Page - 146 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

146 6 ComputingIntegralsandTestingCode Remark The trapezoidal and midpoint methods are just two examples in a jungle of numerical integration rules. Other famous methods are Simpson’s rule and Gaussquadrature.Theyall work in the sameway: ∫ b a f(x)dx≈ n−1∑ i=0 wif(xi). That is, the integral is approximatedby a sum of functionevaluations,where each evaluationf(xi) is given a weight wi. The different methods differ in the way they construct the evaluation points xi and the weights wi. Higher accuracycanbeobtainedbyoptimizing the locationofxi. 6.4 VectorizingtheFunctions Thefunctionsmidpointandtrapezoidalusuallyrunfast inPythonandcompute an integral to satisfactory precision within a fraction of a second. However, long loops in Pythonmay runslowly in more complicated implementations.To increase speed, the loopscanbereplacedbyvectorizedcode.The integrationfunctionsoffer simpleandgoodexamplesonhowtovectorize loops. We have already seen simple examples on vectorization in Sect.1.5, when we evaluateda mathematical functionf(x) fora largenumberofx values stored in an array.Basically,we canwrite def f(x): return exp(-x)*sin(x) + 5*x from numpy import exp, sin, linspace x = linspace(0, 4, 101) # coordinates from 100 intervals on [0, 4] y = f(x) # all points evaluated at once The resulty is an array that, alternatively, could have been computed by runninga for loopover the individualxvaluesandcalledf for eachvalue.Vectorizationes- sentiallyeliminates thisexplicit loopinPython(i.e., the loopingoverxandapplica- tionoff toeachxvalueareinsteadperformedinalibrarywithfast,compiledcode). 6.4.1 VectorizingtheMidpointRule We start by vectorizing themidpoint function, sincetrapezoidal is not equally straightforwardtovectorize.Inbothcases,ourvectorizationwill removetheexplicit loop.The fundamental ideasof thevectorizedalgorithmare to 1. computeandstoreall theevaluationpoints in onearrayx 2. callf(x) toproduceanarrayofcorrespondingfunctionvalues 3. use thesum function to sumup thef(x)values