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Chapter 5: Numerical Integration and Differentiation

Chapter 5: Numerical Integration and DifferentiationPART I: Numerical IntegrationNewton-Cotes Integration FormulasThe idea of Newton-Cotes formulas is to replace a complicated function or tabu-lated data with an approximating function that is easy to baf(x)dx bafn(x)dxwherefn(x) =a0+a1x+a2x2+..+ Trapezoidal RuleUsing the first order Taylor series to approximatef(x),I= baf(x)dx baf1(x)dxwheref1(x) =f(a) +f(b) f(a)b a(x a)1 ThenI ba[f(a) +f(b) f(a)b a(x a)]dx= (b a)f(b) +f(a)2 The trapezoidal rule is equivalent to approximating the area of the trapezoidalFigure 1: Graphical depiction of the trapezoidal ruleunder the straight line connectingf(a)andf(b).

This is to use a third-order Lagrange polynomial to fit to four points of f(x) ... 6480 f(4)(») where » is between a and b. 12. 3 Integration of Equations Newton-Cotes algorithms for equations Compare the following two Pseudocodes for multiple applications of the trape-zoidal rule. Pseudocode 1: Algorithm for multiple applications of the ...

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