Seite - IX - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

Preface ix Target Audience and Background Knowledge This book was written for students, teachers, engineers, and scientists who know nothingaboutprogramming and numerical methods from before but who seek a minimum of the fundamental skills required to get started with programming as a tool for solving scientific and engineering problems. Some knowledge of one- and multivariable calculus is assumed. The basic programming concepts are presented in Chaps.1–5 (about 150 pages), before practical applications of these concepts are demonstrated in important mathematical subjects addressed in the remaining parts of the book (Chaps.6–9).Eachchapter is followedbyasetofexercisesthatcoversawiderange of application areas, e.g., biology, geology, statistics, physics, and mathematics. The exercises were particularly designed to bring across important points from the text. Learning the very basics of programming should not take long, but as with any other craft, mastering the skill requires continued and extensive practice. Some beginningpracticeisgainedthroughChaps.6–9,but theauthorsstronglyemphasize that this is only a start. Students should continue to practice programming in subsequentcourses,while thosewhoexerciseself-studyshouldkeepupthelearning processthroughcontinuedapplicationof thecraft.Thebookisagoodstartingpoint when teaching computer programming as an integrated part of standard university courses in mathematics and natural science. In our experience, such an integration isdoableand indeed rewarding. Numerical Methods An overall goal with this book is to motivate computer pro- grammingasaverypowerfultoolfordoingmathematics.Allexamplesarerelatedto mathematicsanditsuseinengineeringandscience.However, tosolvemathematical problemsthroughcomputerprogramming,weneednumericalmethods.Explaining basic numerical methods is therefore an integral part of the book. Our choice of topics is governed by what is most needed in science and engineering, as well as in the teaching of applied natural science courses. Mathematical models are then central, with differential equations constituting the most frequent type of models. Consequently, the numerical focus in this book is on differential equations.As soft pedagogicalstartersfortheprogrammingofmathematics,wehavechosenthetopics of numerical integration and root finding. We remark that the book is deliberately briefonnumericalmethods.This isbecauseourfocusison implementingnumerical algorithms, and to develop reliable, working programs, the programmer must be confidentabout thebasic ideasof the numericalapproximations involved. The Computer Language: Python We have chosen to use the programming language Python, because this language gives a very compact and readable code that closely resembles the mathematical recipe for solving the problem at hand. Pythonalsohasagentle learningcurve. Other computer languages, like Fortran, C, and C++, have a strong position in science and engineering. During the last two decades, however, there has been a significantshift inpopularityfromthesecompiledlanguagestomorehigh-leveland easier-to-read languages, for instance, MATLAB, Python, R, Maple, Mathematica, andIDL.This latterclassof languages iscomputationally lessefficientbutsuperior with respect to overall human problem-solving efficiency. This book emphasizes