chap08_8up - Numerical Integration Numerical...

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Unformatted text preview: Numerical Integration Numerical Differentiation Richardson Extrapolation Scientic Computing: An Introductory Survey Chapter 8 Numerical Integration and Differentiation Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientic Computing 1 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Outline 1 Numerical Integration 2 Numerical Differentiation 3 Richardson Extrapolation Michael T. Heath Scientic Computing 2 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Integration For f : R R , denite integral over interval [ a, b ] I ( f ) = Z b a f ( x ) dx is dened 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 well-conditioned because of its smoothing effect Michael T. Heath Scientic Computing 3 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Numerical Quadrature Quadrature rule is weighted sum of nite 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 Scientic Computing 4 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Quadrature Rules An n-point 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 Scientic 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 nite 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 Scientic Computing 6 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Method of Undetermined Coefcients...
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chap08_8up - Numerical Integration Numerical...

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