Mathematical Methods for Physics and Engineering - Matematica.NET

27.4 NUMERICAL INTEGRATIONy = f(x)y = cx = ax = bxFigure 27.5 A simple rectangular figure enclosing the area (shown shaded)which is equal to ∫ bf(x) dx.aand the x-axis in the range a ≤ x ≤ b. It may not be possible to do thisanalytically, but if, as shown in figure 27.5, we can enclose the curve in a simplefigure whose area can be found trivially then the ratio of the required (shaded)area to that of the bounding figure, c(b − a), is the same as the probability that arandomly selected point inside the boundary will lie below the line.In order to accommodate cases in which f(x) can be negative in part of thex-range, we treat a slightly more general case. Suppose that, for a ≤ x ≤ b, f(x)is bounded and known to lie in the range A ≤ f(x) ≤ B; then the transformationz = x − ab − awill reduce the integral ∫ baf(x) dx to the formA(b − a)+(B − A)(b − a)∫ 10h(z) dz, (27.52)whereh(z) = 1 [f ((b − a)z + a) − A] .B − AIn this form z lies in the range 0 ≤ z ≤ 1andh(z) lies in the range 0 ≤ h(z) ≤ 1,i.e. both are suitable for simulation using the standard random-number generator.It should be noted that, for an efficient estimation, the bounds A and B shouldbe drawn as tightly as possible –preferably, but not necessarily, they should beequal to the minimum and maximum values of f in the range. The reason forthis is that random numbers corresponding to values which f(x) cannot reachadd nothing to the estimation but do increase its variance.It only remains to estimate the final integral on the RHS of equation (27.52).This we do by selecting pairs of random numbers, ξ 1 and ξ 2 , and testing whether1015