Numerical Calculations Using Maple: Why & How?

9 Pages·0.41 MB·English

by  E. V. Correa Silva

Numerical Calculations Using Maple: Why & How?
E.V. Corrˆea Silva∗, L.G.S. Duarte†L.A. da Mota‡and J.E.F. Skea§
Abstract: The possibility of interaction between Maple and numeric compiled languages in
performing extensive numeric calculations is exemplified by the Ndynamics package, a tool for
studying the (chaotic) behavior of dynamical systems. Programming hints concerning the con-
structionofNdynamicsarepresented. Thesystemcommand,togetherwiththeapplication ofthe
black-box concept, is used to implement a powerful cooperation between Maple code and some
other numeric language code.
Keywords: Dynamical systems, chaos, numeric calculations, non-symbolic compiled languages,
9 black-box
9
9 Introduction
1 dX
In this paper we explore the possibility of interaction =F(X,t), (1)
n dt
a between Maple and numeric languages, exemplified by
J the Ndynamics package [8] — a tool for studying the whereX=(X1(t),X2(t),...,Xn(t))andF=(F1(X,t),
3 (chaotic) behavior of dynamical systems. F2(X,t),...,Fn(X,t)). The variables Xi(t), where i =
1 Maple interaction facilities may benefit users and 1,2,...,n,representtherelevantquantitiesinsomephys-
ical model, and t is a continuous parameter (time, for
programmers of both symbolic and numeric languages.
A] Asfarastheimplementationofextensivenumericalcal- instance). Each Fi(X,t) is an arbitrary function of the
variables X and of the parameter t.
N culationsisconcerned,non-symboliccompiledlanguages
Roughly speaking, “chaoticity” means extreme sen-
like FORTRAN, PASCAL or C are still more popular than
h. anysymbolicenvironment;theexistingcollectionsofef- sitivity to small changes in the initial conditions. This
t is the tipical case of 3-dimensionalnonlinear dynamical
a ficient programswritten in such languagescould not be
systems, among whichchaotic behavior is a rule, rather
m simply overlooked.
than an exception: due to nonlinearity, small fluctua-
[ This paper is organizedas follows: first,the concept
of dynamical systems is introduced to the reader, as tionsintheinitialconditionspropagatedramatically,so
1 thattwoneighbouringinicialconditionsmayyieldorbits
well as to the concepts of fractal dimension and bound-
v with completely distinct asymptotic behavior.
ary [4], extensively used in the paper; the commands of
5
Chaotic systems have a richer structure than “well-
5 the package are then briefly described and an example
behaved” ones, hence their interest. Strange attractors
0 of utilization follows. Finally we point out some use-
1 ful considerations about programming design, as to the andrepellers,aswellasfractalboundariesareexamples
0 of peculiarities of chaotic systems. A measure of the
possibility of interaction between Maple and other lan-
9 degree of chaoticity of a system can be obtained by the
guages.
9
evaluation of the fractal dimension of the boundaries.
/
h Dynamical Systems In the next section, we are going to elaborate fur-
t ther the concept of fractals and introduce a method to
a
m Systems of ordinary differential equations (or dynam- calculate the associated fractal dimension.
ical systems) play a central role in a large number of
:
v problems in all areas of scientific research, mainly in Fractals
i
X physics. Hence the importance of developing tools to
Fractal Dimension
study thesesystems—inparticular,nonlinearsystems.
r
a A n-dimensional dynamical system can be generically The idea of fractal will always lack a precise definition
represented by [1]. However,whenonereferstoasetFasafractal,one
∗CentroBrasileirodePesquisasF´ısicas,R.Dr. XavierSigaud, typically has the following in mind:
150, Urca, CEP 22290–180, Rio de Janeiro, RJ, Brazil. E-mail:
[email protected] 1. F has a “fine structure”; i.e., has complex details
†UniversidadedoEstadodoRiodeJaneiro,InstitutodeF´ısica, on arbitrarily small scales;
Departamento de F´ısica Teo´rica, R. Sa˜o Francisco Xavier, 524,
Maracana˜, CEP 20550–013, Rio de Janeiro, RJ, Brazil. E-mail: 2. Fistooirregulartobedescribedinthetraditional
[email protected]
‡Idem. E-mail: [email protected] geometrical language, both locally and globally;
§Idem. E-mail: [email protected]
MapleTech Vol.?, No. ?, pp. 1–9; ISSN 1061-5733 $6.00 (cid:13)c Birkh¨auser Boston 2008 1
3. often, F has some form of self-similarity, perhaps
approximate or statistical;
4. usually, the “fractal dimension” of F (defined in
some way) is greater than its “topological dimen-
sion”;
5. in most cases of interest, F is defined in a very
simple way, perhaps recursively.
There are a lot of manners to define the dimension
of a fractal. An important one is called the Hausdorff
dimension [2], onepossiblegeneralizationofthe “primi-
tive”notionofdimension. Supposewehaveahypercube
of edge a, its hypervolume V being given by
d
V =a , (2)
where d is the hyperspace dimension. Let us divide the
hypercube into N hypercubic cells of edge ǫ, we have
d
V =Nǫ . (3)
Dividing (3) by (2) we have:
ǫ d
1=N (4)
a
(cid:16) (cid:17)
ǫ
Defining δ = and expressing the number of cells as a
a
function of δ (i.e., N =N(δ)),
d
1=N(δ)δ (5)
Solving (5) for d, we obtain the standard definition of
dimension:
ln(N(δ))
d=− (6)
ln(δ)
Letus now measurethe dimensionofaCantorset, con-
structedthrougharecursiveprocedurepicturedinfigure
.
In figure 1 we take a segment of length ǫ0 and divide it
into three parts of equal length ǫ1 = ǫ0/3; the middle
segment is then replaced by two segments of length ǫ1.
Figure 1: Cantor set after 0, 1, 2 and 6 iterations, re-
We then apply the same procedure to each segment of
spectively.
length ǫ1, dividing them into segments of length ǫ2 =
ǫ1/3, and so on. After M iterations, we will have a
M M
total of N = 4 segments of length ǫM = ǫ0/3 each.
Defining
ǫM 1
δ ≡ = , (7)
ǫ0 3M
we may write
ln(δ) ln(4)
ln(N(δ))=− . (8)
ln(3)
2 MapleTech
Commands of the Package
Taking the limit M → ∞ (or δ → 0), the Hausdorff
dimension (6) of the Cantor set is defined by the limit
Inthissectionwepresentabriefdescriptionofthecom-
mands of the Ndynamicspackage. A more complete de-
ln(N(δ))
d=−lim (9) scriptioncanbefoundintheon-linehelp,providedwith
δ→0 ln(δ)
the package 1.
which, in the case considered, yields
• Nsolve allows the user to specify a system of dif-
ln(4) ferential equations and initial conditions, calcu-
d= ∼1.26 (10)
ln(3) lating trajectories in phase space and generating
2D/3Dplots. Also,randominitialconditionsmay
Fractal Dimension of Boundaries
bechosenfromauser-definedregionofphasespace.
LetRbe a D-dimensionalfinite region—whichwewill
• View allows quick and comfortable visualinspec-
taketobeahypercubeofedgea,withoutlossofgeneral-
tionof the trajectoriescalculatedby Nsolve,and
ity—dividedintoN hypercubic“cells”ofedgeǫ. Letus
the identification of regions of interest in phase
chooseanarbitrarycellC. Ifthereareanypairofpoints
space (e.g., chaos). Zooming in and out is sup-
(P,Q) in C so that the orbits of P and Q have distinct
ported, as well as choosing the variables to be
asymptoticbehavior,C issaidtobeaboundary-cell. In
shown (e.g., 2D or 3D graphs).
the limiting case where N is infinite (ǫ→0), the union
of all boundary-cells constitute a boundary.
• Boxcount performs small random perturbations
Let NB be the number of boundary-cells in R. Ac- in the initial conditions and analyzes their effect
cordingtoeq.(9), thefractaldimensionofthe boundary
upon the evolution of trajectories. Perturbed ini-
[4]canbe inferredfromthe behaviorofNB asǫ→0or, tialconditionsarechosenfromsmallregionssized
equivalently, as δ →0, where δ =ǫ/a.
by a user-defined perturbation parameter,around
However,the total number of cells
each umperturbed initial condition. In the neigh-
D borhoodofaboundary,twosmallbutdistinctper-
1
N(δ)= (11) turbations of the same initial condition may yield
(cid:18)δ(cid:19)
radicallydifferentperturbedtrajectories;Boxcount
rapidly increases as δ → 0, and the computation of determinesthe fractionofthe totalnumber ofini-
NB(δ) results unpractical. An alternative approach is tial conditions for which this is so.
∗
that of picking up N random cells in R and counting
• FdimensionmanagestheexecutionofBoxcount
∗ ∗
thenumberofboundary-cellsN . IfN islargeenough
B for a row of values of the perturbationparameter,
to be statiscally meaningful, we expect that
and analyzes the results obtained to evaluate the
NB∗ NB fractal dimension of the boundary.
→ . (12)
N∗ N
Examples
Supposing that, indeed,
∗ Commands of the package
NB NB
= (13)
N∗ N Ndynamicscontainshelpfultoolsforthedetectionofin-
teresting regions (such as strange attractors, repellers,
we have
andfractalboundaries)in the phase spaceof a dynami-
∗ ∗
ln(NB) ln(NB/N ) ln(N) cal system. We will take the well-known Lorenz system
− =− − . (14)
lnδ lnδ lnδ as an example.
dB D The package is loaded by the standard command
| {z } | {z } with. The precision of calculations will be set to 16
Defining digits, using Maple environment variable Digits 2.
∗ ∗
ln(N /N )
B
α≡D−dB = (15) > with(Ndynamics);
lnδ
> Digits := 16;
we have ∗ We define the Lorenzsystem,andassignsome valuesto
N
ln B =α lnδ (16) its constants:
(cid:18)N∗(cid:19)
1 http://www.dft.if.uerj.br/symbcomp.html
∗ ∗
Bothδ andNB/N canbe measured,allowingus tode- 2In what follows, the output of command lines has been oca-
termineαandthefractaldimensiondB oftheboundary. sionallyomitted.
Vol. ?, No. ?, 2008 3
> Lorenz := {diff(x(t),t)=sigma*(y(t)-x(t)),
> diff(y(t),t)=-x(t)*z(t)+R*x(t)-y(t),
> diff(z(t),t)=x(t)*y(t)-b*z(t)};
> sigma:=10;b:=8/3;R:=28;
The command Nsolve generates a set of random initial
conditions within a user-defined region in phase space, 40
and then calculates the orbit of each individual initial
condition. The number of initial conditions to be gen- 30
erated is controled by the global variable number ic.
z(t)
In addition, the color of each individual orbit can be 20
defined by abooleanexpressionregardingthe finalcon-
dition of the orbit, to be assignedto the globalvariable 10
Colouring. According to the value of that expression,
an orbit may be plotted in one out of two predefined
0 -15 -10 -5 0 5 10 15
colors. x(t)
> number_ic:=8;
> Colouring:=x(t)<-4 and x(t) > -11;
If the user is interested in detailing some particular re-
For the sakeof syntaxclarity only,the following assign-
gionofphasespace,thecommandViewofNdynamicsis
ments are performed: the variable init region defines
recommended:
rangesforeachdependentvariable,thuslimitingthere-
gion in phase space for initial conditions; t range sets > t0 := time();
> View([x=-10..0,z=20..30]); tf := time()-t0;
the independent variable range; steps to the indepen-
dent variable are defined by t calc step (for calcula-
tionpurposes)andt plot step(forplottingpurposes).
30
> init_region := [x=-0.37717..-0.37716,
> y=0.48685..0.48686, z=-0.29894..-0.29893];
> t_range := 0..37; 28
> t_calc_step := .005;
> t_plot_step := .01; 26
The command Nsolve may now be run. The standard
z(t)
Maple function time is employed to give the reader an
24
estimatecomputationtime(whichismachine-dependent
of course).
22
> t0:=time():
> graph := Nsolve(Lorenz, [init_region,
20 -8 -6 -4 -2 0
> [t=t_range,t_plot_step]],[x(t),z(t)], x(t)
> random,method=[rk5C,t_calc_step]):
> tf := time()-t0;
tf :=51.569
(...some ommited output ...)
Inorderto exemplify the use ofthe commands Box-
tf :=123.880
countandFdimension,letusslightlymodifytheLorenz
We have used the option method=[rk5C,...], i.e., we system parameters:
have used the ‘plugged module’ in C to perform the
> sigma:=10;b:=8/3;R:=20;
numeric calculations. The result can be plotted, for in-
and choose another region of initial conditions, as well
stance, by using the command displayof the standard
as other range and steps for the independent variable:
Maplepackageplots(previouslyloadedbyNdynamics):
> init_region2 := [x= -1.001 .. 1.001,
> y = -1.001 .. 1.001,
> display(graph); > z = 21.999 ..22.001];
> t_range2 := 0..16;
> t_calc_step2 := 0.01;
> t_plot_step2 := 0.05;
Also,we willchooseamuchlargenumber ofinitialcon-
ditions, and a different coloring criteria. Besides, all
4 MapleTech
three dependent variables (x,y,z) are to be shown in linear correlation = ,.9993933454816381
the resulting 3-D plot:
> number_ic:=10000;
> Colouring:= x(t)<0:
> frame := [x(t),y(t),z(t)];
Once more Nsolve generates random initial conditions -2.8
and calculates their orbits: -3
> t0 := time():
-3.2
> Nsolve(Lorenz,[init_region2,[t=t_range2,
> t_plot_step2]],frame,initial,random); -3.4
> tf := time()-t0;
-3.6
(...some deleted output ...)
-3.8
tf :=75.725
-15.4 -15.2 -15 -14.8 -14.6 -14.4 -14.2 -14 -13.8
The command Boxcount is now able to count the
numberofhypercubesovertheboundary. Basically,this
command takes perturbed initial conditions (those gen-
tf :=964.940
erated by Nsolve) according to a user-defined parame-
ter, say, epsilon, and calculates their orbits. The final Where the slope of the straightline above (α) is de-
value of the independent variable is representedhere by fined by 16.
final time, and the integration step by integ step. Performance of the Package
(Once more, these variables are assigned just for the
Inthecalculationsabove,theoptionmethod=[rk5C,...]
sake of clarity of the Boxcountcommand line.)
allowed us to employ the Ndynamics“plugged module”
> epsilon := 0.0000002;
—thefilerk5.c—containingtheCsourcecodeforthe
> final_time := 16;
> integ_step := 0.02; algorithm of the 5-th order Runge-Kutta method.
> t0 := time(): To point out the advantages of our hybrid symbol-
> Boxcount(epsilon,final_time,method= ic/numeric approach, which uses Maple to manage the
> [rk5C,integ_step]);
source code generation and compilation for the num-
> tf := time()-t0;
ber crunching, while maintaining Maple’s flexibility, ta-
ble 1 presents a comparison of the elapsed time taken
reading rk5 output
to perform the same calculation using the C interface
From the, 10000, points (that were testable), , 200, and performing the entire calculation in Maple, using
some of its numerical integration routines. In order to
of them were close to the boundary.
obtain a fair comparison, particular care was taken to
[200, 10000] use methods of the same order and adjusting parame-
ters to produce solutions with the same precision. The
tf :=170.560
following procedure was used: for a given set of initial
Inits turn,the commandFdimensioncalculatesthe conditions,Maple’s built-inhigh-precisionTaylorSeries
fractaldimensionoftheboundary,bypickingupagiven integratorwasusedtointegratetheLorenzsysteminthe
number of distinct values of the perturbation paraneter interval 0 ≤ t ≤ 11, until convergence was obtained to
withinauser-definedrange,applyingBoxcountforeach an accuracy of 13 decimal places; subsequently Maple’s
case. global variable Digits was set equal to 13 (higher val-
> epsilon_range := 0.0000002..0.000001; ues of Digits seemed to cause problems with Maple’s
> n_epsilons := 5; integrator),andMaple’sinbuilt5th-orderRunge-Kutta-
> t0 := time(): Fehlbergintegrator,rkf45,wasuseduptot=11. This
> Fdimension(epsilon_range, final_time, n_epsilons,
integrationwasfoundtobeaccurateto4decimalplaces.
> method=[rk5C,integ_step]);
> tf := time()-t0; ThestepsizeintheC routinewasthenadjustedtogive
the same3 precision, resulting in a step size of 0.002.
(...some deleted output...) Having matched the precision of both results, the aver-
ages of computation times for both methods was then
Fractal dimension = ,2.213198116174276,
taken for 200 integrations.
statistical error = 1.868128109494962, % 3infact,theCroutinewasslightlymoreprecise
Vol. ?, No. ?, 2008 5
TheresultsbelowwereobtainedwithMapleV.4run- much comfortable by the interactive user-interface of
ning in Windows 98 on an AMD-K6 266 with 64Mb of Maple. (Changing parameters and checking out results
SDRAM. The C compiler used was Delorie’s implemen- is faster and easier.)
tation of GNU’s gcc 4, with level 3 optimization. In one aspect, the question is analogue to “Why us-
ing a high-level language instead of Assembly ?” No
Integration C-interface Maple matter how fast Assembly programs may be, the kind
Method rk5C rkf45 of details involved in their construction (most of them
Seconds per Trajectory 0.1 7.0 totally unrelated to the problem at hand) certainly dis-
Ratio to the Fastest Case 1 70 courage its use. The more a given language allows its
programmer to focus on “problem-related” details and
Table 1: Comparative performances of Maple’s inbuilt ignore the rest,the most suitable the languagebecomes
numerical integrators and Ndynamics C interface, for to find a solution to the problem. The very nature
the Lorenz system. of ever-evolvingproblems in scientificalresearchrecom-
mends the use of “higher-level” languages.
It would be highly desirable to combine the features
ofsymbolicandnumericlanguages,forthetaskinview;
Programming and Design
thatwasoneofthe guidelinestotheconstructionofour
Considerations package. “High-level”tasks,sotospeak,arehandledby
code written in Maple, whereas numerical calculations
Building computer programs demands methodical ap-
maybe(optionally)performedbyapieceofcodewritten
proach,iftimeandeffortofconstructionaretobesaved,
in some other language. The present implementation of
and coherence to be kept. Furthermore, debugging and
the program contains an extension written in C; future
modificationofaprogramrelatesdirectlytohowappar-
versions will allow the user to simply “plug” his own
entthepurposeofeachpieceofcodeturnsout. Software
(numerical) code. The implementation of these ideas
design is, in itself, a rich subject and field of research,
in Maple was achieved through the utilization of the
barelytouchedhere. Yet,wehopethereadermayprofit standard command system 5 — which allows one to
from reading our present considerations.
sendcommandsdirectlytotheoperationalsystem—as
Interface with Compiled Languages well as the possibility of exchanging data through files.
In the current implementation of Ndynamics, the
The numerical integration of a system of differential
command system is used for:
equationsexemplifiesatipicalsituationinvolvingexten-
sive numericalcalculations. As far as the efficiency (ba- • compiling 6 two source code files: derivs.c, con-
sically,memoryusageandprocessingtime)ofnumerical taining the definition of the dynamical system;
algorithmsis concerned,general-purposesymbolic com- and rk5.c, containing the code for the 5-th order
puting systems (based on interpreted languages) gener- Runge-Kuttamethod. Theexecutablefilerk5.exe
ally offer less attractive means of implementation than is then produced.
non-symboliccompiledlanguagessuchasC,FORTRAN,
and PASCAL. The long history of non-symbolic lan- • executing rk5.exe to perform calculations. The
guages has already produced a large collection of effi- filesrk5.inandrk5.outare,respectively,the in-
cient programs for a wide variety of specific problems put and output files for rk5.exe.
— well-known examples are the NAG library of FOR-
All these operations are performed automatically, en-
TRAN programs, and the collection of C programs in
tirely from within Maple. The user needs not to be
[7]. Using a symbolic environment for numerical calcu-
aware of any such processes.
lations,discardingexistingsolutions(basedoncompiled
The attentive Maple programmer will notice that
languages)mightsoundlikeanattemptto“reinventthe
the utilization ofrk5.exe—assumingthatrk5.inand
wheel”.
rk5.out have the proper formats — does not depend
Thepointisthatsymboliclanguagesprovideflexibil-
whether its original source code was in C or any other
ity and interactivity which lacks in “numeric-oriented”
language. This is an example of a “ black-box”, a con-
languages. In other words, symbolic code and data
cept to be explored in what follows.
structures tend to be more friendly and versatile than
numeric ones. For instance, in our specific case, the 5PleaserefertotheMapleon-linehelp,bytyping ?system.
6Currently,theDJGPP-32GNUcompilerisused. Pleaserefer
identification of chaotic regions in phase space is made
toNdynamics on-linehelpformoredetails.
4http://www.delorie.com
6 MapleTech
Black-Boxes andParameter PassingTech- > member(x,[y,y,x,y],’pos’);
niques > # pos is an output parameter, to be
> # assigned to the position of the
Awell-knowconceptinsoftwaredesignisthatofablack- > # first occurence of x in [y,y,x,y]
box [5]. For our present purposes, it can be pictured
as a Maple procedure or an executable file which can true
be used without any assumption on its internal logic:
all we need to know relates either to input and output
data (meaning, form, values) or to what the procedure > pos;
does (rather than how it is done). A good example of
3
a black-boxis the standardMaple function sin: allone
needs to know to use sin is that it accepts a single Anotherexampleisthecommandirem,similartoiquo.
input argument (a valid algebraic expression, assumed Let us make our point clear with the help of a “toy
to be expressed in radians) and that its output is, also, procedure” f:
an algebraic expression. Using the function sin does
> f := proc(a::posint,sm_,pr_)
not require the knowledge of details concerning actual
> local i;
computations. > sm_ := convert([seq(i,i=1..a)],‘+‘);
Ideal packages, made out of black-box-like proce- > pr_ := convert([seq(i,i=1..a)],‘*‘);
> NULL;
dures,wouldbemosteasilyunderstood,andhencemost
> end:
easily tested, debugged and modified.
Given a positive integer a, we compute the sum (1+
Exchange of data among procedures is a crucial as-
2+...+ a) and the product (1∗2∗... ∗ a), which are
pect in the construction of a package. A Maple proce-
assigned to the parameters sm and pr , respectively.
dure may exchange data by
An example of usage is
• using global or environment variables (input and > f(5,’s,p’); # null output
output); > s,p;
15, 120
• using the result of its last executed line (output);
Now compare the procedure f to a slightly (but signifi-
• using the RETURN command (output); cantly)modifiedversiong,whereoutputparametersare
not used:
• reading from or writing to a file (input and out-
put); > g := proc(a::posint)
> local i, s, p;
• usingitsparametersorarguments(inputandout- > s := convert([seq(i,i=1..a)],‘+‘);
> p := convert([seq(i,i=1..a)],‘*‘);
put).
> s,p;
> end:
Ofalltheseways,thelastoneappearstousasthemost
Thisversionemploysthe“lastexecutedline”technique.
interesting, as far the implementation of black-boxes is
It would run as
concerned. Using arguments of a procedure as a means
of conveying input data is a common practice, needing > g(5);
nofurthercomments. Wewouldliketoinvitethereader 15, 120
toattendmorecloselytothepossibilityofpassingoutput
What is the practical difference between the two proce-
data through the arguments of a procedure — which is
dures? The reader should notice that, in order to inter-
also a knowntechnique [6, ?] found, for instance, in the
pret(andevaluatethecorrectnessof)theoutputofg,he
standard commands iquo and member:
mustholdtheextrainformationthatthefirstoperandof
> iquo(11,2,’r’); theoutputsequencecorrespondsto1+2+3+4+5,and
> # r is an output parameter, to be
the second one correspondsto 1∗2∗3∗4∗5 — i.e., the
> # internally assigned to the
> # remainder of the integer division interpretation of the output depends on its particular
> # 11/2 (and arbitrary) form. In this example, it is a sequence,
5 but it might as well be a list, or even a table. Were it a
sequence or a list, the first operand might be the prod-
> r; uct, instead of the sum ... Well, whatever the choice
1 as to the output form— which has to be made —, it
wouldcertainlyhavelittletodowiththeactualpurpose
Vol. ?, No. ?, 2008 7
of the procedure (i.e., calculating sums and products), ComputationalPhysicsanditsrelationtotheteach-
andthus wouldbe superfluous detail. A procedurecall- ing of Physics.
ing g (and its poor programmer) would have to know
• L.G.S.Duarte is currently a member of staff at
all about these details, though; it (or he, or she) would
theUniversidadeEstadualdoRiodeJaneiro(UERJ).
be free from such concern, if f were used instead. The
He obtained his PhD in Physics at the Universi-
procedurefcanbeusedasablack-box,whereasgcan’t.
dade Federal do Rio de Janeiro, in 1997. His in-
When it comes to developing a large number of in-
terests concentrate on mathematical physics and
tegrated Maple procedures, the black-box is a valuable
hehasrecentlyparticipatedonaprojectthatlead
concept to keep in mind, and so is the “output param-
to the creation of a differential equation solver,
eter” technique. That doesn’t mean that all procedures
writen on Maple, that was finally introduced on
should be written in that way, though. The “last line”
thecomercialreleasenumber5ofsuchmathemat-
and “RETURN” techniques, for instance, seem to be
ical package.
the mostsuitable touser-levelprocedures. The “output
parameter” technique, in its turn, fits best on internal
• L.A.C.P.da Motais currentlya member ofstaff
procedures, where all the “hard work” is carried out.
at the Universidade Estadual do Rio de Janeiro
(UERJ). He obtained his PhD in Physics at the
Conclusion
University of Oxford, U.K, in 1993. He then held
ThepowerfulcombinationofMapleflexibilityandinter- anoneyearpositionasavisitingresearcheratthe
activity with the high speed of numeric languages has same University. His interests were then Elemen-
been used in the implementation of Ndynamics,a pack- tary Particle Phenomenology. Most recently, his
age for the numerical study of (chaotic) behaviour in work has concentrated on mathematical physics
dynamical systems. A more detailed description of the and participated on the same project (mentioned
functionsandphysicalapplicationsofNdynamicscanbe on the biography above).
found on [8].
• J.E.F.SkeagraduatedfromtheUniversityofGlas-
We have also presented some of the programming
gow with a joint honours degree in Physics and
hints and strategies used in its construction, hoping to
Astronomy. He first worked with computer alge-
have shown that there can be a a stronger link between
bra as a research tool at the University of Sussex
users (and programmers) of two kinds: those who are
where, under the supervision of Roger Tayler and
interested in heavy numerical calculations, and those
JohnBarrow,he completedhis Ph.D. inthe areas
whose interest lies in symbolic computations.
of general relativity and cosmology. Subsequently
PartofthecurrentimplementationofNdynamicshas
he worked for 3 years as a postdoc at the School
been written in C and has been “plugged”, so to speak,
of Mathematical Sciences of Queen Mary College
into the main package. Currently, the user himself can
London with Malcolm MacCallum, mainly on the
write and use his own integration routine in C, as long
CAsystemsSHEEPandREDUCE.Heleftscience
asthesameformofinputandoutputdataiskept7. Fu-
foracoupleofyearstoworkinSciencePolicywith
ture versions of Ndynamics will allow the user to “plug
Ben Martin at SPRU in the University of Sussex,
and play” his own numerical piece of code and run the
on the analysis of performance indicators, but re-
compiler corresponding to the language of his prefer-
turned to physics for his third postdoc, this time
ence, provided that the input and output data follow
at CBPF (Brazilian Centre for Physics Research)
the proper conventions.
in Rio. He moved back to Scotland when he fi-
nallygotateachingpositionintheDepartmentof
Biographies
MathematicsoftheUniversityofAberdeen,where
• E.V.CorrˆeaSilvaiscurrentlyaPh.D.studentof heinitiatedacourseonthetheoryofcomputeral-
Physics (Quantum Field Theories) at the Centro gebra, using Maple in practical sessions. In 1996
Brasileiro de Pesquisas F´ısicas, in Rio de Janeiro, he moved to the Theoretical Physics department
Brazil. He obtained his Master degree in Physics ofUERJ(RioStateUniversity)whereheseemsto
at the Universidade Federal do Rio de Janeiro, in have settled down ... for now.
1994. His research interests include the Quantum
Hall Effect, Quantum Field Theory and Chaos, References
as well as System Design, Artificial Intelligence,
[1] B.B.Mandelbrot,TheFractalGeometryofNature.
7Formoredetailedinstructions,pleaserefertotheon-linedoc- Freeman, San Francisco, 1982.
umentation.
8 MapleTech
[2] F.Hausdorff,Dimension unda¨usseres Mass,Math.
Annalen 79, 157 (1918).
[3] E. N. Lorenz, Deterministic Nonperiodic Flow, J.
Atmos. Sci. 20, 130 (1963).
[4] E. Ott, Chaos in Dynamical Systems, Cambridge
University Press, 1997.
[5] Yourdon, E. and Constantine, L.: Structured De-
sign. Yourdon Press, 2nd ed. New York, 1978.
[6] Heck, A.: Introduction to Maple. Springer-Verlag,
New York, 1993.
[7] Press, W.H., Teukolsky, S.A., Vetterling, W.T.,
Flannery,B.P.,NumericalRecipesinC,Cambridge
University Press, second edition.
[8] L. G. S. Duarte, L. A. C. P. da Mota, H. P. de
Oliveira, R. O. Ramos and J. E. F. Skea: Numeri-
cal Analysis of Dynamical Systems and the Fractal
Dimension of Boundaries. Submitted to Computer
Physics Communications, chao-dyn/9812029.
Vol. ?, No. ?, 2008 9