Mathematica Navigator ®

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Mathematica ^® Navigator

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Mathematica ^® Navigator

Mathematics, Statistics, and Graphics

THIRD EDITION

Heikki Ruskeepää

Department of Mathematics University of Turku, Finland

AMSTERDAM•BOSTON•HEIDELBERG•LONDON NEW YORK•OXFORD•PARIS•SAN DIEGO SAN FRANCISCO•SINGAPORE•SYDNEY•TOKYO

Academic Press is an imprint of Elsevier

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The book is produced from PDF files prepared by the author withMathematica . Academic Press is an imprint of Elsevier

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then “Copyright and Permission” and then “Obtaining Permissions.”

Library of Congress Cataloging-in-Publication Data Ruskeepää, Heikki.

Mathematic navigator : mathematics, statistics, and graphics / Heikki Ruskeepää. – 3rd ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-12-374164-6 (pbk. : alk. paper) 1. Mathematics–Data processing. 2. Mathematica (Computer file) I. Title.

QA76.95.R87 2009 510.285'5–dc22

2008044637 British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

ISBN: 978-0-12-374164-6

For information on all Academic Press publications visit our Web site at www.elsevierdirect.com

Printed in the United States of America

09 10 11 9 8 7 6 5 4 3 2 1

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To Marjatta

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Preface

What is the difference between an applied mathematician and a pure mathematician?

An applied mathematician has a solution for every problem, while a pure mathematician has a problem for every solution.

Welcome

The goals of this book, the third edition ofMathematica Navigator: Mathematics, Statistics, Graphics, and Programming, are as follows:

•to introduce the reader toMathematica; and

•to emphasize mathematics (especially methods of applied mathematics), statistics, graphics, programming, and writing mathematical documents.

Accordingly, we navigate the reader throughMathematica and give an overall introduction. Often we slow down somewhat when an important or interesting topic of mathematics or statistics is encountered to investigate it in more detail. We then often use both graphics and symbolic and numerical methods.

Here and there we write small programs to make the use of some procedures easier. One chapter is devoted toMathematica as an advanced environment of writing mathematical documents.

The online version of the book, which can be installed from the enclosed CD-ROM, makes the material easily available when working withMathematica.

Changes in this third edition are numerous and are explained later in the Preface. The current edition is based on Mathematica 6. On the CD-ROM, there is material that describes the new properties of Mathematica 7.

‡ Readership

The book may be useful in the following situations:

•for courses teachingMathematica;

•for several mathematical and statistical courses (given in, for example, mathematics, engineering, physics, and statistics); and

•for self-study.

Indeed, the book may serve as a tutorial and as a reference or handbook ofMathematica, and it may also be useful as a companion in many mathematical and statistical courses, including the following:

differential and integral calculus • linear algebra • optimization • differential, partial differential, and difference equations • engineering mathematics • mathematical methods of physics • mathematical modeling • numerical methods • probability • stochastic processes • statistics • regression analysis • Bayesian statistics

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‡ Previous Knowledge

No previous knowledge ofMathematica is assumed. On the other hand, we assume some knowledge of various topics in pure and applied mathematics. We study, for example, partial differential equations and statistics without giving detailed introductions to these topics. If you are not acquainted with a topic, you can simply skip the chapter or section of the book considering that topic.

Also, to understand the numerical algorithms, it is useful if the reader has some knowledge about the simplest numerical methods. Often we introduce briefly the basic ideas of a method (or they may become clear from the examples or other material presented), but usually we do not derive the methods.

If a topic is unfamiliar to you, consult a textbook about numerical analysis, such as Skeel and Keiper (2001).

‡ Recommendations

If you are a newcomer toMathematica, then Chapter 1, Starting, is mandatory, and Chapter 2, Sightsee- ing, is strongly recommended. You can also browse Chapter 3, Notebooks, and perhaps also Chapter 4, Files, so that you know where to go when you encounter the topics of these chapters. After that you can proceed more freely. However, read Section 13.1, “Basic Techniques,” because it contains some very common concepts used constantly for expressions.

If you have some previous knowledge ofMathematica, you can probably go directly to the chapter or section you are interested in, with the risk, however, of having to go back to study some background material. Again, be sure to read Section 13.1.

The next two chapters consider files, particularly files created by Mathematica, which are called notebooks. We show howMathematica can be used to write mathematical documents. We also explain how to load packages, how to export and import data and graphics into and fromMathematica, and how to manage memory and computing time. You may skip these two chapters until you need them.

Then we go on to graphics. One of the finest aspects ofMathematica is its high-quality graphics, and one of the strongest motivations for studying Mathematica is to learn to illustrate mathematics with figures. We consider separately graphics for functions and graphics for data. In addition, we have chapters about graphics primitives and graphics options.

New inMathematica 6 are the built-in data sources, covering topics such as chemistry, astronomy, particles, countries, cities, finance, polyhedrons, graphs, words, and colors.

The main new topic inMathematica 6 is dynamics. This allows us to easily build interactive interfaces.

The user of such an interface can choose some parameters or other options and the output will be changed dynamically, in real time. This helps in studying various models and phenomena.

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Then we study various types of expressions, from numbers to strings, mathematical expressions, lists, tables, and patterns.

We have two chapters relating to programming. The first studies functions and the next various styles of programming. Four styles are considered: procedural, functional, rule-based, and recursive.

‡ Mathematics and Statistics

In the remaining 12 chapters, we study different areas of pure and applied mathematics and statistics.

The mathematical chapters can be divided into four classes, with each class containing chapters of more or less related topics. Descriptions of these classes follow.

Topics of traditional differential and integral calculus include derivatives, Taylor series, limits, integrals, sums, and transforms.

Then we consider vectors andmatrices; linear, polynomial, and transcendentalequations; and global, local, and classicaloptimization.

In interpolation we have the usual interpolating polynomial, a piecewise-calculated interpolating polynomial, and splines. Inapproximation we distinguish the approximation of data and functions. For the former, we can use the linear or nonlinear least-squares method, whereas for the latter we have, for example, minimax approximation.

Mathematica solvesdifferential equations both symbolically and numerically. We can solve first- and higher-order equations, systems of equations, and initial and boundary value problems. For partial differential equations, we show how some equations can be solved symbolically, how to handle series solutions, and how to numerically solve problems with the method of lines or with the finite difference method. Then we consider difference equations. For linear difference equations, we can possibly find a solution in a closed form, but most nonlinear difference equations have to be investigated in other ways, such as studying trajectories and forming bifurcation diagrams.

Lastly, we study probability and statistics. Mathematica contains information about most of the well-known probability distributions. Simulation of various random phenomena (e.g., stochastic processes) is done well with random numbers. Statistical topics include descriptive statistics, frequen- cies, confidence intervals, hypothesis testing, regression, smoothing, and Bayesian statistics.

Special Aspects

The book explains a substantial portion of the topics ofMathematica. However, some topics are emphasized, some are given less emphasis, and some are even excluded. We describe these special aspects of the book here.

‡ Breadth

We have had the goal of studying important topics in some breadth and depth. This may mean detailed explanations, clarifying examples, programs, and applications. It may also mean introducing topics for which there is little or no built-in material.

The headings of the chapters give a list of topics that are emphasized in this book and that are explained in some breadth. However, some emphasized topics cannot be identified from the chapter headings. One of them is numerical methods; they are used in every mathematical chapter. Another is methods relating to data. Indeed, we use several real-life and artificial data sets in chapters about data, graphics for data, approximation, differential and difference equations, probability, and statistics.

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‡ Depth

To give an impression of the depth of various topics, we next describe some special topics in various chapters of the book.

•Chapter 3,Notebooks: An introduction toMathematica as an environment for preparing technical documents; writing mathematical formulas

•Chapter 5,Graphics for Functions: Stereographic figures; graphics for four-dimensional functions

•Chapter 8,Graphics for Data: Visualizations of several real-life data; dot plots; statistical plots

•Chapter 18,Programs: Four styles of programming (procedural, functional, rule-based, and recursive); emphasis on functional programming; many examples of programs

•Chapter 22,Equations: Iterative methods of solving linear equations; programs for nonlinear equations

•Chapter 23,Optimization: A program for numerical minimization; a program for classical optimization with equality and inequality constraints; dynamic programming

•Chapter 25,Approximation: Graphical diagnostics of least-squares fits

•Chapter 26,Differential Equations: Analyzing and visualizing solutions of systems of nonlinear differential equations; study of a predator-prey model, a competing species model, and the Lorenz model; numerical solution of linear and nonlinear boundary value problems; estimation of nonlinear differential equations from data; solving integral equations

•Chapter 27,Partial Differential Equations: Series solutions for partial differential equations; solving parabolic and hyperbolic problems by the method of lines;solving elliptic problems by the finite difference method

•Chapter 28,Difference Equations: The logistic model as an example of nonlinear difference equations; bifurcation diagrams, periodic points, Lyapunov exponents; a discrete-time predator-prey model as an example of a system of nonlinear difference equations; estimation of nonlinear difference equations from data; fractal images; Lindenmayer systems

•Chapter 29,Probability: Simulation of several stochastic processes

•Chapter 30,Statistics: Visualizing confidence intervals and types of errors in statistical tests;

confidence intervals and tests for probabilities; local regression; Bayesian statistics; Gibbs sampling;

Markov chain Monte Carlo

‡ Programs

Mathematica has a large number of ready-to-use commands for symbolic and numerical calculations and for graphics. Nevertheless, in this book we also present approximately 130 of our own programs.

Indeed, programming is one of the strongest points ofMathematica. It is often amazing how concisely and efficiently we can write a program even for a somewhat complex problem. We think that our own programs can be of some value, despite the fact that they are not so fine and powerful asMathematica’s built-in commands. We have included our own programs for the following reasons:

1.A self-made implementation shows clearly how the algorithm works. You know (or should know) exactly what you are doing when you use your own implementation. The ready-made commands are often like black (or gray) boxes because we do not know much about the methods.

2.Writing our own implementations teaches us programming. We present short programs throughout the book (especially in the mathematical chapters). In this way, we hope that you will become steadily more familiar with programming and that you are encouraged to practice program writing.

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3.A self-made implementation can be pedagogically worthwhile. For example, we implement Euler’s method for differential equations. It has almost no practical value, but as the simplest numerical method for initial value problems, it has a certain pedagogical value. Also, programming a simple method first may help us to tackle a more demanding method later.

‡ Other Special Aspects

We have integrated the so-called packages tightly into the material covered in this book. Instead of presenting a separate chapter about packages, each package is explained in its proper context.

We have tried to make the structure of the book such that finding a topic is easy. Usually a topic is considered in one and only one chapter or section so that you need not search in several places to find the whole story. Each numerical routine is also presented in the proper context after the corresponding symbolic methods. This helps you to find material for solving a given problem: It is usually best to try a symbolic method first and, if this fails, to then resort to a numerical method.

Some topics of a “pure” nature, such as finite fields, quaternions, combinatorics, computational geometry, and graph theory, are not considered in this book;Mathematica has packages for these topics.

Commands for box and notebook manipulation are treated only briefly. We do not considerMathLink (a part of Mathematica that enables interaction between Mathematica and external programs), J/Link (a product that integratesMathematica and Java),XML (a metamarkup language for the World Wide Web), orMathML (an XML-based markup language for representing mathematics). Also, we do not consider any of the many other Mathematica-related products, such as webMathematica, gridMathematica, CalculationCenter, or theApplications Library packages.

Mathematica 6

‡ Introduction

Mathematica 6 contains a huge amount of new functionality. The following is a part of an on-line document:

Mathematica 6.0 fundamentally redefinesMathematica and introduces a major new paradigm for computation. Building on Mathematica’s time-tested core symbolic architecture, version 6.0 adds nearly a thousand new functions~almost doubling the total number of functions in the system~

dramatically increasing both the breadth and depth ofMathematica’s capabilities, as well as introducing hundreds of major original algorithms, and perhaps a thousand new ideas, large and small.

To study the new features, see the following on-line documentation (the use of the Documentation Center is explained in Section 1.4.2, p. 17):

•Help@Startup Palette, theWhat’s New in 6link to Wolfram’s website

•Help@Documentation Center, theNew in 6 links in the home page

•Help@Documentation Center, theguideêSummaryOfNewFeaturesIn60 document

•Help@Documentation Center, theguideêNewIn60AlphabeticalListing document

•Help@Function Navigator, theNew In 6item

If you are a new user ofMathematica and would like to study the basics ofMathematica6, see the following documents:

• Help@Startup Palette: theFirst Five Minutes with Mathematica button

•Help@Virtual Book: theIntroductionitem

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‡ New Properties of Version 6

Because the new features are numerous, we do not list them all here. However, we mention some of the most remarkable new commands and features, classified according to the chapters of the book:

•Chapter 1, Starting: documentation is on-line in the form of Documentation Center, Function Navigator and Virtual Book (we do not have a printed manual); documentation is automatically updated via the Internet; writingMathematica inputs is helped by syntax coloring

•Chapter 3, Notebooks:^Style,^Text,^Hyperlink

•Chapter 4, Files: commands of many packages are now built-in; the remaining packages are rebuilt;

look at Compatibility/guide/StandardPackageCompatibilityGuide in the Documentation Center to obtain information about how to replace the functionality of the old packages

•Chapter 5, Graphics for Functions:GraphicsRow,GraphicsGrid,^Tooltip; graphics is handled like other expressions; the default font in graphics is Times instead of Courier; 3D graphics is adaptive;

contours in contour plots have tooltips; density plots, by default, do not have meshes; 2D graphics can be interactively drawn and edited; 3D graphics can be interactively manipulated (e.g., rotated);

for animation, use^Manipulate or^Animate

•Chapter 6, Graphics Primitives:Ârrow,Ôpacity,Înset

•Chapter 7, Graphics Options:^Directive,^BaseStyle,^Filling; the default value ofAspectRatio in Graphics andParametricPlot is^Automatic instead of1/GoldenRatio

•Chapter 8, Graphics for Data:ListLinePlot,GraphPlot; plotting of several data sets

•Chapter 9, Data:ElementData,CountryData,PolyhedronData, etc.

•Chapter 10, Manipulations:^Manipulate (for creating interactive dynamic interfaces)

•Chapter 11, Dynamics:^Dynamic (for advanced dynamic interfaces),^MenuView,^TabView, etc.

•Chapter 15, Tables:^Grid,^Row,^Column

•Chapter 16, Patterns:DictionaryLookup

•Chapter 21, Matrices:^Accumulate,PositiveDefiniteMatrixQ

•Chapter 23, Optimization:FindShortestTour

•Chapter 29, Probability:^RandomReal,RandomInteger,RandomChoice,RandomSample

•Chapter 30, Statistics:^Tally,^BinCounts,FindClusters

In my opinion, the most impressive new commands in version 6 are ^Manipulate, ^Dynamic, GraphPlot, and^Grid.

Note that many familiar commands, such as^NIntegrate or^NDSolve, have also been enhanced in version 6.

In the forthcoming chapters, we mark with (Ÿ6) the properties and commands of Mathematica available for the first time in version 6.

‡ Obsolete Properties in Version 6

Version 6 makes obsolete some old commands and features, especially in graphics. First, here are some changes that relate to the display and arrangement of graphics:

•To prevent the display of graphics, end the plotting command with ^; instead of using the DisplayFunctionoption.

•In programs, enclose a plotting command with^Print if that command is not the last command of the program and you would like the program to show that plot.

•GraphicsArray is obsolete. To show, for example, two plots^p1 and^p2 side by side, use one of the following ways: ^{{p1, p2}},Row[{p1, p2}], or GraphicsRow[{p1, p2}]. UseGraphicsGrid for arrays of plots.

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•To show two plots side by side, you can also simply give a list of plotting commands^{Plot[…], Plot[…]}.

•To show two plots on top of each other, simply write Show[Plot[…], Plot[…]]; the DisplayFunctionoption is no longer needed.

•^Graphics and ^Graphics3D no longer need ^Show to display the graphics. Thus, write Graphics[{…}] instead ofShow[Graphics[{…}].

•UseInset[gr, pos] instead ofRectangle[{x1, y1}, {x2, y2}, gr]. Some changes that relate to plotting of data are as follows:

•To plot data by connecting the points with lines, use ListLinePlot[data] instead of ListPlot[data, PlotJoinedØTrue].

•To plot data by points and connecting lines, use ListLinePlot[data, MeshØAll] instead of ListPlot[data, PlotJoinedØTrue, EpilogØ{PointSize[s], Map[point, data]}].

•To plot data by points and vertical lines, use ListPlot[data, Filling Ø Axis] instead of resorting to^Prolog or^Epilog.

•To plot several data sets, useListPlot[{data1, data2, … }] orListLinePlot[{data1, data2, … }] instead of resorting toMultipleListPlot in a package.

•To plot several points, simply writePoint[points] instead ofMap[Point, points]. Here are some changes that relate to styles and options of graphics:

•Use^Style instead of^StyleForm.

•Use the^BaseStyle option instead of the^TextStyle option or the^$TextStyle global constant.

•Use theMaxRecursion option instead of thePlotDivision option.

•Use the^DataRange option instead of the^MeshRange option.

•Use the^Filling option instead of the^FilledPlot command.

Some other changes are as follows:

•Use RandomReal[…], RandomInteger[…], and RandomComplex[…] instead of Random[Real, …], etc.

•For random numbers from probability distributions, use RandomReal[contDist, n] or RandomInteger[discrDist, n] instead of resorting to^Random orRandomArray.

•Use^Tally instead ofFrequencies in a package.

The Third Edition

‡ Main Changes

The text has been revised throughout. Indeed, Mathematica 6 brings up so much new and changed features that almost every topic has undergone a revision and new topics are included. Recall that the second edition of this book was based onMathematica 5.

The main change in the structure of the book is that we have six new chapters: Chapter 6, Graphics Primitives; Chapter 9, Data; Chapter 10, Manipulations; Chapter 11, Dynamics; Chapter 15, Tables; and Chapter 16, Patterns. On the other hand, some chapters have been merged and the result is that the current edition has but one chapter about the following topics: graphics for functions, graphics for data, and graphics options (the second edition had two chapters for each of these topics, one for two- dimensional and one for three-dimensional graphics).

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The main change in the contents of the book is the transition from version 5 to version 6. In addition, we have some other enhancements. The chapter on programming is much enhanced and enlarged and contains much more examples. The chapter about matrix calculus is also enhanced. The chapter about optimization now includes the method of dynamic programming. Chapters about graphics for data and optimization have undergone a restructuring.

Note that this book fully utilizes the new features ofMathematica 6. Because version 6 differs so much from earlier versions, this book cannot practically be used with older versions of Mathematica. If you haveMathematica 5.2 or an earlier version, please use the second edition ofMathematica Navigator.

The CD-ROM contains Help Browser material that describes the new properties ofMathematica 7.

‡ Some Notes New Features

Some of the new features of version 6 would have warranted a broader and deeper treatment and more examples of use throughout the book. These features include the creation of dynamic interfaces and the use of the built-in data sources. However, to keep the book at a reasonable size, we had to limit the treatment and the number of examples. We suggest that the reader consults the built-in documentation.

The website http://demonstrations.wolfram.com contains thousands of examples of dynamic interfaces.

Environment

During the writing of this book, I used a Macintosh with MacOS X.Mathematica works in much the same way in various environments, but the keyboard shortcuts of menu commands vary among different environments. To some extent, we mention the shortcuts for the Microsoft Windows and Macintosh environments.

Options

Many commands ofMathematica have options for modifying them. All options have a default value, but we can input other values. When listing the options, we give either all possible values of them or some examples of possible values, but we do not explicitly mention the default values, to save space. In the context of this book,the default value of an option is always the first value mentioned. After that are other possible values or examples of other values.

Simulations

In several places in the book, we simulate various random phenomena. Usually, each time a simulation is run, a slightly different result is obtained. However, in experimenting with the examples of the book, the reader may want to get exactly the same result as printed in the book. This can be achieved by using a seed to the random number generator withSeedRandom[n] for a given integern. With the same seed, the result of a simulation remains the same in repeated executions. We use^SeedRandom quite often in this book. If you want to get other results of simulation than those of this book, give different seeds or do not executeSeedRandom[n] at all (in the latter case, the default seed is used).

CD-ROM

The entire book is contained on the CD-ROM that comes with it. With a few easy steps you can install the book into the Help Browser ofMathematica (the CD-ROM contains installation instructions). With the Help Browser you can easily find and read sections of the book, experiment with the commands, and copy material from the book to your document. You can see all of the figures of the book in color and interactively study the manipulations and animations. The material about the new properties of Mathematica 7 can also be installed into the Help Browser. In addition, the CD-ROM contains some data files that are used in the book.

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Notation

Throughout the book, the adjectives one-, two-, three-, and four-dimensional are abbreviated 1D, 2D, 3D, and 4D, respectively. The symbolÖ is used as a hyphen forMathematica commands. In addition, we use extensively the following handy short notation:

p Means the same as^Pi. The symbolp can be written asÂ^pÂ.

¶ Means the same as^Infinity. The symbol¶ can be written asÂ^infÂ.

P…T Means the same as^[[…]]. For example,^x[[3]] can also be written as^xP3T. The symbolsP and T can be written asÂ^[[Â andÂ^]]Â.

¨ Means the same as^Transpose. For example,Transpose[x] can also be written as^x_¨. The symbol_¨ can be written asÂ^trÂ.

/@ Means the same as^Map. For example,Map[f[#]&, {a, b, c}] can also be written asf[#]& /@ {a, b, c}. A third way is to writeTable[f[x], {x, {a, b, c}}].

The symbolsp and¶ can also be found from theBasicMathInput palette. For example, instead of Map@Ò^ 2 &, Transpose@88Pi, Infinity<<DD

99p²=,8¶<=

we can write

Ò^ 2 &êü88p,¶<<¨

99p²=,8¶<=

Questions

If you have questions about the use ofMathematica, do not hesitate to contact me. I try to answer when I have the time. Also, please send comments and corrections.

Acknowledgments

In preparing this book, the main source has been the excellent on-line Documentation Center of Mathematica 6. The technical support staff at Wolfram Research, Inc., helped me a lot; I especially thank Eric Bynum, Roberto Cavaliere, Huihua Huang, Yong Huang, Vivec Joshi, and Bruce Miller.

The entire book was written and produced withMathematica; each chapter is aMathematica notebook.

The notebooks were connected into a single project by the AuthorTools package ofMathematica. The package then automatically generated the index (after we had attached the index entries with the cells of the book, also with the package), and the package also prepared the on-line Help Browser version of the book.

I have been lucky enough to enjoy excellent working conditions at the Department of Mathematics of the University of Turku. For this my sincere thanks are due to Professor Marko Mäkelä. I also thank Professor Juhani Karhumäki and Professor Matti Vuorinen for their support and encouragement.

The third edition is also published in India. I am deeply indepted to Professor Ponnusamy Saminathan, Indian Institute of Technology, Madras, Chennai, for suggesting and supporting the Indian edition.

For their review of the manuscript of the second edition, I am very thankful to Donald Balenovich, Indiana University of Pennsylvania; Joaquin Carbonara, Buffalo State University; William Emerson, Metropolitan State University; Jim Guyker, Buffalo State University; Mike Mesterton-Gibbons, Florida State University; and Fred Szabo, Concordia University. Their valuable comments and suggestions greatly improved the second edition.

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I also thank the following people for taking the task of writing a review of the second edition of the book in some journals: Robert M. Lurie (Mathematica in Education and Research), Matti Vuorinen (Zentralblatt MATH), K. Waldhör (Computing Reviews), and John A. Wass (Scientific Computing).

Many readers of the second edition have sent me e-mail, giving feedback and asking questions.

Thank you all! Your support has encouraged me in writing the third edition.

The anecdotes at the beginning of the chapters are from the wonderful book by MacHale (1993) (the anecdotes are reproduced or adapted with the permission of the publishers, Boole Press, 26 Temple Lane, Dublin 2, Ireland).

The editorial staff at Elsevier has done a fine work with the production of the book. Especially I would like to thank Phil Bugeau for efficient project management. I am also very grateful to Dan Hays and Kristen Cassereau Ng for copy-editing and proofreading the manuscript with great care.

For financial support I express my deep gratitude to Elsevier Academic Press and Suomen Tietokirjailijat (The Association of Finnish Non-Fiction Writers).

Lastly, I thank my wife, Marjatta, for her encouragement and support during the work.

Heikki Ruskeepää Department of Mathematics University of Turku FIN-20014 Turku Finland ruskeepa@utu.fi

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1 Starting

Introduction 1

1.1 What IsMathematica? 2 1.1.1 An Example 2

1.1.2 The Structure ofMathematica4 1.2 First Calculations 6

1.2.1 Opening, Calculating, and Quitting 6 1.2.2 Names and Decimals 7%,Out,N

1.2.3 Basic Calculations and Plotting 10Pi,E,Sqrt,Exp,Sin,D,Integrate,Simplify,Plot, etc.

1.3 Important Conventions 12 1.4 Getting Help 15

1.4.1 Palettes 15

1.4.2 On-line Documentation 17 1.4.3 Other Help 20 ? 1.5 Editing 22

Introduction

In 1903 at a meeting of the American Mathematical Society, F. N. Cole read a paper entitled “On the Factorization of Large Numbers.” When called upon to speak, Cole walked to the board and, saying nothing, raised two to its sixty-seventh power and subtracted one from the answer.

Then he multiplied, longhand, 193,707,721 by 761,838,257,287 and the answers agreed. Without having said a word, Cole sat down to a standing ovation. Afterwards he announced that it had taken him twenty years of Sunday afternoons to factorize the Mersenne number 2⁶⁷-1.

This chapter is intended to give you an impression ofMathematica and to teach you some of its basic techniques and commands. A more complete insight is given in the next chapter, in which we briefly present a selection of the most important commands ofMathematica.

Although this book puts some emphasis on the methods of applied mathematics, this chapter begins, in Section 1.1, with a “pure” example: factoring integers. We consider the problem mentioned in the anecdote above and show what we can do nowadays with such powerful systems asMathematica. This example will enlighten you regarding some of the major aspects of the program. We emphasize that it is not intended that you do the calculations of this example, nor that you should understand the commands we use.

In Section 1.2, we give a brief overview of some ofMathematica’s basic techniques and commands, beginning with the classical starting example of calculating 1+2 and ending with calculus and graphics.

Section 1.3 presents and explains the important conventions ofMathematica, which often cause trouble for beginners.

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In Sections 1.4 and 1.5, we discuss how you can get help withinMathematica and how you can correct and edit what you have written. These two sections may give more information than you need now, but you can read the basic points and return to these sections later, when getting help and editing become more relevant concerns.

Parts of this chapter depend on the computer you use. We explain only the Windows and Macintosh environments, although some comments may be found about the basics ofMathematica in a Unix system.

1.1 What Is Mathematica ?

1.1.1 An Example

‡ Verifying the Work of Cole

(Note: It is not intended that you do the calculations of Section 1.1.1. The example is only intended to be read and to demonstrate certain aspects ofMathematica. Your actual lessons begin in Section 1.2.)

Did you read the anecdote about F. N. Cole at the beginning of this chapter? Cole sacrificed every Sunday afternoon for 20 years to study the Mersenne numberM67=2⁶⁷-1:

2 ^ 67-1

147 573 952 589 676 412 927

The first line is the command entered to Mathematica, and the second line is the answer given by Mathematica. At last he found that the number is the product of 193,707,721 and 761,838,257,287:

193 707 721*761 838 257 287 147 573 952 589 676 412 927

M67 is thus not a prime. Cole’s feat was admirable. Now, after 100 years, we haveMathematica, and the situation is totally different. It now takes only a fraction of a second to do the factorization:

FactorInteger@2 ^ 67-1D êêTiming

80.016942,88193 707 721, 1<,8761 838 257 287, 1<<<

Mathematica found that 193,707,721 and 761,838,257,287 are factors of multiplicity 1.

‡ Difficult Factors

However, even today some problems can be surprisingly difficult. WhenMathematica, in my computer (which is not very fast), factorizesM254, it needs approximately 10,000 seconds or 3 hours and approximately 200 megabytes of RAM:

FactorInteger@2 ^ 254-1D êêTiming

810 521.7,883, 1<,856 713 727 820 156 410 577 229 101 238 628 035 243, 1<, 8170 141 183 460 469 231 731 687 303 715 884 105 727, 1<<<

MaxMemoryUsed@D 207 758 616

However, note thatM254 is very big:

2 ^ 254-1

28 948 022 309 329 048 855 892 746 252 171 976 963 317 496 166 410 141 009 864 396 001 978 282Ö 409 983

and two of the three factors are also big. Thus, factoring the number is obviously a difficult task. By the way, the difficulty of factoring large numbers is a key to some cryptographic methods.

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However, forM254,Mathematica can immediately tell that it is not a prime:

PrimeQ@2 ^ 254-1D êêTiming 80.000223, False<

Indeed, we can easily investigate Mersenne numbers for primality up to, for example, index 607:

Hmp=Table@8i, PrimeQ@2 ^ i-1D<,8i, 2, 607<D;L êêTiming 80.417603, Null<

The indices for which the corresponding Mersenne number is a prime are as follows:

Select@mp,Ò@@2DDãTrue &D@@All, 1DD

82, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607<

‡ A Demanding Computation

To further illustrate the use ofMathematica, we now factor the Mersenne numbersM2 toM250. (Note that you are not supposed to do the calculations in this example. Just cast an admiring glance at the commands. Later in this book you will learn such commands as^Total,^/@, and^Range.) We do not show the factors themselves; we only count the number of factors:

Ht=Total@FactorInteger@2 ^Ò -1D@@All, 2DDD&êüRange@2, 250DL êêTiming

85430.6,81, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, 1, 6, 4, 4, 2, 7, 3, 3, 3, 6, 3, 7, 1, 5, 4, 3, 4, 10, 2, 3, 4, 8, 2, 8, 3, 7, 6, 4, 3, 10, 2, 7, 5, 7, 3, 9, 6, 8, 4, 6, 2, 13, 1, 3, 7, 7, 3, 9, 2, 7, 4, 9, 3, 14, 3, 5, 7, 7, 4, 8, 3, 10, 6, 5, 2, 14, 3, 5, 6, 10, 1, 13, 5, 9, 3, 6, 5, 13, 2, 5, 8, 14, 2, 11, 2, 10, 11, 6, 1, 15, 2, 12, 6, 11, 5, 9, 6, 9, 9, 6, 6, 17, 4, 3, 5, 8, 5, 14, 1, 9, 5, 9, 2, 15, 3, 5, 10, 11, 2, 9, 2, 16, 6, 6, 6, 19, 5, 6, 7, 10, 2, 14, 5, 11, 8, 10, 8, 18, 4, 5, 8, 13, 7, 16, 5, 10, 10, 8, 2, 19, 4, 7, 7, 10, 4, 11, 9, 14, 6, 5, 3, 24, 4, 11, 5, 11, 5, 8, 5, 10, 10, 10, 5, 16, 3, 7, 8, 11, 2, 17, 2, 20, 6, 4, 7, 20, 6, 5, 9, 12, 6, 22, 3, 10, 7, 4, 8, 21, 6, 5, 7, 19, 4, 13, 6, 16, 14, 10, 2, 17, 4, 12, 10, 12, 4, 16, 7, 13, 8, 11, 6, 23, 2, 8, 10, 9, 7, 12, 6, 12, 5, 11<<

MaxMemoryUsed@D 83 409 352

Thus, for example,M4 has two factors, andM250 has 11 factors. The computations took approximately 1.5 hours and approximately 80 megabytes of RAM.

‡ A Graphic Illustration

We continue studying Mersenne numbers and now form pairs from the indices and the numbers of factors:

s=8Range@2, 250D, t<¨;

(Here, means a transpose.) Then we find a logarithmic least-squares fit for the number of factors as a function of the index of the Mersenne numbers:

lsq=Fit@s,81, Log@iD<, iD -3.05399+2.25251 Log@iD

We then plot the fit and the numbers of factors:

p1=Plot@lsq,8i, 2, 250<, PlotStyleØBlackD; p2=ListLinePlot@s, PlotStyleØBlack,

MeshØAll, MeshStyleØ8Black, PointSize@SmallD<D;

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Show@p1, p2, AspectRatioØ0.3, PlotRangeØ8-1, 25<, ImageSizeØ420, TicksØ8Join@82<, Range@10, 250, 10DD,81, 5, 10, 15, 20, 24<<D

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 5

10 15 20 24

So, here are the numbers of factors ofM2 toM250, together with the logarithmic fit. In the figure we see the prime Mersenne numbers with indices 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, and 127. These Mersenne primes had all been found by 1913. After the index 127, primes are not very common among Mersenne numbers. The next known primes occur with indices 521 (found in 1952); 607; 1279; 2203;

2281; 3217; 4253; 4423; 9689; 9941; 11,213; 19,937; 21,701; 23,209; 44,497; 86,243; 110,503; 132,049; 216,091;

756,839; 859,433; 1,257,787; 1,398,269; 2,976,221; 3,021,377; 6,972,593; 13,466,917; 20,996,011; 24,036,583;

25,964,951; 30,402,457 (found in 2005); 32,582,657 (found in 2006).

‡ Lessons Learned

The preceding examples show how easy it now is to do long and complicated calculations and to visualize the results. Mathematica is one of the more popular systems for doing such calculations.

However, even today, with powerful mathematical systems and machines, some problems remain very time-consuming.

The example also illustrates some aspects ofMathematica, such as working with exact and approxi- mate quantities, using graphics, and making programs. In general, Mathematica integrates symbolic calculation, numerical calculation, graphics, and programming into one system.

Mathematica contains still another aspect: a document-making environment (in versions ofMathemat- ica that support the notebook interface). In this environment, you can do symbolic and numerical calculations, produce graphics, and add text to explain what you have done. The result is a complete document of your work; in fact, this book was written withMathematica. In addition, the document is interactive. You can change parameters and functions, redo calculations, show animations, create interactive graphical interfaces, and continuously develop the document. The notebook interface is included in both Windows and Macintosh versions ofMathematica. In plain Unix, the interface is text- based and thus all features of Mathematica are not supported, but the X Window System supports notebooks.

1.1.2 The Structure ofMathematica

Mathematica is a large system: The total installation ofMathematica takes up approximately 1.1 gigabytes (Gb) of space on the hard drive. The following are the essential parts ofMathematica:

•thekernel, with the nameMathKernel;

•thefront end, with the nameMathematica; and

• thepackages, with the namePackages.

Next we consider each of these three components individually.

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‡ The Kernel

The main component ofMathematica is the kernel; it does all of the computations. It is written mainly with the C programming language and is one of the largest mathematical systems ever written. The total number of lines of code in the kernel is approximately 2.5 million. An important thing to understand is that the kernel is the same in all environments; this means that you get the same results in all environments (except possibly for the precision of floating-point calculations).

Mathematica commands are easy to use and quite versatile. However, behind the commands there is a huge amount of mathematical knowledge and a vast amount of work. For example, behind the single command^Integrate, there are approximately 600 pages of C code and 500 pages ofMathematica code.

‡ The Front End

The front end is an environment for communicating with the kernel. When you openMathematica, you open the front end. Commands are entered into the front end, and they are sent automatically to the kernel (the communication between the front end and the kernel is done withMathLink, a system that handles communication between parts ofMathematica and betweenMathematica and other programs).

The result of a calculation is then displayed by the front end. Figures come from the kernel in the form of PostScript code, and the front end then creates a screen image from this code. The front end contains approximately 700,000 lines of system-independent C++ source code.

There are two types of front ends:notebook andtext-based.

A notebook front end is, for example, in the Windows, Macintosh, and X Window versions of Mathematica. A notebook is an interactive document. It contains the commands you have entered and their results, graphics (including animations), dynamic graphical interfaces, and comments you have added. You can do any kind of correcting, editing, and formatting in a notebook. In fact, you can make a notebook into a whole document of your work, even a book. (The chapters of this book were made as separate notebooks.)

The text-based front end is not nearly so handy and versatile as the notebook front end. This type of front end is found in the plain Unix version ofMathematica. With this kind of front end you can enter commands and see the results, but editing is very limited. Another disadvantage is that pictures are displayed in separate windows.

‡ The Packages

The packages supplement the kernel. They are not normally loaded whenMathematica is loaded; you have to manually load the packages you want to use. Packages contain commands considered not to be as central and as common as the commands in the kernel. Packages are written using the programming capabilities ofMathematica.

To the packages that come withMathematica you can also add the vast collection of packages and other material found at http://library.wolfram.com/. In addition, you can write your own packages and programs.

Furthermore, there are collections of Wolfram application packages and third-party application packages; see www.wolfram.com/products.

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1.2 First Calculations

1.2.1 Opening, Calculating, and Quitting

‡ Opening Mathematica

To openMathematica, do the following:

•In Windows and Macintosh environments: openMathematica as other programs

•In Unix with X Window: typemathematica

•In plain Unix: type^math

When you open Mathematica in a notebook environment, you get an empty notebook. A cursor appears once you have pressed the first key.

‡ Calculating

Now, are you ready to begin? Let us start modestly and try to calculate 1+2. Press these three keys:

1+2

The command is executed (i.e., sent to the kernel for processing) by pressing a special key or key combination.

To execute a command, pressÛ or˜÷Á

In Windows and Macintosh interfaces, both ways work (Û is the key at the bottom right corner of the keyboard;˜÷Á means that you are holding down the Shift key while at the same time pressing the Return key¿). In Unix, it may be that˜÷Á is the only possibility.

Note that although we have spoken about “executing a command,” this is not in keeping with the official terminology. We should speak about “evaluating an input.” You should know the official term evaluation, but we ourselves feel free to use the more concrete termexecution.

So, after typing the command or input^{1 + 2}, press any of the executing or evaluating keys or key combinations. Now the kernel begins to evaluate the input. In most microcomputers, the kernel is not loaded until you ask it to do a calculation. This means that the first calculation takes some time, even if it is as simple as^{1 + 2}. After you have entered the input, the label^In[1]:= appears before the input, and the result has the label^Out[1]=. You get

1+2 3

Note that here we do not show the In and Out labels. Indeed, by adjusting the preferences of Mathematica, we can turn the labels off. In a Macintosh, the preferences can be found from theMathemat- ica@Preferences… menu. Go then into theEvaluation preferences and clickShow In/Out names to be off.

The input and the output can be distinguished by the font: The input is in boldface Courier, whereas the output is in plain Courier. Try the next command (note that although you again cannot see a cursor in the notebook, just start typing; the new command appears where you see a horizontal line):

18ê4 9 2

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Fractions are automatically simplified, and they are written in a 2D form (instead of 9ê2). If the expression contains a decimal number, then the result is also a decimal number:

10ê3.

3.33333

‡ Correcting

You may observe an error in a command when you write it, or after executing the commandMathemat- ica may give you an error message telling you that there is something wrong with the command.

In a notebook environment (e.g., Windows, Macintosh, or X Window), you can do standard editing as with a word processor. For example, to delete an incorrect character, just move the cursor to the desired location by pressing the arrow keys or with the mouse and press the backspace key or the delete key to remove the character to the left or right, respectively. You can also highlight a portion of an input, cut it (‚ÎxÏ; on a Macintosh, useÌ in place of‚) or copy it (‚ÎcÏ), and then paste it (‚ÎvÏ) to a new location.

Once you have a corrected input, execute it. Note that after correcting a command you can leave the cursor where it is and then execute the command; the cursor need not be at the end of the command when you execute it. We consider editing in more detail in Section 1.5, p. 22. (In plain Unix, possibly only the backspace key can be used to edit the input; arrow keys may not function. Using special editing commands such as^Edit together with an editor such as Emacs can help a lot in plain Unix.)

‡ QuittingMathematica

To quitMathematica, do the following:

•In Windows, Macintosh, and X Window: quitMathematica as other programs

•In plain Unix: execute the command^Quit

Do not quit right now. Instead, continue reading and experimenting.

‡ Aborting a Calculation

You may sometimes observe that a calculation is useless (perhaps because there was an error in the input or you do not have time to wait for the answer). You can then abort the calculation.

To abort a calculation, do the following:

• In Windows, Macintosh, and X Window: chooseAbort Evaluation from theEvaluation menu, or press‡Î^.Ï (ÌÎ^.Ï on a Macintosh)

• In plain Unix: press‚ÎcÏ Á aÁ

It may happen that when you try to abort a calculation it seems that the calculation just goes on and on. You can then quit Mathematica (after possibly saving the notebook) and start a new session. In notebook environments, you can also only quit the kernel by choosingEvaluation @Quit Kernel@ Local. Start a new session by executing an input or by choosingEvaluation@Start Kernel@Local.

1.2.2 Names and Decimals

‡ Before Continuing

Now we continue exploringMathematica. Please note the following very important point:

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•When you try the examples of the book with your machine, write the commands exactly as they are printed here.

Mathematica will not forgive even the smallest error in syntax. Be especially careful with small and capital letters: AllMathematica names such as^Sin or^Integrate begin with a capital letter. Also, you have to write all arguments in functions and commands in square brackets^{[ ]}, for example,^Sin[x] and Integrate[a + b x, x]; parentheses^{( )} are not allowed. Parentheses are only used for grouping terms in expressions. Note that Mathematica automatically adds spaces in some places in your input (e.g., around+ or=).

You probably will occasionally get error messages and wrong results because the syntax was not correct. Do not worry. This is normal. Getting used toMathematica takes time, and only by working with the program can you learn to use it efficiently.

During a session, do not just enter one input after another in a hurry. Think carefully about each input and each result~how the input is written and what the result is. In this way, you will learn more effectively.

Also, once you have tried some of the examples in this book, you can try other examples. This is recommended because by trying similar examples you strengthen your skills and get a clearer impression of each command and technique.

‡ Referring to Earlier Results

% Refers to the last result

%% Refers to the next-to-last result

%%...% Refers to thekth previous result if there arek^% marks Out[n] or ^%n Refers to the result in the output line^Out[n]

Often you want to refer to earlier results. The percent mark^% and the output names can be used for this. Try the following commands:

353 ^ 4 15 527 402 881

30 ^ 4+120 ^ 4+272 ^ 4+315 ^ 4 15 527 402 881

%% - % 0

Try also the output names by executing, for example,Out[4] + Out[5] or^{%4 + %5}.

Using^% can become a problem if the command you execute contains errors to be corrected. Suppose you first calculate

16 ^ 2 256

Then you want to calculate 16²-15², but you write

% +15 ^ 2 481

If you now correct the command to read^{% - 15^2}, you do not get what you want because^% now refers to the result 481 of the last (wrong) command. So, you have to correct the command to read^{%% - 15^2}.

Mathematica Navigator ®

Mathematica ^® Navigator

Mathematica ^® Navigator

Mathematics, Statistics, and Graphics

THIRD EDITION

Heikki Ruskeepää

Department of Mathematics University of Turku, Finland

To Marjatta

Contents

Preface

Welcome

Contents

Special Aspects

Mathematica 6

The Third Edition

Acknowledgments

1

Starting

1.1 What Is Mathematica ?

1.2 First Calculations

Mathematica Navigator ®

Mathematica ® Navigator

Mathematica ® Navigator

Mathematics, Statistics, and Graphics

THIRD EDITION

Heikki Ruskeepää

Department of Mathematics University of Turku, Finland

To Marjatta

Contents

Preface

Welcome

Contents

Special Aspects

Mathematica 6

The Third Edition

Acknowledgments

1

Starting

1.1 What Is Mathematica ?

1.2 First Calculations

Mathematica ^® Navigator

Mathematica ^® Navigator