Mathematical Computation with Maple V: Ideas and Applications: Proceedings of the Maple Summer Workshop and Symposium, University of Michigan, Ann Arbor, June 28–30, 1993

201 Pages·1993·7.28 MB·English

by  Yutaka Abe (auth.)

Mathematical Computation
with Maple V:
Ideas and Applications
Mathematicai Computation
with Maple V:
Ideas and Applications
Proceedings of the Maple Summer
Workshop and Symposium,
University of Michigan,
Ann Arbor, June 28-30, 1993
Thomas Lee
Editor
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
ThomasLee
Waterloo Maple Software
Waterloo, Ontario
CANADA N2L 5J2
Printed on acid-free paper
© Springer Science+Business Media New York 1993
Originally published by Birkhauser Boston in 1993
Maple and Maple V are registered trademarks ofWaterloo Maple Software.
Copyright is not claimed for works of U.S. Govemment employees.
AII rights reserved. No part of this publication may be reproduced, stored in a retrieval
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Springer Science+Business Media, LLC for Iibraries and other users registered with
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be addressed directly to Springer Science+Business Media, LLC.
ISBN 978-1-4612-6720-1 ISBN 978-1-4612-0351-3 (eBook)
DOI 10.1007/978-1-4612-0351-3
Camera-ready text prepared by the Authors.
9 8 7 6 5 432 1
Contents
Preface . . . . . . . . . . . . . . . . . . ... vii
I. Maple V in Education 1
Introductory Quantum Mechanics Using Maple V
Yutaka Abe ............... . 3
Combinatorial Objects and their Generating Functions:
A Maple Class Room Environment
John S. Devitt ................. . . ....... 20
Experiences with Maple in Engineering Education at the
University of Waterloo
Mustafa Fofana, Ian LeGrow, Stephen Carr ......... 27
On Integrating Computers into the Physics Curriculum
Ronald L. Greene . . . . . . . ................... 34
Using Maple and the Calculus Reform Material
in the Calculus Sequence
David C. Royster . . . . . . . . . . . . . . . . . . . . . . . 43
Interactive Mathematics Texts: Ideas for Developers
Carol Scheftic ..... . . . . . . . . . . . . . . . . . . . . . . 51
II. Maple V in Mathematics . . . . . . . . . . . . 65
Truncation and Variance in Scale Mixtures
William C. Bauldry and Jaimie L. Hebert ............... 67
Working with Large Matrices in Maple
Reid Pinchback ......... . ........... 77
Using Maple for Asymptotic Convergence Analysis
Noah H. Rhee ....... . . . . . . . . . . . . . . . . . 87
An Algorithm to Compute Floating Point Groebner Bases
Kiyoshi Shimyanagi ........................ 95
III. Maple V in Science and Engineering
Part A: Modeling and Simulation . . . . . . 107
The Use of Maple for Multibody Systems Modeling and Simulation
P. Capolsini . . . . . . . . . . . . . . . . . . . . . . . 109
Sensitivity Analysis of Nonlinear Physical Systems Using Maple
Stephen Carr, Gordon J. Savage . . . . . . . . . . . . . . 118
Exact Calculation of the Kaplan-Meier Bias using Maple Software
Brenda Gillespie, Justine Uro ................ 128
Rotational Energy Dispersions for van der Waals Molecular Clusters
Lawrence L. Lohr, Carl H. Huben ............... 137
Symbolic Computation in Computable General Equilibrium Modeling
Trien T. Nguyen . . . . . . . . . . . . . . . . . . . . . . . 144
Calculation of the State Transition Matrix for Linear Time Varying Systems
J. Watkins, S. Yurkovich ..................... 157
IV. Maple V in Science and Engineering
Part B: Design . . . . . . . . . . . . . . 167
Algebraic Computer Aided-Design with Maple V 2
C. T. Lim, M. T. Ensz, M.A. Ganter, D. W. Storti . 169
The Role of a Symbolic Programming Language in Hardware Verification:
The Case of Maple
Farhad Mavaddat . . . . . . . . . . . . . . . . . . . . . . . . . 176
A Symbolic CSG System Written in Maple V
Darren Thompson, Tom Trias, Laurence LeJJ 188
PREFACE
Developments in both computer hardware and Perhaps the greatest impact has been felt by the
software over the decades have fundamentally education community. Today, it is nearly
changed the way people solve problems. impossible to find a college or university that has
Technical professionals have greatly benefited not introduced mathematical computation in
from new tools and techniques that have allowed some form, into the curriculum. Students now
them to be more efficient, accurate, and creative have regular access to the amount of
in their work. computational power that were available to a very
exclusive set of researchers five years ago. This
Maple V and the new generation of mathematical has produced tremendous pedagogical
computation systems have the potential of challenges and opportunities.
having the same kind of revolutionary impact as
high-level general purpose programming Comparisons to the calculator revolution of the
languages (e.g. FORTRAN, BASIC, C), 70's are inescapable. Calculators have
application software (e.g. spreadsheets, extended the average person's ability to solve
Computer Aided Design - CAD), and even common problems more efficiently, and
calculators have had. Maple V has amplified our arguably, in better ways. Today, one needs at
mathematical abilities: we can solve more least a calculator to deal with standard problems
problems more accurately, and more often. In in life - budgets, mortgages, gas mileage, etc.
specific disciplines, this amplification has taken For business people or professionals, the
excitingly different forms. spreadsheet is quickly becoming the minimum
level of electronic assistance for solving modern
For mathematicians, computer algebra systems problems. Many look towards the next decade
has spawned entire new research and with great anticipation as what is now considered
application directions. In addition, Maple V has advanced mathematical and computing
allowed mathematicians to approach existing techniques (like Maple V) begin influencing the
problems in ways that were not conceivable 20 work of businessmen, social scientists, and
years ago. others who have traditionally not exploited the
tremendous potential of this particular dimension
For scientists and engineers, Maple V of the Information Revolution.
represents a bridge between theory and
practice. Techniques that were traditionally
considered too complex for application are now The Maple Summer Workshop and
beginning to be used by a wide range of Symposium - MSWS '93
scientists and engineers in both academia and
industry. Indeed the concept of "scientific The primary goal of MSWS '93 is to bring
computation" has evolved from a relatively interested people together and to generate
narrow field encompassing primarily numerical ideas and strategies to promote the effective
methods to a much broader field that includes, use of Maple V and other modern computation
numerical methods, algebraic and analytical techniques. This year, the conference was held
methods, and graphics. on June 28-30 at the University of Michigan in
vii
Ann Arbor. This volume summarizes the paper
sessions of the conference. The paper
presentations were part of a varied conference
program designed to encourage interaction
among users from a wide variety of disciplines.
One of the most encouraging aspects of the
paper sessions was the impressive breadth. It is
a clear demonstration of the influence that Maple
V is having on the research and academic
community. As important as the specific
technical topics are, it is equally important to note
the differences in basic approaches to Maple V
based problem solving. The collection includes
a wide range of perspectives - from
-computerization- of classical theory to facilitate
more efficient and accurate problem-solving, to
the development of complete Maple V-based
systems for the automatic solution of problems in
ways that are not possible with conventional
computing systems. If this volume is any
indication of the future of mathematical
computing, then it is very clear that we will
witness an even more impressive proliferation of
computer algebra in the years to come.
MSWS '93 and this volume could not have been
achieved without the invaluable assistance of
many dedicated and enthusiastic people. The
Editor would like to express his gratitude to the
reviewers who served to ensure overall quality in
the papers. Furthermore, the assistance of
Paola O'Alessandro, Karin Turner, Jeff Watling,
Lee Liming, Melanie Mclnness, Ann Kostant and
Edwin Beschler is greatly appreciated.
T. Lee
Waterloo, canada
viii
I
MAPLE V IN EDUCATION
INTRODUCTORY QUANTUM MECHANICS USING
MAPLE V
YutakaAbe
Quantum Instrumentation Laboratory, Hokkaido University, Sapporo, Japan
Abstract 1. INTRODUCTION
This article discusses a set of There is much talk about
Maple programs developed to computer-aided instruction in
reinforce understanding of basic the field of physics and engin
concepts presented in an eering;' but it seems that the
introductory quantum mechanics computer as a learning device
course. It was designed so that has provided little help for
the solutions of the problems students trying to grasp the
were analytically tractable as basic ideas in these fields.
well as numerically obtainable. Of course, various packages for
We note that there exist various numerical computations are now
approaches to a simple quantum available [1], and it is very
mechanical problem. For example, easy for instructors to introduce
for the problem of bound-states these packages in their classes.
with a one-dimensional potential, In my experience, this type of
the transfer matrix method, the instruction has never succeeded
Laplace transform method, and the in extending a student's ability
Feynman path integral method for a further understanding
can equally be applied. We of physics. There are always
believe that this type of program certain barriers between a lec
is extremely useful for igniting turer's analytical description
the student's interest and widening on a blackboard and the output
his or her viewpoint. of numerical computations by a
Several examples, such as the computer. Sometimes the result
solution of various one-dimen in a classroom has been destruc
sional potential problems and the tive, in that the students under
solution of the one-dimensional stand less physics and get fewer
Schroedinger wave equation, are computational skills. Actually,
discussed in order to indicate numerical computations cannot
how constructive reinforcement teach students real physics.
is established. On the other hand, it is true
that there are a few problems in
physics that can be solved analy
Mathematical Computation with Maple Yo
Ideas and App/icalions tically. For example, if you
10m Lee, Editor
wish to explore the behavior of an
©1993 Birkhiiuser Boston
3
electron wave function under a general wave function by matching
certain general potential function, the wave function and its derivative
you may suddenly encounter severe at the discontinuities of the given
difficulty if you stick to some potential function. The transfer
analytical solution. Numerical matrix method is one commonly
methods can treat a much wider used method.
variety of interesting problems Let us consider the Schrodinger
than can be handled by analytical equation for the potential function
methods. in Fig. 1. The Schrodinger equation
The motivation of this paper for regions I, II, III are:
is to explore new methods of
instruction balanced between
analytical and numerical approaches. d2¢ 2
--+a x=O (x<O and x>a)
The author selected Maple V for dx2
(2.1.a),
symbolic (analytical) and numerical
calculations [2] because of its
d2¢ 2
excellent interface with users, --fJ x=O (O<x<a), V>E
distinguished power for mathematical dx2
(2.1.b),
operations, and suitability for
students exploring creative pro
d2¢ 2
gramming with small computers. In --+Il x=O (O<x <a), V<E
dx2
the following sections, we treat
(2.1.c)
several elementary quantum mech
where
anical problems with this approach.
a 2=2mE;1)2, fJ 2 =2m (V -E)/fl 2
112 =2m(E-V)/fJ2
2. ONE-DIMENSIONAL FINITE
(2.2) .
POTENTIAL BARRIER PROBLEMS
We take the wave functions in each
region as:
The problems of one-dimensional
finite potential barriers and
wells are treated universally in ¢I =exp(ia x )+r exp(-iax), x<O
introductory quantum mechanics
textbooks. There are a number of (2.3.a)
ways to attack these problems. In
¢II =A exI( -fJx)+B exp(f3 x), O<x<a
the simple type of potential, the
overall wave function is constructed
(2.3.b)
out of pieces having the form of a
4