Mathematical Methods for Physics and Engineering - Matematica.NET

27.4 NUMERICAL INTEGRATIONwill have a very small variance. Further, any error in inverting the relationshipbetween η and ξ will not be important since f(η)/g(η) will be largely independentof the value of η.As an example, consider the function f(x) =[tan −1 (x)] 1/2 , which is not analyticallyintegrable over the range (0, 1) but is well mimicked by the easily integratedfunction g(x) =x 1/2 (1 − x 2 /6). The ratio of the two varies from 1.00 to 1.06 as xvaries from 0 to 1. The integral of g over this range is 0.619 048, and so it has tobe renormalised by the factor 1.615 38. The value of the integral of f(x) from0to 1 can then be estimated by averaging the value of[tan −1 (η)] 1/2(1.615 38) η 1/2 (1 − 1 6 η2 )for random variables η which are such that G(η) is uniformly distributed on(0, 1). Using batches of as few as ten random numbers gave a value 0.630 for θ,with standard deviation 0.003. The corresponding result for crude Monte Carlo,using the same random numbers, was 0.634 ± 0.065. The increase in precision isobvious, though the additional labour involved would not be justified for a singleapplication.Control variatesThe control-variate method is similar to, but not the same as, importance sampling.Again, an analytically integrable function that mimics f(x) in shape hasto be found. The function, known as the control variate, is first scaled so as tomatch f as closely as possible in magnitude and then its integral is found inclosed form. If we denote the scaled control variate by h(x), then the estimate ofθ is computed ast =∫ 10[f(x) − h(x)] dx +∫ 10h(x) dx. (27.51)The first integral in (27.51) is evaluated using (crude) Monte Carlo, whilst thesecond is known analytically. Although the first integral should have been renderedsmall by the choice of h(x), it is its variance that matters. The method relieson the following result (see equation (30.136)):V [t − t ′ ]=V [t]+V [t ′ ] − 2Cov[t, t ′ ],and on the fact that if t estimates θ whilst t ′ estimates θ ′ using the same randomnumbers, then the covariance of t and t ′ can be larger than the variance of t ′ ,and indeed will be so if the integrands producing θ and θ ′ are highly correlated.To evaluate the same integral as was estimated previously using importancesampling, we take as h(x) the function g(x) used there, before it was renormalised.Again using batches of ten random numbers, the estimated value for θ was foundto be 0.629 ± 0.004, a result almost identical to that obtained using importance1013