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Roughly speaking, the error can be estimated by comparing two approxima-tions obtained with differenth.Consider (1.27). If we halvehwe getI[f] =Th/2[f] +14C2h2+R(h/2).(1.29)Subtracting (1.29) from (1.27) we getC2h2=43(Th/2[f]-Th[f])+43(R(h/2)-R(h)).(1.30)The last term on the right hand side iso(h2). Hence, forhsufficiently small,we haveC2h2≈43(Th/2[f]-Th[f])(1.31)and this could provide a good, computable estimate for the error, i.e.Eh[f]≈43(Th/2[f]-Th[f]).(1.32)The key here is thathhas to be sufficiently small to make the asymptoticapproximation (1.31) valid. We can check this by working backwards. Ifhis sufficiently small, then evaluating (1.31) ath/2 we getC2h22≈43(Th/4[f]-Th/2[f])(1.33)
1.2.AN ILLUSTRATIVE EXAMPLE11and consequently the ratioq(h) =Th/2[f]-Th[f]Th/4[f]-Th/2[f](1.34)should be approximately 4. Thus,q(h) offers a reliable, computable indicatorof whether or nothis sufficiently small for (1.32) to be an accurate estimateof the error.We can now use (1.31) and the idea of error correction to improve theaccuracy ofTh[f] with the following approximation2Sh[f] :=Th[f] +43(Th/2[f]-Th[f]).(1.35)1.2.5Richardson ExtrapolationWe can view theerror correctionprocedure as a way to eliminate theleading order (inh) contribution to the error. Multiplying (1.29) by 4 andsubstracting (1.27) to the result we getI[f] =4Th/2[f]-Th[f]3+4R(h/2)-R(h)3(1.36)Note thatSh[f] is exactly the first term in the right hand side of (1.36) andthat the last term converges to zero faster thanh2.This very useful andgeneral procedure in which the leading order component of the asymptoticform of error is eliminated by a combination of two computations performedwith two different values ofhis calledRichardson’s Extrapolation.Example 2.Consider againf(x) =exin[0,1]. Withh= 1/16we getq116=T1/32[ex]-T1/16[ex]T1/64[ex]-T1/32[ex]≈3.9998(1.37)and the improved approximation isS1/16[ex] =T1/16[ex] +43(T1/32[ex]-T1/16[ex])= 1.718281837561771(1.38)which gives us nearly 8 digits of accuracy (error≈9.1×10-9).S1/32givesus an error≈5.7×10-10. It decreased by approximately a factor of1/16.This would correspond tofourth orderrate of convergence. We will see inChapter 8 that indeed this is the case.2The symbol := means equal by definition.
12CHAPTER 1.INTRODUCTIONIt appears thatSh[f] gives us superior accuracy to that ofTh[f] but atroughly twice the computational cost.If we group together the commonterms inTh[f] andTh/2[f] we can computeSh[f] at about the same compu-tational cost as that ofTh/2[f]:4Th/2[f]-Th[f] = 4h2"12f(a) +2N-1Xj=1f(a+jh/2) +12f(b)#-h"12f(a) +N-1Xj=1f(a+jh) +12f(b)#=h2"f(a) +f(b) + 2N-1Xk=1f(a+kh) + 4N-1Xk=1f(a+kh/2)#.ThereforeSh[f] =h6"f(a) + 2N-1Xk=1f(a+kh) + 4N-1Xk=1f(a+kh/2) +f(b)#.(1.39)The resulting quadrature formulaSh[f] is known as theComposite Simpson’sRuleand, as we will see in Chapter 8, can be derived by approximating theintegrand by quadratic polynomials. Thus, based on cost and accuracy, theComposite Simpson’s Rule would be preferable to the Composite Trapezoidal