Mathematical Methods for Physics and Engineering - Matematica.NET

27.4 NUMERICAL INTEGRATION(a) (b) (c)f(x)f(x)f i+1f if i+1f i+1hhhhf if i−1x i+1x i x i+1/2 x i+1 x i+1x iFigure 27.4 (a) Definition of nomenclature. (b) The approximation in usingthe trapezium rule; f(x) is indicated by the broken curve. (c) Simpson’s ruleapproximation; f(x) is indicated by the broken curve. The solid curve is partof the approximating parabola.x i−1x inumerical evaluations of I are based on regarding I as the area under the curveof f(x) between the limits x = a and x = b and attempting to estimate that area.The simplest methods of doing this involve dividing up the interval a ≤ x ≤ binto N equal sections, each of length h =(b − a)/N. The dividing points arelabelled x i , with x 0 = a, x N = b, i running from 0 to N. The point x i is a distanceih from a. The central value of x in a strip (x = x i + h/2) is denoted for brevityby x i+1/2 , and for the same reason f(x i ) is written as f i . This nomenclature isindicated graphically in figure 27.4(a).So that we may compare later estimates of the area under the curve with thetrue value, we next obtain an exact expression for I, even though we cannotevaluate it. To do this we need to consider only one strip, say that between x iand x i+1 . For this strip the area is, using Taylor’s expansion,∫ h/2−h/2∫ h/2f(x i+1/2 + y) dy =−h/2∞∑==n=0∞∑∞∑f (n) (x i+1/2 ) ynn! dyn=0∫ h/2f (n)i+1/2f (n)i+1/2n even−h/2y nn! dy) n+12 h. (27.35)(n +1)!(2It should be noted that, in this exact expression, only the even derivatives off survive the integration and all derivatives are evaluated at x i+1/2 . Clearly1001