Programming, Modeling and Simulation in Python

I Python programming 15

 

1 Introduction to Python 17

1.1 Installing Python 17 

1.2 Spyder development environment 17 

1.3 Packages and modules 18 

1.4 Interactive runs – Python as a calculator 19 

1.5 Scripts and py-files 20 

1.6 TAB-completion 22 

1.7 Good to know commands 23 

1.8 Variable explorer – read and write to file 24 

1.9 LaTeX – document programs 26 

1.10 Study questions 27 

1.11 Exercises 28

 

2 Numbers and functions 31

2.1 Different types of numbers 31 

2.2 Arithmetic 32 

2.3 Functions of real numbers 35 

2.4 Functions of complex numbers 37 

2.5 Complex numbers – attributes and methods 38 

2.6 Lambda functions 40 

2.7 Significant digits and cancellation 41 

2.8 Study questions 42 

2.9 Exercises 43

 

3 Data types, variables, and assignments 47

3.1 Numbers, strings, lists, and tuples 47 

3.2 Assignments 48 

3.3 Assignments in technical terms 48 

3.4 Assignments – value, type, and reference 49 

3.5 Assignments – stepping 51 

3.6 Tuples and multiple assignments 51

3.7 Assignments of complex expressions 52 

3.8 Variable names 53 

3.9 Reading and writing to the screen 54 

3.10 Application – replacing windows 56 

3.11 Study questions 57 

3.12 Exercises 58

 

4 Lists, tuples and strings 63

4.1 Indexation of lists, tuples and strings 63 

4.2 Slices of lists, tuples and strings 66 

4.3 Slices – how to think 68 

4.4 Lists – mutability 69 

4.5 Lists with numbers – list comprehension 71 

4.6 Classes and methods 73 

4.7 Methods for lists 74 

4.8 Methods for strings 76 

4.9 Study questions 80 

4.10 Exercises 81

 

5 Vectors, matrices and multidimensional arrays 85

5.1 NumPy 85 

5.2 Arrays 85 

5.3 Arrays – attribute 87 

5.4 Vectors 87 

5.5 Range and linspace 89 

5.6 Subvectors 91 

5.7 Vectors – removing elements 93 

5.8 Vectors – mutability 93 

5.9 Matrices 95 

5.10 Zero and one matrices 96 

5.11 Diagonal and banded matrices 97 

5.12 Random matrices 98 

5.13 Reshaping matrices 100 

5.14 Row matrices versus vectors 103 

5.15 Submatrices 104 

5.16 Stacking matrices 106 

5.17 Writing and reading to file 108 

5.18 Arithmetic operations on arrays 112 

5.19 Element-wise functions 114 

5.20 Aggregation and localization functions 116 

5.21 Methods for n ndarray 120 

5.22 Application – temperature 121 

5.23 Study questions 124 

5.24 Exercises 125

 

6 Graphics and visualization 135

6.1 Matplotlib – pyplot 135

6.2 Example gallery 136

6.3 Create and save a figure 137

6.4 Basic plot methods 139

6.5 Axes and scaling 144

6.6 Text and legend 148

6.7 Arrows and explaining text 151

6.8 Polygons and filled areas 152

6.9 Histogram and bar charts 153

6.10 Plots of two-dimensional functions 157

6.11 Plot methods for 3D-graphics 158

6.12 Contour plots 162

6.13 Implicit functions 165

6.14 Matrices and images 167

6.15 Animations 170

6.16 Interaction with images 173

6.17 Graphics – general advice 175

6.18 Study questions 176

6.19 Exercises 177

 

7 Programming 185

7.1 Logical expressions 185 

7.2 If-statements 188

7.3 While-loops 190

7.4 For-loops 193

7.5 Pre-allocation – element counter 197

7.6 Break commands and flags 198

7.7 Break-loops 199

7.8 Nested loops 201

7.9 Application – Madelung constant 202

7.10 Application – signal processing 204

7.11 Application – image processing 206

7.12 Application – heat flow 208

7.13 Study questions 209

7.14 Exercises 210

 

8 Program structure 225

8.1 Program, functions, and modules 225

8.2 Functions 226

8.3 Function calls 227

8.4 Functions – mutability 231

8.5 Functions – undefined local variables 232

8.6 Function names as input variables 233

8.7 Different number of input variables – keyword arguments 235

8.8 Functions collected in modules 237

8.9 The modules search path 238

8.10 Application – symmetries 239

8.11 Application – area on the map 242

8.12 Study questions 245

8.13 Exercises 246

 

9 Vectorization and efficiency 253

9.1 Measuring execution time 253

9.2 Vectorization 254

9.3 Vectorized operations under masks 256

9.4 Extracting indexes and counting 258

9.5 Application – vectorized heat flow 260

9.6 Application – digital image classification 261

9.7 Study questions 263

9.8 Exercises 264

 

10 Object-oriented programming 269

10.1 Classes 269

10.2 Protecting attributes – getters and setters 276

10.3 Special methods 281

10.4 Subclasses – inheritance 287

10.5 Application – functions with parameters 291

10.6 Study questions 293

10.7 Exercises 294

 

II Computer mathematics 297

 

11 Linear algebra 299

11.1 NumPy – linear algebra 299

11.2 Coordinate systems 300

11.3 Computation with vectors 302

11.4 Scalar product 304

11.5 Normalized vectors 305

11.6 Vectors in Python 305

11.7 Matrices 306

11.8 Matrix operations 307

11.9 Inverse matrix 310

11.10 Transpose and Hermitian transpose 312

11.11 Orthogonal and unitary matrices 314

11.12 Symmetric and Hermitian matrices 314

11.13 Determinants 315

11.14 Linear equation systems 317

11.15 Ill-conditioned systems 320

11.16 Eigenvalues and eigenvectors 323

11.17 Numerical method – Gaussian elimination 327

11.18 Numerical method – eigenvalues 330

11.19 Application – dynamical system 332

11.20 Application – free oscillations 337

11.21 Application – forced oscillations 341

11.22 Study questions 344

11.23 Exercises 345

 

12 Functions of one variable 353

12.1 Functions of one variable 353

12.2 Function graph 354

12.3 Continuity and zeros 355

12.4 Derivative 356

12.5 Derivatives of higher order 358

12.6 Taylor polynomials 359

12.7 Maximum and minimum 361

12.8 Grid – finite differences 362

12.9 Finite differences – method of undetermined coefficients 365

12.10 Integrals 368

12.11 The truncation error for the middlesum 370

12.12 Richardson extrapolation 371

12.13 Vector valued functions 375

12.14 Application – graphs of derivatives 377

12.15 Application – zeros 378

12.16 Study questions 380

12.17 Exercises 380

 

13 Functions of several variables 385

13.1 Functions of two variables 385

13.2 Domains in the plane 386

13.3 Function graph 387

13.4 Level curves 389

13.5 Continuity 390

13.6 Derivative 391

13.7 Gradient 392

13.8 Derivatives of higher order 394

13.9 Tangent plane and Taylor polynomials 395

13.10 Maximum and minimum 397

13.11 Grid – finite differences 399

13.12 Double integrals 401

13.13 Double integrals over general domains 403

13.14 Functions of n variables 405

13.15 Vector valued functions of n variables 413

13.16 Study questions 415

13.17 Exercises 416

 

14 Symbolic mathematics 421

14.1 SymPy 421

14.2 Print-out formats and interoperability 422

14.3 Computation with rational numbers 424

14.4 Computation with complex numbers 424

14.5 Symbolic variables 426

14.6 Algebraic manipulations – substitutions 427

14.7 Linear algebra 429

14.8 Functions of one variable 436

14.9 Functions of several variables 444

14.10 Study questions 448

14.11 Exercises 449

 

III Modeling and simulation 453

 

15 Introduction to modeling and simulation with SciPy 455

15.1 Modeling and simulation 455

15.2 Advantages with simulations 456

15.3 Different types of models and simulations 456

15.4 Visualizations 457

15.5 Challenges 458

15.6 SciPy 459

15.7 Import subpackages 460

15.8 Application – changing conditions 461

15.9 Application – epidemiology 462

15.10 Study questions 464

15.11 Exercises 465

 

16 Interpolation 467

16.1 Polynomials 467

16.2 Interpolation with polynomials 470

16.3 Interpolation with splines 475

16.4 Interpolation in two dimensions 477

16.5 Interpolation with periodic functions 482

16.6 Discrete Fourier transform 483

16.7 Nyquist frequency 485

16.8 Fast Fourier Transform (FFT) 486

16.9 Discrete Fourier transform of images 489

16.10 Study questions 495

16.11 Exercises 496

 

17 Non-linear equations 501

17.1 Non-lineare equations 501

17.2 Systems of non-lineare equations 502

17.3 Conditioning 504

17.4 Built-in functions in Python 504

17.5 Numerical methods 511

17.6 Application – Lagrange points 518

17.7 Study questions 520

17.8 Exercises 520

 

18 Optimization 527

18.1 Global and local minima 527

18.2 Univariate optimization 528

18.3 Multivariate optimization 530

18.4 Conditioning 532

18.5 Built-in functions in Python 532

18.6 Numerical methods – univariate minimization 541

18.7 Numerical methods – multivariate minimization 546

18.8 Application – reconstruction of surface 550

18.9 Study questions 553

18.10 Exercises 554

 

19 Modeling of data 559

19.1 Least squares fits 559

19.2 Linear least squares 560

19.3 Polynomial model functions 562

19.4 Linear model functions 564

19.5 Non-linear model functions 566

19.6 Multivariate model functions 570

19.7 Weighted least squares fits – outliers 573

19.8 Non-parametric models 576

19.9 Numerical method – Gauss-Newton 580

19.10 Application – video data 583

19.11 Application – breath of the Earth 585

19.12 Study questions 587

19.13 Exercises 588

 

20 Integrals 597

20.1 Interpretation of integrals 597

20.2 Curve length 598

20.3 Solids of revolution 599

20.4 Integrals in Python 600

20.5 Interpretation of multiple integrals 604

20.6 Double and triple integrals in Python 605

20.7 Numerical methods 608

20.8 Application – corrugated plates 617

20.9 Application – diffraction of a telescope 618

20.10 Application – gravitational pull 619

20.11 Study questions 621

20.12 Exercises 622

 

21 Differential equations 629

21.1 Ordinary differential equations 629

21.2 Systems of ordinary differential equations 630

21.3 Higher order differential equations 631

21.4 Initial value problems in Python 632

21.5 Initial value problems – step methods 638

21.6 Boundary value problems 643

21.7 Eigen value problems 647

21.8 Application – population dynamics 651

21.9 Application – wave functions for hydrogen 654

21.10 Study questions 656

21.11 Exercises 658

 

22 Monte Carlo methods 669

22.1 Initial example 669

22.2 Probability density functions 670

22.3 Random number generation 671

22.4 Measure of a domain 675

22.5 Multiple integrals 677

22.6 Application – Earth’s moment of inertia 679

22.7 Application – error propagation 681

22.8 Application – radiation transfer 683

22.9 Application – ideal gas in 2D 686

22.10 Study questions 689

22.11 Exercises 690

 

Appendix

A Installing anaconda 695

 

B Jupyter Notebook 697

B.1 Start Jupyter Notebook 697

B.2 Save Jupyter Notebook 701

B.3 Inserting images and links 702

B.4 Upload data files 702

B.5 Opening a Jupiter Notebook file 703

 

C Numeric data types 705

C.1 Built-in numeric data types in Python 705

C.2 Built-in numeric data types in NumPy 706

 

D Solutions to selected exercises 707

 

Index 711