Mathematical Methods for Physics and Engineering - Matematica.NET

27.4 NUMERICAL INTEGRATIONso, provided g(x) is a reasonably smooth function, the approximation is a goodone.Taking 3-point integration as an example, the three x i are the zeros of P 3 (x) =12 (5x3 − 3x), namely 0 and ±0.774 60, and the corresponding weights are21 × ( − 2) 3 2= 8 2and9(1 − 0.6)× ( )6 2= 5 9 .2Table 27.8 gives the integration points (in the range −1 ≤ x i ≤ 1) and thecorresponding weights w i for a selection of n-point Gauss–Legendre schemes.◮Using a 3-point formula in each case, evaluate the integral∫ 11I =0 1+x dx, 2(i) using the trapezium rule, (ii) using Simpson’s rule, (iii) using Gaussian integration. Alsoevaluate the integral analytically and compare the results.(i) Using the trapezium rule, we obtainI = 1 × [ ( 12 2 f(0) + 2f 1) ]2 + f(1)[= 1 4 1+8+ ] 15 2 =0.7750.(ii) Using Simpson’s rule, we obtainI = 1 × [ ( 13 2 f(0) + 4f 1) ]2 + f(1)[= 1 6 1+16+ ] 15 2 =0.7833.(iii) Using Gaussian integration, we obtainI = 1 − 0 ∫ 1dz2 −1 1+ 1 (z +1)24{}= 1 0.555 56 [f(−0.774 60) + f(0.774 60)] +0.888 89f(0)2{}= 1 0.555 56 [0.987 458 + 0.559 503] +0.888 89 × 0.82=0.785 27.(iv) Exact evaluation gives∫ 1dxI =0 1+x = [ tan −1 x ] 1= π =0.785 40.2 04In practice, a compromise has to be struck between the accuracy of the result achievedand the calculational labour that goes into obtaining it. ◭Further Gaussian quadrature procedures, ones that utilise the properties of theChebyshev polynomials, are available for integrals over finite ranges when theintegrands involve factors of the form (1 − x 2 ) ±1/2 . In the same way as decreasinglinear and quadratic exponentials are handled through the weight functions inGauss–Laguerre and Gauss–Hermite quadrature, respectively, the square root1007