Python科学编程入门教程(第5版)：蒙特卡洛模拟与数值计算

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"《Python科学编程入门指南(第5版)》是一本由Hans Petter Langtangen编写的书籍，旨在介绍如何使用Python进行科学计算。书中涵盖了蒙特卡洛模拟、Python编程基础、数值微积分、数值方法、面向对象编程、常微分方程以及向量化等主题。该书属于 Springer 出版社的 'Texts in Computational Science and Engineering' 系列。" 在科学计算领域，Python是一种广泛使用的语言，因为它提供了强大的库和工具，使得复杂的数学和工程问题变得易于处理。本书《Python科学编程入门指南》第五版，是学习如何利用Python进行科学研究和工程计算的绝佳资源。 1. **蒙特卡洛模拟**：这是一种基于随机抽样或统计试验的计算方法，用于解决各种复杂问题，如金融建模、物理模拟等。书中会介绍如何在Python环境中实现蒙特卡洛模拟，以及如何评估其结果的准确性和效率。 2. **Python编程**：Python是易学且功能丰富的编程语言，尤其适合初学者。书中将教授基本的Python语法、数据结构、控制流和函数定义，为后续的科学计算打下坚实的基础。 3. **数值微积分**：包括求解积分、微分方程和优化问题的数值方法。读者将学习如何使用Python库（如NumPy和SciPy）进行这些计算，并理解误差分析和收敛性。 4. **数值方法**：这些是用于解决不能通过解析方法求解的数学问题的算法。书中可能涵盖线性代数、非线性方程组求解、插值和拟合等内容。 5. **面向对象编程**：Python支持面向对象编程，这使得代码组织更有序，可维护性更强。书中会解释类和对象的概念，以及如何创建和使用它们来构建复杂系统。 6. **常微分方程**：这是科学研究中常见的问题类型，例如在生物动力学、物理和工程中的动态系统建模。作者将展示如何使用Python求解常微分方程组，可能包括欧拉法、龙格-库塔方法等。 7. **向量化**：Python中的向量化操作可以极大地提高代码执行速度，特别是在处理大量数据时。书中将探讨NumPy数组和矩阵操作，以及如何利用向量化进行高效的科学计算。此书适合对科学计算感兴趣的初学者，也适用于有一定Python基础但想深入科学计算领域的读者。通过学习，读者不仅可以掌握Python编程技术，还能了解到科学计算的常用方法和技术，为自己的研究或工程实践提供有力工具。

xvi Contents

6.1.7 Example: Reading and Plotting Data Recorded at Speciﬁc

Dates ................................. 347

6.2 Strings ..................................... 351

6.2.1 Common Operations on Strings ................. 351

6.2.2 Example: Reading Pairs of Numbers .............. 355

6.2.3 Example: Reading Coordinates ................. 358

6.3 Reading Data from Web Pages ...................... 360

6.3.1 About Web Pages .......................... 361

6.3.2 How to Access Web Pages in Programs ............ 362

6.3.3 Example: Reading Pure Text Files ............... 363

6.3.4 Example: Extracting Data from HTML ............ 365

6.3.5 Handling Non-English Text .................... 366

6.4 Reading and Writing Spreadsheet Files ................. 369

6.4.1 CSV Files .............................. 369

6.4.2 Reading CSV Files ......................... 370

6.4.3 Processing Spreadsheet Data ................... 371

6.4.4 Writing CSV Files ......................... 372

6.4.5 Representing Number Cells with Numerical Python Arrays 373

6.4.6 Using More High-Level Numerical Python Functionality . 374

6.5 Examples from Analyzing DNA ..................... 375

6.5.1 Computing Frequencies ...................... 375

6.5.2 Analyzing the Frequency Matrix ................. 382

6.5.3 Finding Base Frequencies ..................... 385

6.5.4 Translating Genes into Proteins ................. 388

6.5.5 Some Humans Can Drink Milk, While Others Cannot ... 393

6.6 Making Code that is Compatible with Python 2 and 3 ........ 394

6.6.1 More B asic Differences Between Python 2 and 3 ...... 394

6.6.2 Turning Python 2 Code into Python 3 Code .......... 396

6.7 Summary ................................... 396

6.7.1 Chapter Topics ...........................396

6.7.2 Example: A File Database ....................398

6.8 Exercises ................................... 402

7 Introduction to Classes .............................. 409

7.1 Simple Function Classes .......................... 409

7.1.1 Challenge: Functions with Parameters ............. 410

7.1.2 Representing a Function as a Class

............... 412

7.1.3 The Self Variable .......................... 417

7.1.4 Another Function Class Example ................ 419

7.1.5 Alternative Function Class Implementations ......... 420

7.1.6 Making Classes Without the Class Construct ......... 422

7.1.7 Closures ............................... 424

7.2 More Examples on Classes ........................ 426

7.2.1 Bank Accounts ........................... 426

7.2.2 Phone Book ............................. 428

7.2.3 A Circle ................................ 430

7.3 Special Methods ............................... 432

7.3.1 The Call Special Method .....................432

Contents xvii

7.3.2 Example: Automagic Differentiation .............. 433

7.3.3 Example: Automagic Integration ................ 438

7.3.4 Turning an Instance into a String ................ 440

7.3.5 Example: Phone Book with Special Methods .........441

7.3.6 Adding Objects ........................... 443

7.3.7 Example: Class for Polynomials ................. 443

7.3.8 Arithmetic Operations and Other Special Methods ..... 449

7.3.9 Special Methods for String Conversion ............. 449

7.4 Example: Class for Vectors in the Plane ................ 451

7.4.1 Some Mathematical Operations on Vectors .......... 451

7.4.2 Implementation ........................... 452

7.4.3 Usage ................................. 454

7.5 Example: Class for Complex Numbers ................. 455

7.5.1 Implementation ........................... 455

7.5.2 Illegal Operations .......................... 457

7.5.3 Mixing Complex and R eal Numbers .............. 457

7.5.4 Dynamic, Static, Strong, Weak, and Duck Typing ...... 459

7.5.5 Special Methods for “Right” Operands ............. 460

7.5.6 Inspecting Instances ........................ 461

7.6 Static Methods and Attributes ....................... 463

7.7 Summary ................................... 464

7.7.1 Chapter Topics ........................... 464

7.7.2 Example: Interval Arithmetic .................. 466

7.8 Exercises ................................... 470

8 Random Numbers and Simple Games ....................489

8.1 Drawing Random Numbers ........................ 489

8.1.1 The Seed ............................... 490

8.1.2 Uniformly Distributed Random Numbers ........... 491

8.1.3 Visualizing the Distribution .................... 492

8.1.4 Vectorized Drawing of Random Numbers ........... 493

8.1.5 Computing the Mean and Standard Deviation ......... 494

8.1.6 The Gaussian or Normal Distribution .............. 496

8.2 Drawing Integers ............................... 497

8.2.1 Random Integer Functions ....................

498

8.2.2 Example: Throwing a Die ..................... 498

8.2.3 Drawing a Random Element from a List ............ 501

8.2.4 Example: Drawing Cards from a Deck ............. 502

8.2.5 Example: Class Implementation of a Deck .......... 504

8.3 Computing Probabilities .......................... 507

8.3.1 Principles of Monte Carlo Simulation ............. 507

8.3.2 Example: Throwing Dice .....................508

8.3.3 Example: Drawing Balls from a Hat .............. 511

8.3.4 Random Mutations of Genes ................... 513

8.3.5 Example: Policies for Limiting Population Growth ..... 519

8.4 Simple Games ................................ 522

8.4.1 Guessing a Number ......................... 522

8.4.2 Rolling Two Dice .......................... 523

xviii Contents

8.5 Monte Carlo Integration .......................... 526

8.5.1 Derivation of Monte Carlo In tegratio n ............. 526

8.5.2 Implementation of Standard Monte Carlo Integration .... 528

8.5.3 Area Computing by Throwing Random Points ........ 531

8.6 Random Walk in One Space Dimension ................ 534

8.6.1 Basic Implementation ....................... 534

8.6.2 Visualization ............................. 535

8.6.3 Random Walk as a Difference Equation ............ 536

8.6.4 Computing Statistics of the Particle Positions ......... 536

8.6.5 Vectorized Implementation .................... 537

8.7 Random Walk in Two Space Dimensions ................ 539

8.7.1 Basic Implementation ....................... 539

8.7.2 Vectorized Implementation .................... 541

8.8 Summary ................................... 542

8.8.1 Chapter Topics ...........................542

8.8.2 Example: Random Growth ....................544

8.9 Exercises ................................... 549

9 Object-Oriented Programming ........................ 567

9.1 Inheritance and Class Hierarchies ....................567

9.1.1 A Class for Straight Lines ..................... 568

9.1.2 A First Try on a Class for Parabolas .............. 569

9.1.3 A Class for Parabolas Using Inheritance ............ 569

9.1.4 Checking the Class Type ..................... 571

9.1.5 Attribute vs Inheritance: has-a vs is-a Relationship ..... 572

9.1.6 Superclass for Deﬁning an Interface ..............574

9.2 Class Hierarchy for Numerical Differentiation .............576

9.2.1 Classes for Differentiation .................... 577

9.2.2 Veriﬁcation .............................. 579

9.2.3 A ﬂexible Main Program ..................... 581

9.2.4 Extensions .............................. 582

9.2.5 Alternative Implementation via Functions ...........585

9.2.6 Alternative Implementation via Functional Programming . 586

9.2.7 Alternative Implementation via a Single Class ........587

9.3 Class Hierarchy for Numerical Integration

............... 589

9.3.1 Numerical Integration Methods ................. 589

9.3.2 Classes for Integration ....................... 590

9.3.3 Veriﬁcation .............................. 594

9.3.4 Using the C lass Hierarchy .................... 595

9.3.5 About Object-Oriented Programming ............. 597

9.4 Class Hierarchy for Making Drawings ................. 599

9.4.1 Using the Object Collection ................... 600

9.4.2 Example of Classes for Geometric Objects .......... 609

9.4.3 Adding Functionality via Recursion .............. 614

9.4.4 Scaling, Translating, and Rotating a Figure .......... 618

9.5 Classes for DNA Analysis ......................... 620

9.5.1 Class for Regions .......................... 620

9.5.2 Class for Genes ........................... 621

Contents xix

9.5.3 Subclasses .............................. 626

9.6 Summary ................................... 627

9.6.1 Chapter Topics ........................... 627

9.6.2 Example: I nput Data Reader ...................629

9.7 Exercises ................................... 635

A Sequences and Difference Equations ..................... 645

A.1 Mathematical Models Based on Difference Equations ........ 646

A.1.1 Interest Rates ............................ 647

A.1.2 The Factorial as a Difference Equation ............. 649

A.1.3 Fibonacci Numbers ......................... 650

A.1.4 Growth of a Population ...................... 651

A.1.5 Logistic Growth ........................... 652

A.1.6 Payback of a Loan ......................... 654

A.1.7 The Integral as a Difference Equation ............. 655

A.1.8 Taylor Series as a Difference Equation ............. 657

A.1.9 Making a Living from a Fortune ................. 658

A.1.10Newton’s Method .......................... 659

A.1.11The Inverse of a Function ..................... 663

A.2 Programming with Sound ......................... 665

A.2.1 Writing Sound to File ....................... 666

A.2.2 Reading Sound from File ..................... 667

A.2.3 Playing Many Notes ........................667

A.2.4 Music of a Sequence ........................ 668

A.3 Exercises ................................... 671

B Introduction to Discrete Calculus ....................... 683

B.1 Discrete Functions .............................. 683

B.1.1 The Sine Function ......................... 684

B.1.2 Interpolation ............................. 685

B.1.3 Evaluating the Approximation .................. 686

B.1.4 Generalization ............................ 687

B.2 Differentiation Becomes Finite Differences .............. 688

B.2.1 Differentiating the Sine Function ................ 689

B.2.2 Differences on a Mesh ....................... 690

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

692

B.3 Integration Becomes Summation .....................693

B.3.1 Dividing into Subintervals .................... 693

B.3.2 Integration on Subintervals .................... 695

B.3.3 Adding the Subintervals ...................... 696

B.3.4 Generalization ............................ 697

B.4 Taylor Series .................................699

B.4.1 Approximating Functions Close to One Point ......... 699

B.4.2 Approximating the Exponential Function ........... 699

B.4.3 More Accurate Expansions .................... 700

B.4.4 Accuracy of the Approximation ................. 702

B.4.5 Derivatives Revisited ........................ 704

B.4.6 More Accurate Difference Approximations .......... 705

xx Contents

B.4.7 Second-Order Derivatives ..................... 707

B.5 Exercises ................................... 709

C Introduction to differential equations ....................715

C.1 The simplest case .............................. 716

C.2 Exponential Growth ............................. 718

C.3 Logistic Growth ............................... 723

C.4 A Simple Pendulum ............................. 724

C.5 A Model for the Spreading of a Disease ................ 727

C.6 Exercises ................................... 729

D A Complete Differential Equation Project ................. 731

D.1 About the Problem: Motion and Forces in Physics ..........731

D.1.1 The Physical Problem ....................... 731

D.1.2 The Computational Algorithm .................. 733

D.1.3 Derivation of the Mathematical Model ............. 734

D.1.4 Derivation of the Algorithm ................... 736

D.2 Program Development and Testing ....................737

D.2.1 Implementation ........................... 737

D.2.2 Callback Functionality ....................... 740

D.2.3 Making a Module ..........................742

D.2.4 Veriﬁcation .............................. 743

D.3 Visualization ................................. 746

D.3.1 Simultaneous Computation and Plotting ............ 746

D.3.2 Some Applications ......................... 748

D.3.3 Remark on Choosing t ..................... 749

D.3.4 Comparing Several Quantities in Subplots ........... 750

D.3.5 Comparing Approximate and Exact Solutions ........ 751

D.3.6 Evolution of the Error as t Decreases ............ 752

D.4 Exercises ................................... 755

E Programming of Differential Equations ................... 757

E.1 Scalar Ordinary Differential Equations ................. 758

E.1.1 Examples on Right-Hand-Side Functions ........... 758

E.1.2 The Forward Euler Scheme .................... 759

E.1.3 Function Implementation .....................

760

E.1.4 Verifying the Implementatio n .................. 761

E.1.5 From Discrete to Continuous Solution ............. 763

E.1.6 Switching Numerical Method .................. 764

E.1.7 Class Implementation ....................... 764

E.1.8 Logistic Growth via a Function-Based Approach ...... 769

E.1.9 Logistic Growth via a Class-Based Approach ......... 769

E.2 Systems of Ordinary Differential Equations .............. 772

E.2.1 Mathematical Problem ....................... 773

E.2.2 Example of a System of ODEs .................. 774

E.2.3 Function Implementation ..................... 775

E.2.4 Class Implementation ....................... 777

E.3 The ODESolver Class Hierarchy .....................779

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