(It is in Mathematica 3 as well.) 
The answer is that
these tools allow for the production of ``interactive
engineering handbooks", ``interactive mathematical and scientific
textbooks".

What extras are there in Maple Vr5? Answer: lots of things
which have been popularised by Mathematica 3.0 and more.
\begin{itemize}
\item Export to HTML has already been already mentioned.
\item
There is a MATLAB link (which, however, I have not tested as
the PC did not have Matlab).
\item
There are spreadsheet mechanisms, which could be used in
on-screen emulation of the style of older printed tables
of integral transforms, for example.
%There are no doubt many other uses, and I have heard that
%UNSW are considering teaching uses. 
%(We believe these are related to setting variants
%of similar questions to different students,
%but do not know the details).
\item
There are palettes - to assist in the input of mathematical symbols.
So far, I
have not  used these seriously in either
Maple Vr5 or in Mathematica 3.0. 
(Older mathematician users, like myself,
who have formed their habits in earlier periods, are probably a bit less
experimental on these matters than others. 
The reviewer, and most of his colleagues on the staff at UWA Maths,
when using the package to ``do sums" for their research,
would confess to using maple, ordinarily, rather than xmaple when using 
Unix systems.)
\item
There are {\tt smartplots} which allow `drag and drop' from
output selections.
Though I could have continued to live without them,
smartplots will be fun to use (in moderation) when showing off 
in front of students.
The easier changes of ranges will have educational uses.
(Another early user of Vr5 wrote:
`Smartplots are nice.  Being able to alter plot properties of
individual plots is great.  
But when I'm done, I want a record of just what properties
I've settled on so I can create the plot directly from the keyboard.'
Shouldn't be hard, Waterloo, to record a history, or better,
if possible, refer users to some tool at operating system level
which will do this.)
\item
There are now context-sensitive menus.
\item
There is now automatic command completion, an item which has
been in Mathematica for a very long time.
More on this towards the end of the article.
% in \S\ref{sec:WMS}.
\end{itemize}

\subsection*{New mathematical capabilities in Vr5, etc.}

There are some small improvements to the programming
language.

There is a change in that the ditto " meaning 
`preceding output' has been replaced with \%,
the convention used in Mathematica, MuPAD, Macsyma ... .

There are a few additions to the `algebra',
but rather more to the `computer applied calculus'.
A short sample of the latter follows.

\subsubsection*{int and special functions}

Greatly enhanced capabilities with special functions
are provided in Vr5, along with more functions,
MeijerG, LerchPhi, and many many more.

I have a paper in Journal of Fluid Mechanics,
Nov 97, where the  $w=0$ and $w=1$ special cases of
\begin{verbatim}
assume(epsilon>0):
int(exp(-epsilon*k^3)*(BesselJ(1,k)/k)*exp(I*w*k),k=0 .. infinity);
\end{verbatim}
were needed.
Mathematica 2.2 did it, Maple Vr4 did not, but Maple Vr5 does.

\subsubsection*{dsolve}

In Vr5 {\tt dsolve} can solve many more d.e.s.
Vr4 could solve, when $a=0$,
$$     y^{''} + (x-a) y =0, 
$$
but failed when $a=1$, say.
There were numerous other examples in this genre.
This is fixed in Vr5. More generally, Vr5
appears to work quite hard -- and well --
 at using symmetries in solving d.e.s.

There appear to be a few remaining errors 
(that were also present in Vr4).
Here is one
(which I encountered in July 97 when doing a problem for
Bob O'Malley during his visit to Perth).
The problem is one of several o.d.e.s popularised by
John Mahony: see page 283 of \cite{FS}.
(For more details, other related problems, and general methods,
see \cite{MS}.)
Let 
$\epsilon>0$, and solve
$$\epsilon y' = x^2 ( x^2 - y^2 ) ,
$$
with initial condition $y(-1)$ prescribed. 
Maple Vr4's answer -- easily plotted -- was discontinuous at $x=0$. 
I asked Bob O'Malley to report this to Waterloo Maple,
but I note that it is still wrong in the Vr5-beta.
In spite of the error in Maple, Maple's output, used
intelligently, does guide working mathematicians to the correct solution.

There are test suites in this area, e.g.\\
\verb$http://www.loria.fr/~zimmerma/ComputerAlgebra/$\\
I will leave it to people at this site to provide the
systematic record. However, progress since Vr4 is significant.

\section*{Requests of Waterloo Maple}%\label{sec:WMS}

I am becoming concerned about the size and complexity of the 
software, especially as regards user-interface and presentation
matters.
I favour a world where a set of orthogonal tools enable users
to get the work done.

Here are some things in recent releases I have liked.
\begin{itemize}
\item
I liked the Export to LaTeX facility introduced at Vr3.
It is easy to use and reliable.
I like the idea of Export to other widely used formats too.
I anticipate that the Export to HTML facility from Vr5
will be important (though it looks set to be revised in
major ways, perhaps in Vr6).
(Export to HTML makes up for the absence of a the free MathReader
which has been available from Mathematica, in the sense that
read-only forms of the Worksheets can now be readily delivered
to students with Netscape, but without Maple.
RMIT has done this sort of delivery via pdf files, but this
seems to be a bit fiddly to set up, and the pdf files may be
unnecessarily big.)

\item
I approve of the cooperation between Matlab and Maple
which is emerging. It may be that there is need for greater
awareness of Matlab in the {\tt codegen} package, or 
in something like {\tt MacroC} from the {\tt share} library.
One can imagine Matlab's optimization package being run
with Maple as pre-processor, where the user provides the
function $f$ and Maple produces the Matlab code for $f$,
$g$ the gradient of $f$, and $h$ the hessian, all in a form
ready for Matlab's package to call.
(Perhaps, Maple producing C codes to mex into Matlab is
better.)

\item
Though I haven't used it yet, I approve of the idea of OLE2
interconnectivity with Microsoft products.
\end{itemize}

I am less confident about describing how I would like the
products to develop.
I hope Maple's word-processing and presentation
capabilities will not grow more complex.
\begin{itemize}
\item
I speculate that interconnectivity with Web browsers like
Netscape, combined with MathML extensions to HTML, might 
be a good way to provide `interactive engineering handbooks'.
Perhaps there is even room for different CAS to use
the same display engine, such as TechExplorer.
(TechExplorer -- or some close relative --
 is the display engine of AXIOM 2.$n$, for some $n$.)

\item
Maple is available on Unix and on Mac as well as on PC.
In the PC CAS marketplace,
Scientific Workplace, MathOffice and Mathcad all have
Maple as their algebra engines.
(Scientific Workplace is TeX oriented, or at least
the company had Scientific Word as its first product.
MathOffice is for PCs with Microsoft products.) 
I hope that it will become clearer which market segments
are appropriate to which product.
Perhaps some tools for conversion between the different
formats may become available.
\end{itemize}

\subsection*{Word processing? Presentation?}

I have heard from another user who is working with both
Vr5 and Scientific Word/Scientific Workplace. 
The user has an Engineering Mathematics text in preparation,
its difference from those that dominate the market place
at the moment being that it has been designed for students
with CAS.
This user wants `Maple-exported Latex to be compatible with Scientific
Workplace'. In this case, presumably the designers and
programmers of both products need to work together on this.


\subsection*{An easy change?}

%Although I have to admit in the Vr5 beta I have I still haven't
%got the automatic command name completion going (and haven't found
%the files relevant to this in the places where the help screen says
%they shoud be), I think the 
Automatic command name completion allows the possibility of
better command names. Mathematica has done a good job with 
these. Given that the world has these two large CAS packages, 
it would be better, I think,
if, where they have the same function, it has the
same name.
(There is the matter of backwards compatability, to consider,
but a translation tool %- something lex or even sed like might do -
could be provided.) 
There appears to be little gain in Maple using
abbreviations like 
{\tt eigenvals} and 
{\tt eigenvects} rather than the complete
names that Mathematica uses. Of course, Maple should stay
with its own conventions on use of 
upper and lower case.

Until the harmonisation is achieved,
I would welcome someone providing a file to be read
into Maple which gave a set of aliases so I could
use the full-word Mathematica form
(though without the initial letter capitalization used
in Mathematica).
(A problem at UWA is that certain groups -
Electrical and Electronic Engineering, Physics
and Chemistry - requested that their students
were taught using Mathematica rather than Maple.
UWA Mathematics staff put
effort into making sure that undergraduates
never need to learn more than one CAS, and
that large classes are taught with just one
wherever possible. 
Inevitably it happens that some 3rd and 4th year
courses have some Maple-using students  and
others who are Mathematica users.
Materials then need to be provided in both,
and if this could be made easier it would be
appreciated.)

\section*{Conclusion and Acknowledgement}

I very much like Maple Vr5
and look forward to hearing from other Australian Maple users
(e-mail: \verb$keady@maths.uwa.edu.au$)
on points raised in this article.

The PC on which Maple Vr5 was tested was provided by
Andrew Barry of Environmental Engineering. Thanks.
\begin{thebibliography}{999}
% REFERENCES

\bibitem[1]{B1}
S. Bedding,
Teaching Mathematics with mathematical software,
{\it Gazette of the Austral. Math. Soc.}
{\bf 22} (March 95), 43-44

\bibitem[2]{FS}
N. Fowkes and J. Silberstein,
Obituary: John Mahony,
{\it Australian Academy of Sciences, Historical Records of
Australian Science},
{\bf 10}(3) (June 95), 265-291.

\bibitem[3]{MS}
J.J. Mahony and J.J. Shepherd,
Stiff systems of ordinary differential equations, Parts I, II and III,
{\it Bulletin of the Austral. Math. Soc., Series B}
{\bf 23} (1981), 17-51, 136-172 and 310-331.

\end{thebibliography}

\end{document}
\end