Page - 60 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

60 3 Computing Integrals Bysimplifying the righthandsideof (3.15)weget bZ a f.x/dx h 2 Œf.x0/C2f.x1/C2f.x2/C : : :C2f.xn 1/Cf.xn/ (3.16) which ismorecompactlywrittenas bZ a f.x/dx h " 1 2 f.x0/C n 1X iD1 f.xi/C 1 2 f.xn/ # : (3.17) Composite integrationrules Theword composite is often usedwhen a numerical integrationmethod is ap- pliedwithmore thanonesub-interval. Strictly speaking then,writing, e.g., “the trapezoidalmethod”, should imply theuseofonlyasingle trapezoid,while “the composite trapezoidalmethod”is themostcorrectnamewhenseveral trapezoids areused.However, thisnamingconventionisnotalwaysfollowed,sosayingjust “thetrapezoidalmethod”maypointtoasingletrapezoidaswellasthecomposite rulewithmany trapezoids. 3.2.2 Implementation Specific orgeneral implementation? Supposeour primarygoalwas to compute the specific integral R1 0 v.t/dt with v.t/ D 3t2et3. First we played aroundwith a simple hand calculation to see what the methodwas about, before we (as one often does inmathematics) developed a general formula (3.17) for the general or “abstract” integral Rb a f.x/dx. To solve our specific problem R1 0 v.t/dt wemust thenapplythegeneral formula(3.17) to thegivendata (functionandintegral limits) in ourproblem. Although simple in principle, thepractical steps are confusing for manybecausethenotationintheabstractproblemin(3.17)differsfromthenotation in our special problem. Clearly, thef ,x, andh in (3.17) correspond to v, t, and perhaps t for the trapezoidwidth inour special problem. Theprogrammer’sdilemma 1. Shouldwewrite a special program for the special integral, using the ideas fromthegeneral rule (3.17),but replacingf byv,xby t, andhby t? 2. Should we implement the general method (3.17) as it stands in a general functiontrapezoid(f, a, b, n) and solve the specific problemat hand byaspecializedcall to this function? Alternative2 isalways thebest choice! The first alternative in the box above sounds less abstract and thereforemore attractive tomany. Nevertheless, as we hopewill be evident from the examples, the secondalternative is actually the simplestandmost reliable fromboth amath- ematical andprogrammingpointof view. Theseauthorswill claim that the second