A Primer on scientific programming with Python [3rd ed] 9783642302923, 9783642302930, 3642302920

Table of contents :
Cover......Page 1
Title Page......Page 5
Preface......Page 7
Contents......Page 8
Changes to the First Edition......Page 10
Acknowledgments......Page 11
Contents......Page 13
List of Exercises......Page 25
1.1 The First Programming Encounter: A Formula......Page 35
1.1.2 About Programs and Programming......Page 36
1.1.3 Tools for Writing Programs......Page 37
1.1.4 Using Idle to Write the Program......Page 38
Warning About Typing Program Text......Page 40
1.1.5 How to Run the Program......Page 41
1.1.7 Using Variables......Page 42
1.1.8 Names of Variables......Page 43
1.1.10 Comments......Page 44
Printf Syntax......Page 45
Format String Syntax......Page 47
1.2 Computer Science Glossary......Page 48
Straightforward Coding of the Formula......Page 53
1.3.2 Objects in Python......Page 54
1.3.3 Avoiding Integer Division......Page 55
1.3.4 Arithmetic Operators and Precedence......Page 56
Problem......Page 57
Two Ways of Importing a Module......Page 58
1.4.2 Example: Using More Mathematical Functions......Page 59
1.4.3 A First Glimpse of Round-Off Errors......Page 60
1.5.1 Using the Python Shell......Page 61
1.5.2 Type Conversion......Page 62
1.5.3 IPython......Page 63
Quick Recovery of Previous Output......Page 64
Remark......Page 65
1.6 Complex Numbers......Page 66
1.6.2 Complex Functions in Python......Page 67
1.6.3 Unified Treatment of Complex and Real Functions......Page 68
Program Files......Page 69
Object Types......Page 70
Common Mathematical Functions......Page 71
Terminology......Page 72
Solution......Page 73
1.7.3 About Typesetting Conventions in This Book......Page 74
What Does It Mean to Solve an Exercise?......Page 75
2.1.1 A Naive Solution......Page 83
2.1.2 While Loops......Page 84
2.1.3 Boolean Expressions......Page 86
2.1.4 Loop Implementation of a Sum......Page 88
2.2.1 Basic List Operations......Page 89
The Nature of For Loops......Page 92
Making the Table......Page 93
2.3.2 The Range Construction......Page 94
2.3.3 For Loops with List Indices......Page 95
2.3.5 List Comprehension......Page 97
2.4 Nested Lists......Page 98
2.4.1 A Table as a List of Rows or Columns......Page 99
Modules for Pretty Print of Objects......Page 100
2.4.3 Extracting Sublists......Page 101
2.4.4 Traversing Nested Lists......Page 103
2.5 Tuples......Page 105
While Loops......Page 106
Nested Lists......Page 107
For Loops......Page 108
Problem......Page 109
Solution......Page 110
2.6.3 How to Find More Python Information......Page 112
2.7 Exercises......Page 113
3.1.1 Functions of One Variable......Page 121
3.1.2 Local and Global Variables......Page 123
3.1.3 Multiple Arguments......Page 125
3.1.4 Multiple Return Values......Page 127
3.1.5 Functions with No Return Values......Page 129
3.1.6 Keyword Arguments......Page 130
Example: Function with Default Parameters......Page 131
Example: Computing a Sum with Default Tolerance......Page 132
3.1.7 Doc Strings......Page 133
3.1.9 Functions as Arguments to Functions......Page 134
The Behavior of the Numerical Derivative as h → 0......Page 135
3.1.10 The Main Program......Page 136
3.1.11 Lambda Functions......Page 137
3.2.1 If-Else Blocks......Page 138
3.2.2 Inline If Tests......Page 140
List Iteration......Page 141
Program Flow......Page 142
While Loops......Page 144
List Comprehensions......Page 145
Using Python's Library......Page 146
Generating Random DNA Strings......Page 147
Measuring CPU Time......Page 148
Keyword Arguments......Page 149
Problem......Page 150
Solution......Page 151
3.5 Exercises......Page 154
4 Input Data and Error Handling......Page 171
4.1.1 Reading Keyboard Input......Page 172
4.1.2 The Magic “eval” Function......Page 173
4.1.3 The Magic “exec” Function......Page 177
4.1.4 Turning String Expressions into Functions......Page 178
4.2.1 Providing Input on the Command Line......Page 179
4.2.2 A Variable Number of Command-Line Arguments......Page 180
4.2.3 More on Command-Line Arguments......Page 181
4.2.4 Option-Value Pairs on the Command Line......Page 182
4.3 Handling Errors......Page 186
4.3.1 Exception Handling......Page 187
Testing for a Specific Exception......Page 188
Examples on Exception Types......Page 189
Example......Page 190
4.4 A Glimpse of Graphical User Interfaces......Page 192
4.5.1 Example: Interest on Bank Deposits......Page 195
4.5.2 Collecting Functions in a Module File......Page 196
Test Block......Page 197
Flexible Test Blocks......Page 198
4.5.3 Using Modules......Page 201
Question and Answer Input......Page 203
Generating Code on the Fly......Page 204
Handling Exceptions......Page 205
Problem......Page 206
Solution......Page 207
Making a Function......Page 210
Making a Module......Page 211
A Flexible Program for Solving f(x) = 0......Page 212
4.7 Exercises......Page 214
5 Array Computing and Curve Plotting......Page 221
5.1.1 The Vector Concept......Page 222
5.1.2 Mathematical Operations on Vectors......Page 223
5.1.3 Vector Arithmetics and Vector Functions......Page 225
5.2.1 Using Lists for Collecting Function Data......Page 227
5.2.2 Basics of Numerical Python Arrays......Page 228
5.2.3 Computing Coordinates and Function Values......Page 229
5.2.4 Vectorization......Page 230
5.3.1 Matplotlib; Pylab......Page 232
Decorating the Plot......Page 233
Plotting Multiple Curves......Page 234
Placing Several Plots in One Figure......Page 235
5.3.2 Matplotlib; Pyplot......Page 236
Importing SciTools and Easyviz......Page 238
Decorating the Plot......Page 240
Plotting Multiple Curves......Page 241
Placing Several Plots in One Figure......Page 242
Example......Page 243
Animation in Matplotlib......Page 245
Remarks on Filenames......Page 246
Movie Formats......Page 247
5.3.5 Curves in Pure Text......Page 248
5.4 Plotting Difficulties......Page 249
Example: The Heaviside Function......Page 250
Example: A Hat Function......Page 251
5.4.2 Rapidly Varying Functions......Page 252
5.5 More Advanced Vectorization of Functions......Page 253
5.5.1 Vectorizing StringFunction Objects......Page 254
Loop......Page 255
Manual Vectorization......Page 256
5.5.3 Vectorization of a Hat Function......Page 258
5.6.1 Copying Arrays......Page 260
5.6.2 In-Place Arithmetics......Page 261
5.6.4 Generalized Indexing......Page 262
Example: Vectorizing a Constant Function......Page 263
5.6.7 Shape Manipulation......Page 264
5.7.1 Matrices and Arrays......Page 265
5.7.2 Two-Dimensional Numerical Python Arrays......Page 266
5.7.4 Two-Dimensional Arrays and Functions of Two Variables......Page 269
5.7.5 Matrix Objects......Page 270
5.8.1 Chapter Topics......Page 271
Making Movies......Page 272
Solution......Page 273
Scaling......Page 275
5.9 Exercises......Page 277
6.1 Reading Data from File......Page 291
For Loop over Lines......Page 292
Reading a File into a String......Page 294
6.1.2 Reading a Mixture of Text and Numbers......Page 295
6.1.3 What Is a File, Really?......Page 296
Pure Text Files......Page 297
Image Files......Page 298
Remarks......Page 299
6.2.1 Making Dictionaries......Page 300
6.2.2 Dictionary Operations......Page 301
Remark......Page 302
6.2.3 Example: Polynomials as Dictionaries......Page 303
Dictionaries with Default Values......Page 305
Ordered Dictionaries......Page 306
Problem......Page 307
Algorithm......Page 308
Implementation......Page 309
Dissection......Page 310
Problem......Page 312
Solution......Page 313
6.3.1 Common Operations on Strings......Page 316
Substitution......Page 317
Strings Are Constant......Page 318
Joining Strings......Page 319
Solution......Page 320
Problem......Page 322
Solution 1: Substring Extraction......Page 323
Solution 3: String Split......Page 324
6.4.1 About Web Pages......Page 325
6.4.2 How to Access Web Pages in Programs......Page 326
6.4.3 Example: Reading Pure Text Files......Page 327
6.4.4 Example: Extracting Data from HTML......Page 329
Solution......Page 330
6.5.2 Standard Input and Output as File Objects......Page 332
Redirecting Standard Input, Output, and Error......Page 333
6.5.3 Reading and Writing Spreadsheet Files......Page 334
Reading CSV Files......Page 335
Remark......Page 337
Representing Number Cells with Numerical Python Arrays......Page 338
6.6.1 Computing Frequencies......Page 339
Separate Frequency Lists......Page 340
Nested List......Page 341
Numerical Python Array......Page 342
Dictionary of Lists......Page 343
Remark......Page 344
Using Arrays and Vectorization......Page 345
6.6.2 Analyzing the Frequency Matrix......Page 346
Dict of Dicts Frequency Matrix......Page 347
6.6.3 Finding Base Frequencies......Page 349
6.6.4 Translating Genes into Proteins......Page 351
6.6.5 Some Humans Can Drink Milk, While Others Cannot......Page 356
File Operations......Page 357
Dictionaries......Page 358
Problem......Page 359
Solution......Page 360
6.8 Exercises......Page 363
7 Introduction to Classes......Page 375
Problem......Page 376
A Bad Solution: Global Variables......Page 377
7.1.2 Representing a Function as a Class......Page 378
Implementation......Page 379
Usage and Dissection......Page 380
The self Variable......Page 381
Remark......Page 382
Doc Strings......Page 383
7.1.3 Another Function Class Example......Page 384
7.1.4 Alternative Function Class Implementations......Page 385
7.1.5 Making Classes Without the Class Construct......Page 387
Second Remark......Page 389
7.2.1 Bank Accounts......Page 390
7.2.2 Phone Book......Page 392
7.2.3 A Circle......Page 393
7.3 Special Methods......Page 394
Problem......Page 395
Solution......Page 396
Application: Newton's Method......Page 397
7.3.3 Example: Automagic Integration......Page 398
A Simple Implementation......Page 399
7.3.4 Turning an Instance into a String......Page 400
7.3.5 Example: Phone Book with Special Methods......Page 401
Remark......Page 402
Implementation......Page 403
Usage......Page 405
Pretty Print of Polynomials......Page 406
7.3.8 Arithmetic Operations and Other Special Methods......Page 407
7.3.9 Special Methods for String Conversion......Page 408
7.4 Example: Class for Vectors in the Plane......Page 409
7.4.2 Implementation......Page 410
7.4.3 Usage......Page 412
7.5 Example: Class for Complex Numbers......Page 413
7.5.1 Implementation......Page 414
7.5.2 Illegal Operations......Page 415
7.5.3 Mixing Complex and Real Numbers......Page 416
Computer Science Discussion......Page 417
7.5.4 Special Methods for “Right” Operands......Page 418
Remark......Page 419
7.5.5 Inspecting Instances......Page 420
7.6 Static Methods and Attributes......Page 421
Classes......Page 422
7.7.2 Example: Interval Arithmetics......Page 423
Solution......Page 425
7.8 Exercises......Page 429
8 Random Numbers and Simple Games......Page 447
8.1.1 The Seed......Page 448
8.1.2 Uniformly Distributed Random Numbers......Page 449
8.1.3 Visualizing the Distribution......Page 450
Warning......Page 451
8.1.5 Computing the Mean and Standard Deviation......Page 452
8.1.6 The Gaussian or Normal Distribution......Page 453
8.2 Drawing Integers......Page 454
8.2.1 Random Integer Functions......Page 455
8.2.3 Drawing a Random Element from a List......Page 456
8.2.4 Example: Drawing Cards from a Deck......Page 457
8.2.5 Example: Class Implementation of a Deck......Page 459
8.3.1 Principles of Monte Carlo Simulation......Page 462
Straightforward Solution......Page 463
A Game......Page 464
Decide If a Game Is Fair......Page 465
8.3.3 Example: Drawing Balls from a Hat......Page 466
Vectorized Version......Page 468
A Markov Chain Mutation Model......Page 470
8.3.5 Example: Policies for Limiting Population Growth......Page 473
The Implementation......Page 476
The Implementation......Page 477
A Class Version......Page 478
8.5.1 Standard Monte Carlo Integration......Page 480
8.5.2 Area Computing by Throwing Random Points......Page 482
8.6 Random Walk in One Space Dimension......Page 484
8.6.1 Basic Implementation......Page 485
8.6.3 Random Walk as a Difference Equation......Page 486
8.6.4 Computing Statistics of the Particle Positions......Page 487
8.6.5 Vectorized Implementation......Page 488
8.7.1 Basic Implementation......Page 490
8.7.2 Vectorized Implementation......Page 491
Typical Probability Computation......Page 493
Problem......Page 494
Solution......Page 495
8.9 Exercises......Page 500
9.1 Inheritance and Class Hierarchies......Page 517
9.1.1 A Class for Straight Lines......Page 518
9.1.3 A Class for Parabolas Using Inheritance......Page 519
9.1.4 Checking the Class Type......Page 521
9.1.5 Attribute Versus Inheritance......Page 522
9.1.6 Extending Versus Restricting Functionality......Page 523
9.1.7 Superclass for Defining an Interface......Page 524
9.2 Class Hierarchy for Numerical Differentiation......Page 526
9.2.1 Classes for Differentiation......Page 527
Computer Science Remark......Page 529
9.2.2 A Flexible Main Program......Page 530
9.2.3 Extensions......Page 531
Application......Page 532
9.2.4 Alternative Implementation via Functions......Page 534
9.2.5 Alternative Implementation via Functional Programming......Page 535
9.2.6 Alternative Implementation via a Single Class......Page 536
Remark......Page 537
9.3.1 Numerical Integration Methods......Page 538
9.3.2 Classes for Integration......Page 539
9.3.3 Using the Class Hierarchy......Page 543
9.3.4 About Object-Oriented Programming......Page 545
9.4 Class Hierarchy for Making Drawings......Page 547
Basic Drawing......Page 548
Groups of Objects......Page 549
The Figure Composition as an Object Hierarchy......Page 550
Animation: Translating the Vehicle......Page 553
Animation: Rolling the Wheels......Page 555
Simple Geometric Objects......Page 557
Class Curve......Page 559
Compound Geometric Objects......Page 560
Basic Principles of Recursion......Page 562
Explaining Recursion......Page 563
Scaling......Page 565
Translation......Page 566
Rotation......Page 567
9.5.2 Class for Genes......Page 568
Basic Features of class Gene......Page 569
Flexible Constructor......Page 570
Other Methods......Page 571
9.5.3 Subclasses......Page 573
Acknowledgments......Page 574
Subclassing in General......Page 575
Problem......Page 576
Solution......Page 577
Reading from File......Page 578
Reading from the Command Line......Page 579
More Flexibility in the Superclass......Page 580
Demonstrating the Tool......Page 581
9.7 Exercises......Page 582
A Sequences and Difference Equations......Page 591
A.1 Mathematical Models Based on Difference Equations......Page 592
A.1.1 Interest Rates......Page 593
A.1.2 The Factorial as a Difference Equation......Page 595
A.1.3 Fibonacci Numbers......Page 596
A.1.4 Growth of a Population......Page 597
A.1.5 Logistic Growth......Page 598
A.1.6 Payback of a Loan......Page 600
A.1.7 The Integral as a Difference Equation......Page 601
A.1.8 Taylor Series as a Difference Equation......Page 603
A.1.10 Newton's Method......Page 605
A.1.11 The Inverse of a Function......Page 609
A.2 Programming with Sound......Page 611
A.2.1 Writing Sound to File......Page 612
A.2.2 Reading Sound from File......Page 613
Problem......Page 614
Solution......Page 615
A.3 Exercises......Page 617
B.1 Discrete Functions......Page 627
B.1.1 The Sine Function......Page 628
B.1.3 Evaluating the Approximation......Page 630
B.1.4 Generalization......Page 631
B.2 Differentiation Becomes Finite Differences......Page 633
B.2.2 Differences on a Mesh......Page 634
B.2.3 Generalization......Page 636
B.3 Integration Becomes Summation......Page 637
B.3.1 Dividing into Subintervals......Page 638
B.3.2 Integration on Subintervals......Page 639
B.3.3 Adding the Subintervals......Page 640
B.3.4 Generalization......Page 641
B.4.2 Approximating the Exponential Function......Page 643
B.4.3 More Accurate Expansions......Page 644
B.4.4 Accuracy of the Approximation......Page 646
B.4.5 Derivatives Revisited......Page 648
B.4.6 More Accurate Difference Approximations......Page 649
B.4.7 Second-Order Derivatives......Page 651
B.5 Exercises......Page 653
C Introduction to Differential Equations......Page 659
C.1 The Simplest Case......Page 660
C.2 Exponential Growth......Page 662
Numerical Solution......Page 664
C.3 Logistic Growth......Page 667
C.4 A Simple Pendulum......Page 668
C.5 A Model for the Spread of a Disease......Page 671
C.6 Exercises......Page 673
D.1.1 The Physical Problem......Page 675
The Key Quantities......Page 677
D.1.3 Derivation of the Mathematical Model......Page 678
D.1.4 Derivation of the Algorithm......Page 680
D.2.1 Implementation......Page 682
D.2.2 Callback Functionality......Page 685
D.2.3 Making a Module......Page 686
D.2.4 Verification......Page 687
D.3.1 Simultaneous Computation and Plotting......Page 689
Fixing Axes......Page 690
D.3.3 Remark on Choosing Δt......Page 692
D.3.4 Comparing Several Quantities in Subplots......Page 693
D.3.5 Comparing Approximate and Exact Solutions......Page 694
D.3.6 Evolution of the Error as Δt Decreases......Page 695
D.4 Exercises......Page 699
E Programming of Differential Equations......Page 701
E.1.1 Examples on Right-Hand-Side Functions......Page 702
E.1.2 The Forward Euler Scheme......Page 704
E.1.4 Verifying the Implementation......Page 705
E.1.5 From Discrete to Continuous Solution......Page 706
E.1.6 Switching Numerical Method......Page 707
Class Wrapping of a Function......Page 708
A More Flexible Class......Page 709
E.2 Systems of Ordinary Differential Equations......Page 711
E.2.1 Mathematical Problem......Page 712
E.2.2 Example of a System of ODEs......Page 714
Function Implementation......Page 715
Class Implementation......Page 716
E.2.4 Numerical Methods......Page 718
E.2.5 The ODE Solver Class Hierarchy......Page 719
The Superclass......Page 720
Remark......Page 721
E.2.6 The Backward Euler Method......Page 722
E.2.7 Application 1: u' = -u......Page 725
Basic Problem and Solver Classes......Page 727
Computing an Appropriate Δt......Page 729
Dealing with Time-Dependent Coefficients......Page 731
Reading Input......Page 732
E.2.9 Application 3: An Oscillating System......Page 734
E.2.10 Application 4: The Trajectory of a Ball......Page 736
E.2.11 Further Developments of ODESolver......Page 738
E.3 Exercises......Page 739
F.1 Using a Debugger......Page 769
F.2.1 A Recipe for Program Writing and Debugging......Page 772
Problem......Page 774
Solution......Page 775
Refining the User Interface......Page 784
G Migrating Python to Compiled Code......Page 787
G.1.2 A Scalar Python Implementation......Page 788
G.1.3 A Vectorized Python Implementation......Page 789
G.2.1 A Plain Cython Implementation......Page 791
G.2.2 A Better Cython Implementation......Page 793
G.3.1 Writing a C Program......Page 795
G.3.2 Migrating Loops to C Code via F2PY......Page 796
G.3.3 Migrating Loops to C Code via Cython......Page 798
G.3.4 Comparing Efficiency......Page 799
H.1.2 Executing Python Programs on Unix......Page 801
H.1.3 Executing Python Programs on Windows......Page 803
Gnuplot Plots on Windows......Page 804
H.2 Integer and Float Division......Page 805
H.3 Visualizing a Program with Lumpy......Page 806
H.4 Doing Operating System Tasks in Python......Page 808
H.5 Variable Number of Function Arguments......Page 810
Example......Page 811
Example......Page 813
H.6.1 Making Time Measurements......Page 815
Hardware Information......Page 816
H.6.2 Profiling Python Programs......Page 817
References......Page 819
Index......Page 821