Computer Algebra with SymbolicC++

598 Pages·2008·4.171 MB·English

by  Yorick Hardy, Willi-Hans Steeb, Kiat Shi Tan

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Yorick Hardy
University of Johannesburg
South Africa
Kiat Shi Tan
Ilog Co., Ltd., Singapore
Willi-Hans Steeb
es. University of Johannesburg
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World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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Algebra on 07/2
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CoVE Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from
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E ISBN-13 978-981-283-360-0
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A ISBN-10 981-283-360-9
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b ISBN-10 981-283-361-7 (pbk)
Printed in Singapore.
LaiFun - ComputerAlgebra.pmd 1 7/18/2008, 3:39 PM
es. Preface
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cientifid, exce ematics and physics.
orldsmitte In the most restrictive sense, computer algebra is used for the manipulation of sci-
w.wper enti(cid:12)c and engineeringformulae. Usually, a mathematical formula described in the
wot programming languages such as C, C++ and Java can only be evaluated numeri-
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wnloutio additiontothenumericalmanipulations. Thisisthereforeanindispensabletoolfor
C++ Dod distrib research and scienti(cid:12)c computation.
SymbolicRe-use an csOcabileljyne,ccoet-baojsreiciettn-htoaerdsiebnpetreeondgrspaurmgoggmersaintmegdmhaianssgaciprseoaasntseiibdmleapsonoreltuwatnieotrnaaptfoporsroopaftcrwhogatrroaeamdnemavlieynlzogipnimngepcnrotom.bpBleuamtsesir-,
Algebra with on 07/25/19. dseoxeftsteiwgnnasiirnbeglep.sryosdtuemctss abnedcobmueildleisnsgesrorlourtipornosn.eB, yeaaspieprlytoingmtahinistamine,thmoodree(cid:11)reecutsiavbellye,atnhde
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NI Among the many object-oriented programming languages available nowadays, we
U haveselectedC++asourprogramminglanguage. Itisthemostwidelyusedobject-
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In addition to the reasons stated above, we have selected C++ over other object-
oriented languagesbecause of its e(cid:14)ciency in execution speed and its utilization of
pointers and templates. The Standard Template Library provided by C++ is very
helpful for the implementation of a computer algebra system.
Chapter 1 introduces the general notion of Computer Algebra. We discuss the es-
sential properties and requirements of a computer algebra system. Some pitfalls
v
vi PREFACE
and limitationsarealsolisted forreference. Finally, wepresentacomputeralgebra
system | SymbolicC++. This new system has many advantages over existing
computer algebra systems.
Chapter 2 presents the general mathematics for a computer algebra system. We
describe how fundamental mathematical quantities are built up to form morecom-
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cientifid, exce cInlasCsehsa,pctoerns4t,ruwcetoirnst,roadnudcetetmhpelalatnesg.uaWgeetdoeoslscriinbeCe+rr+orsuhcahndalsintghetetchhinsiqupeosinatenrd,
orldsmitte introduce the concept of exception handling. Examples are also given for demon-
w.wper stration purposes. Wealsodescriberecursion. A numberof programsillustratethe
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C++ Dod distrib use of this class. This stringclass of C++ will be used in SymbolicC++.
SymbolicRe-use an TltahhrrgeeeeSnftoauunmndbdaearrdtiooTfneaemxlpaitmlaemtpelseL:s.icboArnattraytihn(eSerTcsoL,r)ealigosofirntihtthreomSd,tuacanendddairindteCrTahetamoprptsel.artT6ehLteoisbgeertaithreyemraswrewitohthrkae
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Computer VERSITY Cvehraypltoenrg7ingtievgeesras,croaltlieocntiaolnnuomf ubseerfsu,lqculaastseersnifoonrsc,oemxapcuttdereraivlgaetbivreas.,Wveectionrvse,smtiagtartie-
NI ces, arrays, bit vectors, (cid:12)nite sets and polynomials. They are the building blocks
U of mathematics as described in Chapter 2. The internal structures and external
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duced and its internal representations and public interfaces are described. Several
examples are also presented to demonstrate the functionalities of the system. A
symbolic numeric interface is also described.
In Chapter 9, we apply the classes developed in Chapters 7 and 8 to problems in
mathematicsandphysics. Applicationsarecategorizedaccordingtoclasses. Several
classes may be used simultaneously to solve a particular problem. Many interest-
PREFACE vii
ing problems are presented, such as ghost solutions, Pad(cid:19)e approximant, Lie series
techniques, Picard’s method, Mandelbrot set, etc.
In Chapter 10, we discuss how the programming language Lisp can be used to im-
plement a computeralgebrasystem. We implement an algebraicsimpli(cid:12)cation and
di(cid:11)erentiation program.
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SymbolicRe-use an Steeb (1994) [50], Steeb (2005) [54] and Steeb et al. (2004) [55].
h 9. The C++ programshavebeen tested with all newer C++ compilers which comply
wit5/1 with the C++ Standard and include an implementation of the Standard Template
Algebra on 07/2 Library.
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mputRSI License. We omit the following comment (or its equivalent) in all header (cid:12)le and
CoVE program (cid:12)le listings in the interest of brevity:
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P SymbolicC++ : An object oriented computer algebra system written in C++
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y This program is free software; you can redistribute it and/or modify
b it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
viii PREFACE
You should have received a copy of the GNU General Public License along
with this program; if not, write to the Free Software Foundation, Inc.,
51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*/
The license is included with SymbolicC++ which is available from the web site
s. of the International School for Scienti(cid:12)c Computing as described below.
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orldscientifimitted, exce Sinygm. [e-mail]Siecnhtoio(cid:12)lcfCoromScpieuntitni(cid:12)gcaCreomput-
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Contents
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O Preface v
cientific.comd, except for 1 I1n.1troWduhcattioisnComputer Algebra? . . . . . . . . . . . . . . . . . . . . . . 11
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ww.wot per 1.4 Design of a Computer Algebra System . . . . . . . . . . . . . . . . . 5
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C++ Dod distrib 22..45 RRaeatiloNnaulmNbuemrsbe.rs. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1270
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wit5/1 2.9 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Algebra on 07/2 22..1110 GPorlo(cid:127)ybnnoemriBalasse.s. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4375
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mputRSI 2.13 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
CoVE 2.14 Risch Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
NI 2.15 Commutativity and Non-Commutativity. . . . . . . . . . . . . . . . 63
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3.2 Reduce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.2 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.1 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.2 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4 Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
ix
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3.4.1 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.2 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.1 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.2 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.6 MuPAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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arti 3.6.2 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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wit5/1 4.13 Wrapper Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Algebra on 07/2 44..1154 ERrercourrsHioannd.li.ng. T.e.ch.n.iq.ue.s. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 112179
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mputRSI 4.17 Run-Time Type Identi(cid:12)cation . . . . . . . . . . . . . . . . . . . . . 129
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6 Standard Template Library 145
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2 Namespace Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3 Vector Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.4 List Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.5 Stack Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.6 Queue Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156