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Unformatted text preview: Numerical Integration Numerical Differentiation Richardson Extrapolation Scientific Computing: An Introductory Survey Chapter 8 Numerical Integration and Differentiation Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Outline 1 Numerical Integration 2 Numerical Differentiation 3 Richardson Extrapolation Michael T. Heath Scientific Computing 2 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Integration For f : R R , definite integral over interval [ a, b ] I ( f ) = Z b a f ( x ) dx is defined by limit of Riemann sums R n = n X i =1 ( x i +1 x i ) f ( i ) Riemann integral exists provided integrand f is bounded and continuous almost everywhere Absolute condition number of integration with respect to perturbations in integrand is b a Integration is inherently wellconditioned because of its smoothing effect Michael T. Heath Scientific Computing 3 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Numerical Quadrature Quadrature rule is weighted sum of finite number of sample values of integrand function To obtain desired level of accuracy at low cost, How should sample points be chosen? How should their contributions be weighted? Computational work is measured by number of evaluations of integrand function required Michael T. Heath Scientific Computing 4 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Quadrature Rules An npoint quadrature rule has form Q n ( f ) = n X i =1 w i f ( x i ) Points x i are called nodes or abscissas Multipliers w i are called weights Quadrature rule is open if a < x 1 and x n < b closed if a = x 1 and x n = b Michael T. Heath Scientific Computing 5 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Quadrature Rules, continued Quadrature rules are based on polynomial interpolation Integrand function f is sampled at finite set of points Polynomial interpolating those points is determined Integral of interpolant is taken as estimate for integral of original function In practice, interpolating polynomial is not determined explicitly but used to determine weights corresponding to nodes If Lagrange is interpolation used, then weights are given by w i = Z b a ` i ( x ) , i = 1 , . . . , n Michael T. Heath Scientific Computing 6 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Method of Undetermined Coefficients Alternative derivation of quadrature rule uses...
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