Page - 132 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

132 6 ComputingIntegralsandTestingCode Calculatingan integral is traditionallydoneby ∫ b a f(x)dx=F(b)−F(a), (6.1) where f(x)= dF dx . The major problem with this procedure is that we need to find an anti-derivative F(x) corresponding to a givenf(x). For some relatively simple integrandsf(x), findingF(x) isadoabletask.Often,however,it isreallychallenging,andsometimes even impossible! Themethod(6.1)providesanexactoranalyticalvalueof theintegral. Ifwerelax the requirement of computing an exact value for the integral, and instead look for approximate values, produced by numerical methods, integration becomes a very straightforward task for almost any given f(x)! In particular, we do not need an anti-derivativeF(x) at all, since it is just the known integrandf(x) that enters the calculations. The (apparent) downside of a numerical method is that it can only find an approximate answer. Leaving the exact for the approximate is a mental barrier in the beginning, but remember that most real applications of integration will involve anf(x) function that containsphysicalparameters,whichare measuredwith some error. That is, f(x) is very seldom exact, and it does not make sense trying to compute the integralwitha smaller error than theonealreadypresent inf(x). Another advantage of numerical methods is that we can easily integrate a functionf(x) that is only known as samples, i.e., discrete valuesat somex points, and not as a continuous function ofx expressed through a formula. This is highly relevantwhenf is measured inaphysical experiment. 6.1 BasicIdeasofNumerical Integration We consider the integral ∫ b a f(x)dx. (6.2) Most numerical methods for computing this integral split up the original integral into a sum of several integrals, each covering a smaller part of the original integration interval [a,b]. This re-writing of the integral is based on a selection of integration points xi, i = 0,1,.. .,n that are distributed on the interval [a,b]. Integration points may, or may not, be evenly distributed. An even distribution simplifies expressions and is often sufficient, so we will mostly restrict ourselves to thatchoice.The integrationpointsare thencomputedas xi =a+ ih, i=0,1,.. .,n, (6.3)