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8CHAPTER 1.INTRODUCTIONBut (1.20) gives us two important properties of the approximation methodin question.First, (1.20) tell us thatEh[f]!0 ash!0.That is, thequadrature ruleTh[f]convergesto the exact value of the integral ash!01.Recallh= (b-a)/N, so as we increaseNour approximation to theintegral gets better and better. Second, (1.20) tells us how fast the approx-imation converges, namely quadratically inh. This is the approximation’srate of convergence.If we doubleN(or equivalently halveh) then theerror decreases by a factor of 4. We also say that the error is orderh2andwriteEh[f] =O(h2). The Big ‘O’ notation is used frequently in NumericalAnalysis.Definition 1.We say thatg(h)is orderh↵, and writeg(h) =O(h↵), ifthere is a constantCandh0such that|g(h)|Ch↵for0hh0, i.e. forsufficiently smallh.Example 1.Let’s check the Trapezoidal Rule approximation for an integralwe can compute exactly.Takef(x) =exin[0,1].The exact value of theintegral ise-1.Observe how the error|I[ex]-T1/N[ex]|decreases by aTable 1.1: Composite Trapezoidal Rule forf(x) =exin [0,1].NT1/N[ex]|I[ex]-T1/N[ex]|Decrease factor161.7188411285799945.593001209489579⇥10-4321.7184216603163271.398318572816137⇥10-40.250012206406039641.7183167868500943.495839104861176⇥10-50.2500030517238101281.7182905680834788.739624432374526⇥10-60.250000762913303factor of (approximately) 1/4 asNis doubled, in accordance to (1.20).1.2.4Error bounds and Computable Error EstimatesWe can get an upper bound for the error using (1.20) and thatf00is boundedin [a, b], i.e.|f00(x)|M2for allx2[a, b] for some constantM2. Then|Eh[f]|112(b-a)h2M2.(1.22)1Neglecting round-o↵errors introduced by finite precision number representation andcomputer arithmetic.