Page - VII - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

Preface vii Targetaudienceandbackgroundknowledge Thisbookwaswrittenforstudents, teachers, engineers and scientists that know nothing about programming and nu- mericalmethods frombefore, butwho seek aminimum of the fundamental skills required to get startedwith programmingas a tool for solving scientific and engi- neeringproblems. Someknowledgeofone-andmulti-variablecalculus isassumed. The basic programming concepts are presented in only 50 pages (Chapters 1 and 2), before practical applications of these concepts are demonstrated in important mathematical subjects addressed in the remainingpartsof thebook(Chapters3-6). Eachchapter is followedbyasetofexercises thatcoverawiderangeofapplication areas,e.g.biology,geology,statistics,physicsandmathematics. Theexerciseswere particularlydesignedtobringacross importantpointsfromthetext. Thereaderwill realize that themodest content of thefirst 50pages can in fact bring youquite far inpowerfulproblemsolving! Learning theverybasics of programmingshouldnot take long, but aswith any other craft, mastering the skill requires continued and extensive practice. Some beginningpractice isgained throughChapters3-6,but theauthorsstronglyempha- size that this is only a start. Students should continue to practice programming in subsequent courses,while thosewhoexercise self-study, shouldkeepup the learn- ingprocess throughcontinuedapplicationof the craft. Thebook is agoodstarting pointwhen teaching computer programmingas an integratedpart of standard uni- versity courses inmathematics and physical sciences. In our experience, such an integration isdoableand indeed rewarding. Numericalmethods Anoverall goalwith this book is tomotivate computer pro- grammingasaverypowerful tool fordoingmathematics. All examplesare related tomathematics and its use in engineering and science. However, to solvemath- ematical problems through computer programming, we need numericalmethods. Explainingbasic numericalmethods is therefore an integral part of the book. Our choiceof topics isgovernedbywhat ismost needed in scienceandengineering, as well as in the teaching of applied physical science courses. Mathematicalmodels are then central, with differential equations constituting themost frequent type of models. Consequently, thenumerical focusin thisbookisondifferentialequations. Asasoftpedagogical starter for theprogrammingofmathematics,wehavechosen the topic of numerical integration. There is also a chapter on root finding, which is important for the numerical solution onnonlinear differential equations. We re- mark that thebook is deliberatelybrief onnumericalmethods. This is becauseour focus is on implementing numerical algorithms, but to develop reliable, working programs, theprogrammermustbeconfidentabout thebasic ideasof thenumerical approximations involved. The computer language: Python Wehave chosen to use the programming lan- guage Python, because this language gives very compact and readable code that closely resembles themathematical recipe for solving theproblemathand. Python also has a gentle learning curve. There is aMATLAB/Octave companion of this book in case that language is preferred. Comparing these twoversionsof thebook provides an excellent demonstration of how similar these languages are. Other computer languages, like Fortran, C, and C++, have a strong position in science andengineering.During the last twodecades,however, therehasbeenasignificant