An introduction to programming with Mathematica

556 Pages·2005·6.481 MB·English

by  Paul R. Wellin, Richard J. Gaylord, Samuel N. Kamin

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AnIntroductiontoProgrammingwithMathematica(cid:1)r
An Introduction to Programming with Mathematica(cid:1)r is de-
signedtointroducetheMathematicaprogramminglanguage
to a wide audience. Since the last edition of this book was
published, significant changes have occurred in Mathemat-
icaanditsuseworldwide.Keepingpacewiththesechanges,
this substantially larger, updated version includes new and
revised chapters on numerics, procedural, rule-based, and
front end programming, and gives significant coverage to
the latest features up to, and including, version 5.1 of the
software.
Mathematicanotebooks,availablefromwww.cambridge.org/
0521846781, contain examples, programs, and solutions to
exercises in the book. Additionally, material to supplement
laterversionsofthesoftwarewillbemadeavailable.Thisis
theidealtextforallscientificstudents,researchers,andpro-
grammers wishing to deepen their understanding of Math-
ematica, or even those keen to program using an interac-
tive language that contains programming paradigms from
all major programming languages: procedural, functional,
recursive,rule-based,andobject-oriented.

An Introduction to
Programming
with
Mathematica(cid:1)r
Third Edition
PaulR.Wellin|RichardJ.Gaylord|SamuelN.Kamin
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521846783
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2005
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Cambridge University Press has no responsibility for the persistence or accuracy of
s for external or third-party internet websites referred to in this book, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Mathematica, Mathlink and Mathsource are registered trademarks of Wolfram
Research, Inc. All other trademarks used herein are the property of their respective
owners. Mathematica is not associated with Mathematica Policy Research, Inc. or
MathTech, Inc.
Wolfram Research is the holder of the copyright to the Mathematica software system
described in this document, 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 an otherwise authorized by law is an
infringement of the copyright.
1 An introduction to Mathematica
Mathematica is a very large and seemingly complex system. It contains hundreds of
functions for performing various tasks in science, mathematics, and engineering,
including computing, programming, data analysis, knowledge representation, and
visualization of information. In this introductory chapter, we introduce the elementary
operations in Mathematica and give a sense of its computational and programming
breadth and depth. In addition, we give some basic information that users of Mathemat-
ica need to know, such as how to startMathematica, how to get out of it, how to enter
simple inputs and get answers, and finally how to use Mathematica’s documentation to
get answers to questions about the system.
1.1 A brief overview of Mathematica
Numerical computations
Mathematica has been aptly described as a sophisticated calculator. With it you can enter
mathematical expressions and compute their values.
.08 12
In[1]:= Sin .86 Log 1
12
Out[1]= 0.481899
You can store values in memory.
In[2]:= rent 350
Out[2]= 350
In[3]:= food 175
Out[3]= 175
In[4]:= heat 83
Out[4]= 83
2 An Introduction to Programming withMathematica
In[5]:= rent food heat
Out[5]= 608
Yet Mathematica differs from calculators and simple computer programs in its ability to
calculate exact results and to compute to an arbitrary degree of precision.
1 1 1
In[6]:=
15 35 63
1
Out[6]=
9
In[7]:= 2500
Out[7]= 3273390607896141870013189696827599152216642046043064789483291
368096133796404674554883270092325904157150886684127560071009
217256545885393053328527589376
In[8]:= N , 500
Out[8]= 3.14159265358979323846264338327950288419716939937510582097494
459230781640628620899862803482534211706798214808651328230664
709384460955058223172535940812848111745028410270193852110555
964462294895493038196442881097566593344612847564823378678316
527120190914564856692346034861045432664821339360726024914127
372458700660631558817488152092096282925409171536436789259036
001133053054882046652138414695194151160943305727036575959195
309218611738193261179310511854807446237996274956735188575272
48912279381830119491
Symbolic computations
One of the more powerful features of Mathematica is its ability to manipulate and compute
with symbolic expressions. For example, you can factor polynomials and simplify trigono-
metric expressions.
In[9]:= Factor x5 1
Out[9]= 1 x 1 x x2 x3 x4
In[10]:= TrigReduce Sin 3
1
Out[10]= 3Sin Sin 3
4
1 An introduction to Mathematica 3
You can simplify expressions using assumptions about variables contained in those expres-
sions. For example, ifk is assumed to be an integer,sin 2 k x simplifies to sin x .
In[11]:= Simplify Sin 2 k x , k Integers
Out[11]= Sin x
This computes the conditions for which a general quadratic polynomial will have both
roots equal to each other.
In[12]:= Reduce x,ax2 bx c 0 y,ay2 by c 0 x y , a, b, c
b2
Out[12]= a 0&&b 0 a 0&&bc 0 a 0&&c
4a
You can create functions that are defined piecewise.
In[13]:= Piecewise 1, x 0 , Sin x x
1 x 0
Out[13]= Sin x True
x
The knowledge base of Mathematica includes algorithms for solving polynomial equations,
and computing integrals.
In[14]:= Solve x3 ax 1 0, x
2 1 3a 9 3 27 4a3 1 3
Out[14]= x 3 ,
1 3 21 332 3
9 3 27 4a3
1 3 a
x
1 3
22 331 3 9 3 27 4a3
1 3
1 3 9 3 27 4a3
,
221 332 3
1 3 a
x
1 3
22 331 3 9 3 27 4a3
1 3
1 3 9 3 27 4a3
221 332 3
1
In[15]:= x
1 x4
1
Out[15]= 2ArcTan 1 2 x 2ArcTan 1 2 x
4 2
Log 1 2 x x2 Log 1 2 x x2
4 An Introduction to Programming withMathematica
Graphics
The ability to visualize functions or sets of data often allows us greater insight into their
structure and properties. Mathematica provides a wide range of graphing capabilities.
These include two- and three-dimensional plots of functions or data sets, contour and
density plots of functions of two variables, bar charts, histograms and pie charts of data
sets, and many packages designed for specific graphical purposes. In addition, theMathemat-
ica programming language allows you to construct graphical images “from the ground up”
using primitive elements, as we will see in Chapter 9.
Here is a simple two-dimensional plot of the function sin x 2 sin x2 .
In[16]:= Plot Sin x 2 Sin x2 , x, ,
1
0.5
3 2 1 1 2 3
0.5
1
Out[16]= Graphics
You can combine two or more plots in a single graphic by enclosing them inside curly
braces.
In[17]:= Plot Sin x , Sin 2x , x, 0, 2 ;
1
0.5
1 2 3 4 5 6
0.5
1