A Student’s Guide to Python for Physical Modeling A Student’s Guide to Python for Physical Modeling Jesse M. Kinder and Philip Nelson Princeton University Press Princeton and Oxford Copyright © 2015 by Princeton University Press. All rights associated with the computer code in this work are retained by Jesse M. Kinder and Philip Nelson. Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved ISBN 978-0-691-16958-3 ISBN (pbk.) 978-0-691-17050-3 British Library Cataloging-in-Publication Data is available. A This book has been composed using the LT X typesetting system. E The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ Printed in the United States of America 1 3 5 7 9 10 8 6 4 2 The front cover shows a Mandelbrot set, an image that you will be able to generate for yourself after you work through this book. Although the authors have sought to provide accurate and up- to-date information, they assume no responsibility for errors or omissions, nor any liability for damages resulting from the use of the information contained within this document. The authors make no claim that suggestions and code samples described herein will work in future versions of Python or its extended environments. For Oliver Arthur Nelson – PN Contents Let’s Go xiii 1 Getting Started with Python 1 1.1 Algorithms and algorithmic thinking 1 1.1.1 Algorithmic thinking 1 1.1.2 States 2 1.1.3 What does a = a + 1 mean? 3 1.1.4 Symbolic versus numerical 4 1.2 Launch Python 4 1.2.1 IPython console 5 1.2.2 Error messages 9 1.2.3 Sources of help 9 1.2.4 Good practice: Keep a log 10 1.3 Python modules 11 1.3.1 import 11 1.3.2 from ... import 11 1.3.3 NumPy and PyPlot 12 1.4 Python expressions 13 1.4.1 Numbers 13 1.4.2 Arithmetic operations and predefined functions 13 1.4.3 Good practice: Variable names 14 1.4.4 More about functions 15 2 Structure and Control 17 2.1 Objects and their methods 17 2.2 Lists, tuples, and arrays 19 2.2.1 Creating a list or tuple 19 2.2.2 NumPy arrays 19 2.2.3 Filling an array with values 21 2.2.4 Concatenation of arrays 22 2.2.5 Accessing array elements 23 2.2.6 Arrays and assignments 24 2.2.7 Slicing 25 vii viii 2.2.8 Flattening an array 26 2.2.9 Reshaping an array 26 2.2.10 Lists and arrays as indices 26 2.3 Strings 27 2.3.1 Formatting strings with the format() method 29 2.3.2 Formatting strings with % 30 2.4 Loops 30 2.4.1 for loops 30 2.4.2 while loops 32 2.4.3 Very long loops 32 2.4.4 Infinite loops 32 2.5 Array operations 33 2.5.1 Vectorizing math 33 2.5.2 Reducing an array 35 2.6 Scripts 36 2.6.1 The Editor 36 2.6.2 Other editors 36 2.6.3 First steps to debugging 37 2.6.4 Good practice: Commenting 39 2.6.5 Good practice: Using named parameters 42 2.6.6 Good practice: Units 43 2.7 Contingent behavior: Branching 43 2.7.1 The if statement 44 2.7.2 On truth 45 2.8 Nesting 45 3 Data In, Results Out 46 3.1 Importing data 46 3.1.1 Obtaining data 47 3.1.2 Bringing data into Python 47 3.2 Exporting data 49 3.2.1 Scripts 50 3.2.2 Data files 50 3.3 Visualizing data 52 3.3.1 The plot command and its relatives 52 3.3.2 Manipulate and embellish 55 3.3.3 Error bars 57 3.3.4 3D graphs 57 3.3.5 Multiple plots 57 3.3.6 Subplots 59 3.3.7 Saving figures 59 3.3.8 Using figures in other applications 60 Jump to Contents Jump to Index T 2 ix 4 First Computer Lab 61 4.1 HIV example 61 4.1.1 Explore the model 61 4.1.2 Fit experimental data 62 4.2 Bacterial example 63 4.2.1 Explore the model 63 4.2.2 Fit experimental data 63 5 More Python Constructions 65 5.1 Writing your own functions 65 5.1.1 Defining functions in Python 66 5.1.2 Updating functions 68 5.1.3 Arguments, keywords, and defaults 68 5.1.4 Return values 69 5.1.5 Functional programming 70 5.2 Random numbers and simulation 71 5.2.1 Simulating coin flips 71 5.2.2 Generating trajectories 72 5.3 Histograms and bar graphs 72 5.4 Contour plots and surfaces 74 5.4.1 Generating a grid of points 74 5.4.2 Contour plots 74 5.4.3 Surface plots 75 5.5 Numerical solution of nonlinear equations 75 5.5.1 General real functions 76 5.5.2 Complex roots of polynomials 77 5.6 Solving systems of linear equations 77 5.7 Numerical integration 78 5.7.1 Integrating a predefined function 78 5.7.2 Integrating your own function 79 5.7.3 Oscillatory integrands 79 5.7.4 Parameter dependence 80 5.8 Numerical solution of differential equations 80 5.8.1 Reformulating the problem 81 5.8.2 Solving an ODE 81 5.8.3 Parameter dependence 83 5.9 Vector fields and streamlines 83 5.9.1 Vector fields 84 5.9.2 Streamlines 85 6 Second Computer Lab 86 6.1 Generating and plotting trajectories 86 Jump to Contents Jump to Index T 2