The Mathematica Book

1301 Pages·2003·8.678 MB·English

by  Stephen Wolfram

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Library of Congress Cataloging-in-Publication Data

Wolfram, Stephen, 1959 –
Mathematica book / Stephen Wolfram. — 5th ed.
p. cm.
Includes index.
ISBN 1–57955–022–3 (hardbound).
1. Mathematica (Computer file) 2. Mathematics—Data processing.
I. Title.
QA76.95.W65 2003
510 .285 5369—dc21 XX–XXXXX
CIP

Comments on this book will be welcomed at:
[email protected]
In publications that refer to the Mathematica
system, please cite this book as:
Stephen Wolfram, The Mathematica Book, 5th ed.
(Wolfram Media, 2003)
First and second editions published by Addison-Wesley Publishing Company
under the title Mathematica: A System for Doing Mathematics by Computer.
Third and fourth editions co-published by Wolfram Media and Cambridge University Press.
Published by:
ISBN 1–57955–022–3
Wolfram Media, Inc. web: www.wolfram–media.com; email: info@wolfram–media.com phone:
[phone]; fax: [phone]
mail: 100 Trade Center Drive, Champaign, IL 61820, USA
Copyright © 1988, 1991, 1996, 1999, 2003 by Wolfram Research, Inc.
All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in
any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior
written permission of the copyright holder.
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Wolfram Research is the holder of the copyright to the Mathematica software system described in this book,
including without limitation such aspects of the system as its code, structure, sequence, organization, “look
and feel”, programming language and compilation of command names. Use of the system unless pursuant
to the terms of a license granted by Wolfram Research or as otherwise authorized by law is an infringement
of the copyright.
The author, Wolfram Research, Inc. and Wolfram Media, Inc. make no representations, express or
implied, with respect to this documentation or the software it describes, including without limita-
tions, any implied warranties of merchantability or fitness for a particular purpose, all of which are
expressly disclaimed. Users should be aware that included in the terms and conditions under which
Wolfram Research is willing to license Mathematica is a provision that the author, Wolfram
Research, Wolfram Media, and their distribution licensees, distributors and dealers shall in no event
be liable for any indirect, incidental or consequential damages, and that liability for direct damages
shall be limited to the amount of the purchase price paid for Mathematica.
In addition to the foregoing, users should recognize that all complex software systems and their
documentation contain errors and omissions. The author, Wolfram Research and Wolfram Media
shall not be responsible under any circumstances for providing information on or corrections to
errors and omissions discovered at any time in this book or the software it describes, whether or not
they are aware of the errors or omissions. The author, Wolfram Research and Wolfram Media do not
recommend the use of the software described in this book for applications in which errors or omis-
sions could threaten life, injury or significant loss.
Mathematica, MathLink and MathSource are registered trademarks of Wolfram Research. J/Link, MathLM,
MathReader, .NET/Link, Notebooks and webMathematica are trademarks of Wolfram Research. All other
trademarks used are the property of their respective owners. Mathematica is not associated with Mathemat-
ica Policy Research, Inc. or MathTech, Inc.
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About the Author
Stephen Wolfram is the creator of Mathematica, and a well-known scientist. He is widely regarded as the most impor-
tant innovator in technical computing today, as well as one of the world's most original research scientists.
Born in London in 1959, he was educated at Eton, Oxford and Caltech. He published his first scientific paper at the age
of fifteen, and had received his PhD in theoretical physics from Caltech by the age of twenty. Wolfram's early scientific
work was mainly in high-energy physics, quantum field theory and cosmology, and included several now-classic
results. Having started to use computers in 1973, Wolfram rapidly became a leader in the emerging field of scientific
computing, and in 1979 he began the construction of SMP—the first modern computer algebra system—which he
released commercially in 1981.
In recognition of his early work in physics and computing, Wolfram became in 1981 the youngest recipient of a
MacArthur Prize Fellowship. Late in 1981, Wolfram then set out on an ambitious new direction in science: to develop
a general theory of complexity in nature. Wolfram's key idea was to use computer experiments to study the behavior of
simple computer programs known as cellular automata. And in 1982 he made the first in a series of startling discover-
ies about the origins of complexity. The publication of Wolfram's papers on cellular automata led to a major shift in
scientific thinking, and laid the groundwork for a new field of science that Wolfram named “complex systems
research”.
Through the mid-1980s, Wolfram continued his work on complexity, discovering a number of fundamental connec-
tions between computation and nature, and inventing such concepts as computational irreducibility. Wolfram's work
led to a wide range of applications—and provided the main scientific foundations for the popular movements known as
complexity theory and artificial life. Wolfram himself used his ideas to develop a new randomness generation system
and a new approach to computational fluid dynamics—both of which are now in widespread use.
Following his scientific work on complex systems research, Wolfram in 1986 founded the first research center and first
journal in the field. Then, after a highly successful career in academia—first at Caltech, then at the Institute for
Advanced Study in Princeton, and finally as Professor of Physics, Mathematics and Computer Science at the University
of Illinois—Wolfram launched Wolfram Research, Inc.
Wolfram began the development of Mathematica in late 1986. The first version of Mathematica was released on June
23, 1988, and was immediately hailed as a major advance in computing. In the years that followed, the popularity of
Mathematica grew rapidly, and Wolfram Research became established as a world leader in the software industry,
widely recognized for excellence in both technology and business. Wolfram has been president and CEO of Wolfram
Research since its inception, and continues to be personally responsible for the overall design of its core technology.
Following the release of Mathematica Version 2 in 1991, Wolfram began to divide his time between Mathematica
development and scientific research. Building on his work from the mid-1980s, and now with Mathematica as a tool,
Wolfram made a rapid succession of major new discoveries. By the mid-1990s his discoveries led him to develop a
fundamentally new conceptual framework, which he then spent the remainder of the 1990s applying not only to new
kinds of questions, but also to many existing foundational problems in physics, biology, computer science, mathematics
and several other fields.
After more than ten years of highly concentrated work, Wolfram finally described his achievements in his 1200-page
book A New Kind of Science. Released on May 14, 2002, the book was widely acclaimed and immediately became a
bestseller. Its publication has been seen as initiating a paradigm shift of historic importance in science.
In addition to leading Wolfram Research to break new ground with innovative technology, Wolfram is now developing
a series of research and educational initiatives in the science he has created.
Other books by Stephen Wolfram:
è Cellular Automata and Complexity: Collected Papers (1993)
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è A New Kind of Science (2002)
Author's website:
www.stephenwolfram.com
Author's address:
email: [email protected]
mail: c/o Wolfram Research, Inc.
100 Trade Center Drive
Champaign, IL 61820, USA
For comments on this book or Mathematica
send email to [email protected]
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About Mathematica
Mathematica is the world's only fully integrated environment for technical computing. First released in 1988, it has had
a profound effect on the way computers are used in many technical and other fields.
It is often said that the release of Mathematica marked the beginning of modern technical computing. Ever since the
1960s individual packages had existed for specific numerical, algebraic, graphical and other tasks. But the visionary
concept of Mathematica was to create once and for all a single system that could handle all the various aspects of
technical computing in a coherent and unified way. The key intellectual advance that made this possible was the
invention of a new kind of symbolic computer language that could for the first time manipulate the very wide range of
objects involved in technical computing using only a fairly small number of basic primitives.
When Mathematica Version 1 was released, the New York Times wrote that “the importance of the program cannot be
overlooked”, and Business Week later ranked Mathematica among the ten most important new products of the year.
Mathematica was also hailed in the technical community as a major intellectual and practical revolution.
At first, Mathematica's impact was felt mainly in the physical sciences, engineering and mathematics. But over the
years, Mathematica has become important in a remarkably wide range of fields. Mathematica is used today throughout
the sciences—physical, biological, social and other—and counts many of the world's foremost scientists among its
enthusiastic supporters. It has played a crucial role in many important discoveries, and has been the basis for thousands
of technical papers. In engineering, Mathematica has become a standard tool for both development and production, and
by now many of the world's important new products rely at one stage or another in their design on Mathematica. In
commerce, Mathematica has played a significant role in the growth of sophisticated financial modeling, as well as
being widely used in many kinds of general planning and analysis. Mathematica has also emerged as an important tool
in computer science and software development: its language component is widely used as a research, prototyping and
interface environment.
The largest part of Mathematica's user community consists of technical professionals. But Mathematica is also heavily
used in education, and there are now many hundreds of courses—from high school to graduate school—based on it. In
addition, with the availability of student versions, Mathematica has become an important tool for both technical and
non-technical students around the world.
The diversity of Mathematica's user base is striking. It spans all continents, ages from below ten up, and includes for
example artists, composers, linguists and lawyers. There are also many hobbyists from all walks of life who use
Mathematica to further their interests in science, mathematics and computing.
Ever since Mathematica was first released, its user base has grown steadily, and by now the total number of users is
above a million. Mathematica has become a standard in a great many organizations, and it is used today in all of the
Fortune 50 companies, all of the 15 major departments of the U.S. government, and all of the 50 largest universities in
the world.
At a technical level, Mathematica is widely regarded as a major feat of software engineering. It is one of the largest
single application programs ever developed, and it contains a vast array of novel algorithms and important technical
innovations. Among its core innovations are its interconnected algorithm knowledgebase, and its concepts of symbolic
programming and of document-centered interfaces.
The development of Mathematica has been carried out at Wolfram Research by a world-class team led by Stephen
Wolfram. The success of Mathematica has fueled the continuing growth of Wolfram Research, and has allowed a large
community of independent Mathematica-related businesses to develop. There are today well over a hundred special-
ized commercial packages available for Mathematica, as well as more than three hundred books devoted to the system.
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New in Version 5
Mathematica Version 5 introduces important extensions to the Mathematica system, especially in scope and scalability
of numeric and symbolic computation. Building on the core language and extensive algorithm knowledgebase of
Mathematica, Version 5 introduces a new generation of advanced algorithms for a wide range of numeric and symbolic
operations.
Numerical computation
† Major optimization of dense numerical linear algebra.
† New optimized sparse numerical linear algebra.
† Support for optimized arbitrary-precision linear algebra.
† Generalized eigenvalues and singular value decomposition.
† LinearSolveFunction for repeated linear-system solving.
† p norms for vectors and matrices.
† Built-in MatrixRank for exact and approximate matrices.
† Support for large-scale linear programming, with interior point methods.
† New methods and array variable support in FindRoot and FindMinimum.
† FindFit for full nonlinear curve fitting.
† Constrained global optimization with NMinimize.
† Support for n-dimensional PDEs in NDSolve.
† Support for differential-algebraic equations in NDSolve.
† Support for vector and array-valued functions in NDSolve.
† Highly extensive collection of automatically-accessible algorithms in NDSolve.
† Finer precision and accuracy control for arbitrary-precision numbers.
† Higher-efficiency big number arithmetic, including processor-specific optimization.
† Enhanced algorithms for number theoretical operations including GCD and FactorInteger.
† Direct support for high-performance basic statistics functions.
Symbolic computation
† Solutions to mixed systems of equations and inequalities in Reduce.
† Complete solving of polynomial systems over real or complex numbers.
† Solving large classes of Diophantine equations.
† ForAll and Exists quantifiers and quantifier elimination.
† Representation of discrete and continuous algebraic and transcendental solution sets.
† FindInstance for finding instances of solutions over different domains.
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† Exact constrained minimization over real and integer domains.
† Integrated support for assumptions using Assuming and Refine.
† RSolve for solving recurrence equations.
† Support for nonlinear, partial and q difference equations and systems.
† Full solutions to systems of rational ordinary differential equations.
† Support for differential-algebraic equations.
† CoefficientArrays for converting systems of equations to tensors.
Programming and Core System
† Integrated language support for sparse arrays.
† New list programming with Sow and Reap.
† EvaluationMonitor and StepMonitor for algorithm monitoring.
† Enhanced timing measurement, including AbsoluteTiming.
† Major performance enhancements for MathLink.
† Optimization for 64-bit operating systems and architectures.
† Support for computations in full 64-bit address spaces.
Interfaces
† Support for more than 50 import and export formats.
† High efficiency import and export of tabular data.
† PNG, SVG and DICOM graphics and imaging formats.
† Import and export of sparse matrix formats.
† MPS linear programming format.
† Cascading style sheets and XHTML for notebook exporting.
† Preview version of .NET/Link for integration with .NET.
Notebook Interface
† Enhanced Help Browser design.
† Automatic copy/paste switching for Windows.
† Enhanced support for slide show presentation.
† AuthorTools support for notebook diffs.
Standard Add-on Packages
† Statistical plots and graphics.
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† Algebraic number fields.
New in Versions 4.1 and 4.2
† Enhanced pattern matching of sequence objects.
† Enhanced optimizer for built-in Mathematica compiler.
† Enhanced continued fraction computation.
† Greatly enhanced DSolve.
† Additional TraditionalForm formats.
† Efficiency increases for multivariate polynomial operations.
† Support for import and export of DXF, STL, FITS and STDS data formats.
† Full support for CSV format import and export.
† Support for UTF character encodings.
† Extensive support for XML, including SymbolicXML subsystem and NotebookML.
† Native support for evaluation and formatting of Nand and Nor.
† High-efficiency CellularAutomaton function.
† J/Link MathLink-based Java capabilities.
† MathMLForm and extended MathML support.
† Extended simplification of Floor, Erf, ProductLog and related functions.
† Integration over regions defined by inequalities.
† Integration of piecewise functions.
† Standard package for visualization of regions defined by inequalities.
† ANOVA standard add-on package.
† Enhanced Combinatorica add-on package.
† AuthorTools notebook authoring environment.
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The Role of This Book
The Scope of the Book
This book is intended to be a complete introduction to Mathematica. It describes essentially all the capabilities of
Mathematica, and assumes no prior knowledge of the system.
In most uses of Mathematica, you will need to know only a small part of the system. This book is organized to make it
easy for you to learn the part you need for a particular calculation. In many cases, for example, you may be able to set
up your calculation simply by adapting some appropriate examples from the book.
You should understand, however, that the examples in this book are chosen primarily for their simplicity, rather than to
correspond to realistic calculations in particular application areas.
There are many other publications that discuss Mathematica from the viewpoint of particular classes of applications. In
some cases, you may find it better to read one of these publications first, and read this book only when you need a more
general perspective on Mathematica.
Mathematica is a system built on a fairly small set of very powerful principles. This book describes those principles,
but by no means spells out all of their implications. In particular, while the book describes the elements that go into
Mathematica programs, it does not give detailed examples of complete programs. For those, you should look at other
publications.
The Mathematica System Described in the Book
This book describes the standard Mathematica kernel, as it exists on all computers that run Mathematica. Most major
supported features of the kernel in Mathematica Version 5 are covered in this book. Many of the important features of
the front end are also discussed.
Mathematica is an open software system that can be customized in a wide variety of ways. It is important to realize that
this book covers only the full basic Mathematica system. If your system is customized in some way, then it may behave
differently from what is described in the book.
The most common form of customization is the addition of various Mathematica function definitions. These may come,
for example, from loading a Mathematica package. Sometimes the definitions may actually modify the behavior of
functions described in this book. In other cases, the definitions may simply add a collection of new functions that are
not described in the book. In certain applications, it may be primarily these new functions that you use, rather than the
standard ones described in the book.
This book describes what to do when you interact directly with the standard Mathematica kernel and notebook front
end. Sometimes, however, you may not be using the standard Mathematica system directly. Instead, Mathematica may
be an embedded component of another system that you are using. This system may for example call on Mathematica
only for certain computations, and may hide the details of those computations from you. Most of what is in this book
will only be useful if you can give explicit input to Mathematica. If all of your input is substantially modified by the
system you are using, then you must rely on the documentation for that system.
Additional Mathematica Documentation
For all standard versions of Mathematica, the following is available in printed form, and can be ordered from Wolfram
Research:
©1988-2003 Wolfram Research, Inc. All rights reserved.