Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSsampling, in both value and precision. Since we knew already that f(x) andg(x)diverge monotonically by about 6% as x varies over the range (0, 1), we couldhave made a small improvement to our control variate by scaling it by 1.03 beforeusing it in equation (27.51).Antithetic variatesAs a final example of a method that improves on crude Monte Carlo, and one thatis particularly useful when monotonic functions are to be integrated, we mentionthe use of antithetic variates. This method relies on finding two estimates t andt ′ of θ that are strongly anticorrelated (i.e. Cov[t, t ′ ] is large and negative) andusing the resultV [ 1 2 (t + t′ )] = 1 4 V [t]+ 1 4 V [t′ ]+ 1 2 Cov[t, t′ ].For example, the use of 1 2[f(ξ) +f(1 − ξ)] instead of f(ξ) involves only twiceas many evaluations of f, and no more random variables, but generally givesan improvement in precision significantly greater than this. For the integral off(x) =[tan −1 (x)] 1/2 , using as previously a batch of ten random variables, anestimate of 0.623 ± 0.018 was found. This is to be compared with the crudeMonte Carlo result, 0.634 ± 0.065, obtained using the same number of randomvariables.For a fuller discussion of these methods, and of theoretical estimates of theirefficiencies, the reader is referred to more specialist treatments. For practical implementationschemes, a book dedicated to scientific computing should be consulted. §Hit or miss methodWe now come to the approach that, in spirit, is closest to the activities that gaveMonte Carlo methods their name. In this approach, one or more straightforwardyes/no decisions are made on the basis of numbers drawn at random – the endresult of each trial is either a hit or a miss! In this section we are concernedwith numerical integration, but the general Monte Carlo approach, in whichone estimates a physical quantity that is hard or impossible to calculate directlyby simulating the physical processes that determine it, is widespread in modernscience. For example, the calculation of the efficiencies of detector arrays inexperiments to study elementary particle interactions are nearly always carriedout in this way. Indeed, in a normal experiment, far more simulated interactionsare generated in computers than ever actually occur when the experiment istaking real data.As was noted in chapter 2, the process of evaluating a one-dimensional integralf(x)dx can be regarded as that of finding the area between the curve y = f(x)∫ ba§ e.g. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C: TheArtofScientificComputing, 2nd edn (Cambridge: Cambridge University Press, 1992).1014