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Unformatted text preview: Scientic Computing Approximations Computer Arithmetic Scientic Computing: An Introductory Survey Chapter 1 Scientic Computing 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 Scientic Computing 1 / 46 Scientic Computing Approximations Computer Arithmetic Outline 1 Scientic Computing 2 Approximations 3 Computer Arithmetic Michael T. Heath Scientic Computing 2 / 46 Scientic Computing Approximations Computer Arithmetic Introduction Computational Problems General Strategy Scientic Computing What is scientic computing ? Design and analysis of algorithms for numerically solving mathematical problems in science and engineering Traditionally called numerical analysis Distinguishing features of scientic computing Deals with continuous quantities Considers effects of approximations Why scientic computing ? Simulation of natural phenomena Virtual prototyping of engineering designs Michael T. Heath Scientic Computing 3 / 46 Scientic Computing Approximations Computer Arithmetic Introduction Computational Problems General Strategy WellPosed Problems Problem is wellposed if solution exists is unique depends continuously on problem data Otherwise, problem is illposed Even if problem is well posed, solution may still be sensitive to input data Computational algorithm should not make sensitivity worse Michael T. Heath Scientic Computing 4 / 46 Scientic Computing Approximations Computer Arithmetic Introduction Computational Problems General Strategy General Strategy Replace difcult problem by easier one having same or closely related solution innite nite differential algebraic nonlinear linear complicated simple Solution obtained may only approximate that of original problem Michael T. Heath Scientic Computing 5 / 46 Scientic Computing Approximations Computer Arithmetic Sources of Approximation Error Analysis Sensitivity and Conditioning Sources of Approximation Before computation modeling empirical measurements previous computations During computation truncation or discretization rounding Accuracy of nal result reects all these Uncertainty in input may be amplied by problem Perturbations during computation may be amplied by algorithm Michael T. Heath Scientic Computing 6 / 46 Scientic Computing Approximations Computer Arithmetic Sources of Approximation Error Analysis Sensitivity and Conditioning Example: Approximations Computing surface area of Earth using formula A = 4 r 2 involves several approximations Earth is modeled as sphere, idealizing its true shape Value for radius is based on empirical measurements and previous computations Value for requires truncating innite process Values for input data and results of arithmetic operations are rounded in computer Michael T. Heath Scientic Computing 7 / 46 Scientic Computing...
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