Page - 65 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

3.3 TheCompositeMidpointMethod 65 Thepresent integralproblemsresult inshortcode. Inmorechallengingengineer- ingproblems the codequicklygrows tohundredsand thousandsof lines. Without abstractions in termsofgeneral algorithms ingeneral reusable functions, the com- plexityof theprogramgrowsso fast that itwill beextremelydifficult tomakesure that theprogramworksproperly. Another advantage of packagingmathematical algorithms in functions is that a functioncanbe reusedbyanyone to solve aproblemby just calling the function with a proper set of arguments. Understanding the function’s inner details is not necessary to compute a new integral. Similarly, youcanfind libraries of functions on the Internet and use these functions to solve your problems without specific knowledgeofeverymathematicaldetail in the functions. This desirable feature has its downside, of course: the user of a functionmay misuse it, and the functionmaycontainprogrammingerrorsand lead towrongan- swers. Testing theoutputofdownloadedfunctions is thereforeextremely important before relyingon the results. 3.3 TheCompositeMidpointMethod The idea Rather thanapproximating theareaundera curveby trapezoids,wecan use plain rectangles (Fig. 3.3). Itmay sound less accurate to use horizontal lines andnotskewlinesfollowingthefunctiontobeintegrated,butanintegrationmethod basedonrectangles (themidpointmethod) is in fact slightlymoreaccurate than the onebasedon trapezoids! Fig.3.3 Computing approximately the integral of a function as the sumof the areas of the rect- angles