CHAPTER 1.INTRODUCTIONSubtracting (1.27) from (1.26) we getC2h2=43(Th/2[f]-Th[f])+43(R(h/2)-R(h)).(1.28)The last term on the right hand side iso(h2). Hence, forhsufficiently small,we haveC2h2⇡43(Th/2[f]-Th[f])(1.29)and this could provide a good, computable estimate for the error, i.e.Eh[f]⇡43(Th/2[f]-Th[f]).(1.30)The key here is thathhas to be sufficiently small to make the asymptoticapproximation (1.29) valid. We can check this by working backwards. Ifhis sufficiently small, then evaluating (1.29) ath/2 we getC2✓h2◆2⇡43(Th/4[f]-Th/2[f])(1.31)and consequently the ratioq(h) =Th/2[f]-Th[f]Th/4[f]-Th/2[f](1.32)should be approximately 4. Thus,q(h) o↵ers a reliable, computable indicatorof whether or nothis sufficiently small for (1.30) to be an accurate estimateof the error.1.2.5Error Correction and Richardson ExtrapolationIfhis sufficiently small, as explained above, we can use (1.30) to improvethe accuracy ofTh[f] with the following approximation2Sh[f] :=Th[f] +43(Th/2[f]-Th[f]).(1.33)2The symbol := means equal by definition.