Page - 131 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

6ComputingIntegralsandTestingCode We now turn our prime attention to the solving of mathematical problems through computer programming. A fundamentally important part of programming, is to checkthat thewrittencodeworksas intended.That is, thecodemustbe tested.This far, we have checked our coding in rather simple ways, e.g., by comparing to hand calculations. Now, we will look at more powerful test strategies, while addressing numericalcomputationof integrals. Therearemanyreasons tochoose integrationasourfirst application. Integration is well known already from high school mathematics. Most integrals are not tractablebypenandpaper,andacomputerizedsolutionapproachisbothverymuch simplerandmuchmorepowerful—youcanessentially treatall integrals ∫b a f(x)dx in10 linesofcomputercode! Integration also demonstrates the difference between exact mathematics by pen and paper and numerical mathematics on a computer. The latter approaches the resultof theformerwithoutanyworriesaboutroundingerrorsduetofiniteprecision arithmetics incomputers(incontrast todifferentiation,wheresucherrorspreventus fromgettinga result as accurateaswe desire). Finally, integration is thought of as a somewhat difficult mathematical concept tograsp,andprogrammingintegrationshouldgreatly improvetheunderstandingof what integrationreally isandhowit works. Not onlyshall we understandhow to use the computer to integrate,but we shall also learn a series of good habits to ensure your computer work is of the highest scientific quality. In particular,we will have a strong focuson how to write Python code that is freeofprogrammingmistakes. ©The Author(s) 2020 S.Linge, H.P. Langtangen, Programming forComputations -Python, Texts in Computational Science and Engineering 15, https://doi.org/10.1007/978-3-030-16877-3_6 131