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Unformatted text preview: Chapter 14 Trapezoidal integration The mathematical analysis of the error, E N , incurred when evaluating an integral via the trapezoidal rule using N interval integraldisplay b a f ( x ) d x = Δ x bracketleftBigg f 2 + f 1 + f 2 + ... + f i + ... + f N − 1 + f N 2 bracketrightBigg + E N (14.1) is presented here. The object is to understand the structure of the error and to explore if this structure can be used advantageously to correct the trapezoidal esstimates. We will first derive the leading error term in section 14.1, using nothing more then integration by parts. We will then attempt a closer inspection of the error as a series, and this can be done in one of two ways: the simple one tries to rely only on Taylor series, sectionsec:Taylor, which is simple but inevitably contains a handwaving argument; the other approach uses Bernoulli polynomials and is more rigorous mathematically, and is done in 14.3. 14.1 Error approximations for trapezoidal inte gration Here we will illustrate how error estimates for the trapezoidal integration rule can be estimated. Let us concentrate on the interval [ x i − 1 x i ] and let us carry out a first manipulation of an integration by part: δA i = integraldisplay x i x i 1 f ( x ) d x (14.2) = integraldisplay x i x i 1 f ( x ) d parenleftBig x x i − 1 2 parenrightBig = bracketleftBigparenleftBig x x i − 1 2 parenrightBig f ( x ) bracketrightBig x i x i 1 integraldisplay x i x i 1 parenleftBig x x i − 1 2 parenrightBig f ′ ( x ) d x = bracketleftBigparenleftBig x i x i − 1 2 parenrightBig f ( x i ) parenleftBig x i − 1 x i − 1 2 parenrightBig f ( x i − 1 ) bracketrightBig integraldisplay x i x i 1 parenleftBig x x i − 1 2 parenrightBig f ′ ( x ) d x 101 102 CHAPTER 14. TRAPEZOIDAL INTEGRATION 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 δA 1 δA 2 δA i x f ( x ) δA 1 δA 2 δA i δA 1 δA 2 δA i Figure 14.1: Trapezoidal approximation of the area under the red curve = bracketleftbigg Δ x 2 f i Δ x 2 f i − 1 bracketrightbigg integraldisplay x i x i 1 parenleftBig x x i − 1 2 parenrightBig f ′ ( x ) d x = Δ x f i − 1 + f i 2 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright δ ˜ A i integraldisplay x i x i 1 parenleftBig x x i − 1 2 parenrightBig f ′ ( x ) d x bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright δE i (14.3) The first term on the right hand side of 14.3, δ ˜ A i , is clearly the trapezoidal formula for the interval [ x i − 1 x i ], and the second term is nothing but the error term, referred to as δE i if the trapezoidal formula is used to approximate the error in that interval. What we need to do know is to find a way to bound this error term in terms of the numerical parameters and the features of the function to be integrated. We will illustrate that in two ways....
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This note was uploaded on 01/08/2012 for the course MSC 321 taught by Professor Staff during the Fall '08 term at University of Miami.
 Fall '08
 Staff

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