C++ for Mathematicians: An Introduction for Students and Professionals

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ISBN: 9781584885849 | 158488584X
Cover: Nonspecific Binding
Copyright: 6/6/2006

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For problems that require extensive computation, a C++ program can race through billions of examples faster than most other computing choices. C++ enables mathematicians of virtually any discipline to create programs to meet their needs quickly, and is available on most computer systems at no cost. C++ for Mathematicians: An Introduction for Students and Professionals accentuates C++ concepts that are most valuable for pure and applied mathematical research.This is the first book available on C++ programming that is written specifically for a mathematical audience; it omits the language's more obscure features in favor of the aspects of greatest utility for mathematical work. The author explains how to use C++ to formulate conjectures, create images and diagrams, verify proofs, build mathematical structures, and explore myriad examples. Emphasizing the essential role of practice as part of the learning process, the book is ideally designed for undergraduate coursework as well as self-study. Each chapter provides many problems and solutions which complement the text and enable you to learn quickly how to apply them to your own problems. An accompanying CD ROM provides all numbered programs so that readers can easily use or adapt the code as needed.Presenting clear explanations and examples from the world of mathematics that develop concepts from the ground up, C++ for Mathematicians can be used again and again as a resource for applying C++ to problems that range from the basic to the complex.

List of Programs

xiii

List of Figures

xvii

Preface

xix

I Procedures

(90)

1 The Basics

(8)

1.1 What is C++?

(1)

1.2 Hello C++

(4)

1.3 Exercises

(3)

2 Numbers

(20)

2.1 The integer types

(3)

2.2 The real number types

(1)

2.3 The bool and char types

(1)

2.4 Checking the size and capacity of the different types

(3)

2.5 Standard operations

(4)

2.6 Comparisons and Boolean operations

(1)

2.7 Complex numbers

(4)

2.8 Naming variables

(1)

2.9 Exercises

(3)

3 Greatest Common Divisor

(22)

3.1 The problem

(1)

3.2 A first approach

(6)

3.3 Euclid's method

(4)

3.4 Looping with for, while, and do

(2)

3.5 An exhaustive approach to the GCD problem

(2)

3.6 Extended gcd, call by reference, and overloading

(4)

3.7 Exercises

(4)

4 Random Numbers

(14)

4.1 Pseudo random number generation

(1)

4.2 Uniform random values

(3)

4.3 More on pseudo random number generation

(3)

4.4 A Monte Carlo program for the GCD problem

(1)

4.5 Normal random values

(2)

4.6 Exercises

(4)

5 Arrays

(24)

5.1 Euler's totient

(2)

5.2 Array fundamentals

(2)

5.3 A procedure to factor integers

(5)

5.4 A procedure to calculate Euler's totient

(2)

5.5 The Sieve of Eratosthenes: new and delete []

(5)

5.6 A faster totient

(2)

5.7 Computing pn for large n

(2)

5.8 The answer

(1)

5.9 Exercises

(3)

II Objects

(176)

6 Points in the Plane

(22)

6.1 Data and methods

(1)

6.2 Declaring the Point class

(3)

6.3 Data hiding

(2)

6.4 Constructors

(1)

6.5 Assignment and conversion

100

(1)

6.6 Methods

101

(2)

6.7 Procedures using arguments of type Point

103

(1)

6.8 Operators

104

(8)

6.9 Exercises

112

(3)

7 Pythagorean Triples

115

(12)

7.1 Generating Pythagorean triples

115

(1)

7.2 Designing a primitive Pythagorean triple class

116

(1)

7.3 Implementation of the PTriple class

117

(4)

7.4 Finding and sorting the triples

121

(4)

7.5 Exercises

125

(2)

8 Containers

127

(30)

8.1 Sets

127

(3)

8.2 Set iterators

130

(3)

8.3 Multisets

133

(1)

8.4 Adjustable arrays via the vector class

134

(4)

8.5 Ordered pairs

138

(1)

8.6 Maps

139

(5)

8.7 Lists, stacks, and assorted queues

144

(7)

8.7.1 Lists

144

(4)

8.7.2 Stacks

8.7.3 Queues

148

(1)

8.7.4 Deques

149

(1)

8.7.5 Priority queues

150

(1)

8.8 Exercises

151

(6)

9 Modular Arithmetic

157

(20)

9.1 Designing the Mod type

157

(1)

9.2 The code

158

(5)

9.3 The default modulus: Static class variables and methods

163

(4)

9.4 Constructors and get/set methods

167

(1)

9.5 Comparison operators

167

(2)

9.6 Arithmetic operators

169

(3)

9.7 Writing Mod objects to output streams

172

(1)

9.8 A main to demonstrate the Mod class

172

(2)

9.9 Exercises

174

(3)

10 The Projective Plane

177

(38)

10.1 Introduction to the projective plane, RP²

177

(1)

10.2 Designing the classes PPoint and PLine

178

(3)

10.3 Inheritance

181

(3)

10.4 Protected class members

184

(2)

10.5 Class and file organization for PPoint and PLine

186

(1)

10.6 The parent class PObject

187

(8)

10.7 The classes PPoint and PLine

195

(5)

10.8 Discovering and repairing a bug

200

(7)

10.9 Pappus revisited

207

(4)

10.10 Exercises

211

(4)

11 Permutations

215

(20)

11.1 Ulam's problem

215

(2)

11.2 Designing the Permutation class

217

(12)

11.2.1 Data

218

(1)

11.2.2 Constructors and destructors

218

(2)

11.2.3 Copy and assign

220

(3)

11.2.4 Basic inspection and modification methods

223

(1)

11.2.5 Permutation operations

224

(1)

11.2.6 Comparison operators

225

(1)

11.2.7 Output

225

(1)

11.2.8 The code file Permutation c

225

(4)

11.3 Finding monotone subsequences

229

(3)

11.4 Exercises

232

(3)

12 Polynomials

235

(32)

12.1 Procedure templates

235

(3)

12.2 Class templates

238

(4)

12.2.1 Using class templates

238

(1)

12.2.2 Creating class templates

239

(3)

12.3 The Polynomial class template

242

(12)

12.3.1 Data

243

(1)

12.3.2 Constructors

243

(1)

12.3.3 Get and set methods

244

(1)

12.3.4 Function methods

245

(1)

12.3.5 Equality

246

(1)

12.3.6 Arithmetic

246

(1)

12.3.7 Output to the screen

247

(1)

12.3.8 GCD

247

(1)

12.3.9 The code

247

(7)

12.4 The GCD problem revisited

254

(4)

12.5 Working in binary

258

(6)

12.5.1 Signed versus unsigned integers

258

(1)

12.5.2 Bit operations

259

(1)

12.5.3 The bit set class template

260

(3)

12.5.4 Class templates with non-type arguments

263

(1)

12.6 Exercises

264

(3)

III Topics

267

(94)

13 Using Other Packages

269

(20)

13.1 Arbitrary precision arithmetic: The GMP package

269

(4)

13.2 Linear algebra

273

(13)

13.2.1 Two-dimensional arrays in C++

273

(1)

13.2.2 The TNT and JAMA packages

274

(8)

13.2.3 The newmat package

282

(4)

13.3 Other packages

286

(1)

13.4 Exercises

287

(2)

14 Strings, Input/Output, and Visualization

289

(44)

14.1 Character arrays

289

(2)

14.2 The string class

291

(6)

14.2.1 Initialization

291

(1)

14.2.2 Fundamental operations

292

(3)

14.2.3 Searching

295

(2)

14.2.4 Converting between string and char* types

297

(1)

14.3 Command line arguments

297

(3)

14.4 Reading and writing data in files

300

(7)

14.4.1 Opening files for input/output

300

(3)

14.4.2 Reading and writing

303

(1)

14.4.3 Detecting the end of an input file

304

(1)

14.4.4 Other methods for input

305

(2)

14.5 String streams

307

(1)

14.6 Formatting

308

(3)

14.6.1 Setting precision

309

(1)

14.6.2 Showing all digits

309

(1)

14.6.3 Setting the width

310

(1)

14.6.4 Other manipulators

311

(1)

14.7 A class to parse files

311

(4)

14.8 Visualization

315

(15)

14.8.1 Introducing and installing the plotutils package

316

(1)

14.8.2 Drawing with plotutils—a first example

317

(5)

14.8.3 Pascal's triangle modulo two

322

(2)

14.8.4 Tracing the motion of a point moving randomly in a triangle

324

(2)

14.8.5 Drawing Paley graphs

326

(4)

14.9 Exercises

330

(3)

15 Odds and Ends

333

(28)

15.1 The switch statement

333

(3)

15.2 Labels and the got o statement

336

(2)

15.3 Exception handling

338

(6)

15.3.1 The basics of try, throw, and catch

338

(4)

15.3.2 Other features of the exception-handling system

342

(2)

15.4 Friends

344

(3)

15.5 Other ways to create types

347

(3)

15.5.1 Structures

347

(1)

15.5.2 Enumerations

348

(1)

15.5.3 Unions

348

(1)

15.5.4 Using typedef

349

(1)

15.6 Pointers

350

(8)

15.6.1 Pointer basics

350

(1)

15.6.2 Dereferencing

351

(2)

15.6.3 Arrays and pointer arithmetic

353

(2)

15.6.4 new and delete revisited

355

(1)

15.6.5 Why use pointers'

356

(2)

15.7 Exercises

358

(3)

IV Appendices

361

(126)

A Your C++ Computing Environment

363

(18)

A.1 Programming with a command window and a text editor

363

(9)

A.1.1 What you need and how to get it (for free)

364

(1)

A.1.2 Editing program files

365

(1)

A.1.3 Compiling and running your program

366

(2)

A.1.4 Compiler options

368

(2)

A.1.5 Introduction to make

370

(2)

A.2 Programming with an integrated development environment

372

(6)

A.2.1 Visual C++ for Windows

373

(3)

A.2.2 Xcode for Macintosh OS X

376

(2)

A.3 General advice on debugging

378

(3)

B Documentation with Doxygen

381

(10)

B.1 Doxygen comments

381

(5)

B.1.1 Documenting files

382

(1)

B.1.2 Documenting procedures

382

(1)

B.1.3 Documenting classes, data, and methods

383

(3)

B.2 Using Doxygen

386

(5)

B.2.1 Configuring Doxygen

386

(3)

B.2.2 Running Doxygen

389

(1)

B.2.3 More features

389

(2)

C C++ Reference

391

(24)

C.1 Variables and types

391

(3)

C.1.1 Fundamental types

391

(1)

C.1.2 Standard classes/templates

391

(1)

C.1.3 Declaring variables

392

(1)

C.1.4 Static variables and scope

392

(1)

C.1.5 Constants and the keyword const

393

(1)

C.1.6 Arrays

393

(1)

C.2 Operations

394

(2)

C.2.1 Assignment

394

(1)

C.2.2 Arithmetic

394

(1)

C.2.3 Comparison operators

394

(1)

C.2.4 Logical operators

394

(1)

C.2.5 Bit operators

395

(1)

C.2.6 Potpourri

395

(1)

C.3 Control statements

396

(2)

C.3.1 if—else

396

(1)

C.3.2 Looping: for, while, and do

396

(1)

C.3.3 switch

397

(1)

C.3.4 goto

398

(1)

C.3.5 Exceptions

398

(1)

C.4 Procedures

398

(3)

C.4.1 File organization

399

(1)

C.4.2 Call by value versus call by reference

399

(1)

C.4.3 Array (and pointer) arguments

400

(1)

C.4.4 Default values for arguments

400

(1)

C.4.5 Templates

400

(1)

C.4.6 inline procedures

401

(1)

C.5 Classes

401

(7)

C.5.1 Overview and file organization

401

(1)

C.5.2 Constructors and destructors

402

(1)

C.5.3 Operators

403

(1)

C.5.4 Copy and assign

404

(1)

C.5.5 static data and methods

405

(1)

C.5.6 this

406

(1)

C.5.7 Friends

406

(1)

C.5.8 Class templates

407

(1)

C.5.9 Inheritance

407

(1)

C.6 Standard functions

408

(7)

C.6.1 Mathematical functions

408

(3)

C.6.2 Mathematical constants

411

(1)

C.6.3 Character procedures

411

(2)

C.6.4 Other useful functions

413

(2)

D Answers

415

(72)

Index

487

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C++ for Mathematicians: An Introduction for Students and Professionals

Summary

Table of Contents

Supplemental Materials