Python科学编程入门：实战解题与计算科学基础

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xiv Contents

5.2 Arraysin Python Programs ........................ 193

5.2.1 Using Lists for Collecting Function Data . . . . . . . . 193

5.2.2 Basics of Numerical Python Arrays . . . . . . . . . . . . 194

5.2.3 Computing Coordinates and Function Values . . . . 195

5.2.4 Vectorization................................ 196

5.3 CurvePlotting.................................... 198

5.3.1 Matplotlib;Pylab............................ 198

5.3.2 Matplotlib;Pyplot........................... 202

5.3.3 SciToolsandEasyviz......................... 204

5.3.4 MakingAnimations.......................... 209

5.3.5 CurvesinPureText ......................... 214

5.4 PlottingDiﬃculties................................ 215

5.4.1 Piecewisely Deﬁned Functions . . . . . . . . . . . . . . . . . 216

5.4.2 RapidlyVaryingFunctions.................... 218

5.5 More Advanced Vectorization of Functions . . . . . . . . . . . 219

5.5.1 Vectorizing StringFunction Objects . . . . . . . . . . . . 220

5.5.2 Vectorization of the Heaviside Function . . . . . . . . . 221

5.5.3 VectorizationofaHat Function ............... 224

5.6 Moreon Numerical PythonArrays .................. 226

5.6.1 CopyingArrays ............................. 226

5.6.2 In-PlaceArithmetics ......................... 227

5.6.3 AllocatingArrays............................ 228

5.6.4 GeneralizedIndexing......................... 228

5.6.5 TestingfortheArray Type ................... 229

5.6.6 Compact Syntax for Array Generation. . . . . . . . . . 230

5.6.7 ShapeManipulation.......................... 230

5.7 Higher-DimensionalArrays ......................... 231

5.7.1 MatricesandArrays ......................... 231

5.7.2 Two-DimensionalNumericalPythonArrays..... 232

5.7.3 ArrayComputing............................ 235

5.7.4 Two-Dimensional Arrays and Functions of Two

Variables ................................... 235

5.7.5 MatrixObjects.............................. 236

5.8 Summary ........................................ 237

5.8.1 ChapterTopics.............................. 237

5.8.2 Example:AnimatingaFunction ............... 239

5.9 Exercises......................................... 243

6 Files, Strings, and Dictionaries ...................... 257

6.1 ReadingData from File ............................ 257

6.1.1 ReadingaFileLine by Line................... 258

6.1.2 Reading a Mixture of Text and Numbers . . . . . . . 261

6.1.3 WhatIsa File, Really?....................... 262

6.2 Dictionaries ...................................... 266

6.2.1 MakingDictionaries.......................... 266

Contents xv

6.2.2 DictionaryOperations........................ 267

6.2.3 Example: Polynomials as Dictionaries . . . . . . . . . . 269

6.2.4 Dictionaries with Default Values and Ordering . . 271

6.2.5 Example: File Data in Dictionaries . . . . . . . . . . . . . 273

6.2.6 Example: File Data in Nested Dictionaries . . . . . . 274

6.2.7 Example:ComparingStockPrices ............. 278

6.3 Strings........................................... 282

6.3.1 CommonOperationsonStrings ............... 282

6.3.2 Example:ReadingPairsofNumbers ........... 286

6.3.3 Example:ReadingCoordinates ................ 288

6.4 ReadingData fromWebPages...................... 291

6.4.1 AboutWebPages ........................... 291

6.4.2 How to Access Web Pages in Programs . . . . . . . . . 292

6.4.3 Example:ReadingPureText Files ............. 293

6.4.4 Example: Extracting Data from HTML . . . . . . . . . 295

6.5 WritingData to File............................... 296

6.5.1 Example:WritingaTable to File .............. 296

6.5.2 Standard Input and Output as File Objects . . . . . 298

6.5.3 Reading and Writing Spreadsheet Files . . . . . . . . . 300

6.6 Examplesfrom Analyzing DNA ..................... 305

6.6.1 ComputingFrequencies....................... 305

6.6.2 AnalyzingtheFrequencyMatrix............... 312

6.6.3 FindingBaseFrequencies..................... 315

6.6.4 TranslatingGenesintoProteins ............... 317

6.6.5 Some Humans Can Drink Milk, While Others

Cannot..................................... 322

6.7 Summary ........................................ 323

6.7.1 ChapterTopics.............................. 323

6.7.2 Example:AFileDatabase .................... 325

6.8 Exercises......................................... 329

7 Introduction to Classes ............................. 341

7.1 SimpleFunctionClasses............................ 342

7.1.1 Problem: Functions with Parameters . . . . . . . . . . . 342

7.1.2 RepresentingaFunctionas aClass............. 344

7.1.3 AnotherFunctionClassExample .............. 350

7.1.4 Alternative Function Class Implementations . . . . . 351

7.1.5 Making Classes Without the Class Construct . . . . 353

7.2 MoreExamples on Classes.......................... 356

7.2.1 BankAccounts.............................. 356

7.2.2 PhoneBook................................. 358

7.2.3 ACircle.................................... 359

7.3 SpecialMethods .................................. 360

7.3.1 TheCallSpecialMethod ..................... 361

7.3.2 Example: Automagic Diﬀerentiation . . . . . . . . . . . . 361

xvi Contents

7.3.3 Example: Automagic Integration . . . . . . . . . . . . . . . 364

7.3.4 TurninganInstanceintoa String.............. 366

7.3.5 Example: Phone Book with Special Methods . . . . 367

7.3.6 AddingObjects ............................. 369

7.3.7 Example:ClassforPolynomials ............... 369

7.3.8 Arithmetic Operations and Other Special

Methods.................................... 373

7.3.9 Special Methods for String Conversion. . . . . . . . . . 374

7.4 Example:Classfor Vectors inthePlane.............. 375

7.4.1 Some Mathematical Operations on Vectors . . . . . . 376

7.4.2 Implementation ............................. 376

7.4.3 Usage ...................................... 378

7.5 Example:Classfor Complex Numbers ............... 379

7.5.1 Implementation ............................. 380

7.5.2 IllegalOperations............................ 381

7.5.3 MixingComplexandReal Numbers............ 382

7.5.4 Special Methods for “Right” Operands . . . . . . . . . 384

7.5.5 InspectingInstances ......................... 386

7.6 StaticMethodsand Attributes...................... 387

7.7 Summary ........................................ 388

7.7.1 ChapterTopics.............................. 388

7.7.2 Example:IntervalArithmetics................. 389

7.8 Exercises......................................... 395

8 Random Numbers and Simple Games ............... 413

8.1 DrawingRandomNumbers......................... 414

8.1.1 TheSeed ................................... 414

8.1.2 Uniformly Distributed Random Numbers . . . . . . . 415

8.1.3 VisualizingtheDistribution................... 416

8.1.4 Vectorized Drawing of Random Numbers . . . . . . . 417

8.1.5 Computing the Mean and Standard Deviation . . . 418

8.1.6 The Gaussian or Normal Distribution . . . . . . . . . . 419

8.2 DrawingIntegers.................................. 420

8.2.1 RandomIntegerFunctions.................... 421

8.2.2 Example:ThrowingaDie..................... 422

8.2.3 Drawing a Random Element from a List . . . . . . . . 422

8.2.4 Example: Drawing Cards from a Deck . . . . . . . . . . 423

8.2.5 Example: Class Implementation of a Deck . . . . . . 425

8.3 Computing Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

8.3.1 Principles of Monte Carlo Simulation . . . . . . . . . . . 428

8.3.2 Example:ThrowingDice ..................... 429

8.3.3 Example: Drawing Balls from a Hat . . . . . . . . . . . . 432

8.3.4 RandomMutationsofGenes .................. 434

8.3.5 Example: Policies for Limiting Population Growth 439

8.4 SimpleGames .................................... 442

Contents xvii

8.4.1 GuessingaNumber .......................... 442

8.4.2 Rolling Two Dice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

8.5 MonteCarlo Integration ........................... 446

8.5.1 StandardMonteCarloIntegration ............. 446

8.5.2 Area Computing by Throwing Random Points . . . 448

8.6 RandomWalk inOneSpaceDimension .............. 450

8.6.1 BasicImplementation ........................ 451

8.6.2 Visualization................................ 452

8.6.3 Random Walk as a Diﬀerence Equation . . . . . . . . . 452

8.6.4 Computing Statistics of the Particle Positions . . . 453

8.6.5 VectorizedImplementation.................... 454

8.7 RandomWalk inTwoSpaceDimensions ............. 456

8.7.1 BasicImplementation ........................ 456

8.7.2 VectorizedImplementation.................... 457

8.8 Summary ........................................ 459

8.8.1 ChapterTopics.............................. 459

8.8.2 Example:RandomGrowth.................... 460

8.9 Exercises......................................... 466

9 Object-Oriented Programming ...................... 483

9.1 Inheritanceand Class Hierarchies ................... 483

9.1.1 AClassfor Straight Lines .................... 484

9.1.2 A First Try on a Class for Parabolas . . . . . . . . . . . 485

9.1.3 A Class for Parabolas Using Inheritance . . . . . . . . 485

9.1.4 CheckingtheClassType ..................... 487

9.1.5 AttributeVersusInheritance .................. 488

9.1.6 Extending Versus Restricting Functionality . . . . . 489

9.1.7 Superclass for Deﬁning an Interface . . . . . . . . . . . . 490

9.2 Class Hierarchy for Numerical Diﬀerentiation . . . . . . . . . 492

9.2.1 ClassesforDiﬀerentiation..................... 493

9.2.2 AFlexibleMainProgram..................... 496

9.2.3 Extensions.................................. 497

9.2.4 Alternative Implementation via Functions . . . . . . . 500

9.2.5 Alternative Implementation via Functional

Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

9.2.6 Alternative Implementation via a Single Class . . . 502

9.3 ClassHierarchy for Numerical Integration ............ 504

9.3.1 NumericalIntegrationMethods................ 504

9.3.2 ClassesforIntegration........................ 505

9.3.3 UsingtheClass Hierarchy .................... 509

9.3.4 About Object-Oriented Programming . . . . . . . . . . 511

9.4 ClassHierarchy for MakingDrawings ................ 513

9.4.1 UsingtheObjectCollection................... 514

9.4.2 Example of Classes for Geometric Objects . . . . . . 523

9.4.3 Adding Functionality via Recursion . . . . . . . . . . . . 528

xviii Contents

9.4.4 Scaling, Translating, and Rotating a Figure . . . . 531

9.5 Classesfor DNA Analysis .......................... 534

9.5.1 Classfor Regions ........................... 534

9.5.2 Classfor Genes............................. 534

9.5.3 Subclasses ................................. 539

9.6 Summary ........................................ 541

9.6.1 ChapterTopics............................. 541

9.6.2 Example: Input Data Reader . . . . . . . . . . . . . . . . . 542

9.7 Exercises......................................... 548

A Sequences and Diﬀerence Equations ................. 557

A.1 Mathematical Models Based on Diﬀerence Equations . . 558

A.1.1 InterestRates.............................. 559

A.1.2 The Factorial as a Diﬀerence Equation . . . . . . . . 561

A.1.3 FibonacciNumbers ......................... 562

A.1.4 Growthofa Population ..................... 563

A.1.5 LogisticGrowth ............................ 564

A.1.6 Paybackofa Loan.......................... 566

A.1.7 The Integral as a Diﬀerence Equation . . . . . . . . . 567

A.1.8 TaylorSeriesasa Diﬀerence Equation......... 569

A.1.9 MakingaLiving from aFortune.............. 571

A.1.10 Newton’sMethod........................... 571

A.1.11 TheInverseofa Function.................... 575

A.2 Programming with Sound . . . . . . . . . . . . . . . . . . . . . . . . . . 577

A.2.1 WritingSoundto File ....................... 578

A.2.2 ReadingSoundfrom File .................... 579

A.2.3 PlayingManyNotes ........................ 580

A.2.4 Musicofa Sequence ........................ 580

A.3 Exercises......................................... 583

B Introduction to Discrete Calculus ................... 593

B.1 DiscreteFunctions................................. 593

B.1.1 TheSineFunction .......................... 594

B.1.2 Interpolation............................... 596

B.1.3 EvaluatingtheApproximation ............... 596

B.1.4 Generalization ............................. 597

B.2 Diﬀerentiation Becomes Finite Diﬀerences . . . . . . . . . . . . 599

B.2.1 DiﬀerentiatingtheSineFunction ............. 600

B.2.2 Diﬀerencesona Mesh....................... 600

B.2.3 Generalization ............................. 602

B.3 Integration Becomes Summation . . . . . . . . . . . . . . . . . . . . 603

B.3.1 Dividing into Subintervals . . . . . . . . . . . . . . . . . . . 604

B.3.2 Integration on Subintervals . . . . . . . . . . . . . . . . . . 605

B.3.3 Adding the Subintervals . . . . . . . . . . . . . . . . . . . . . 606

B.3.4 Generalization ............................. 607

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