chap01_8up

chap01_8up - Scientific Computing Approximations Computer...

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Unformatted text preview: Scientific Computing Approximations Computer Arithmetic Scientific Computing: An Introductory Survey Chapter 1 – Scientific Computing 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 Scientific Computing 1 / 46 Scientific Computing Approximations Computer Arithmetic Outline 1 Scientific Computing 2 Approximations 3 Computer Arithmetic Michael T. Heath Scientific Computing 2 / 46 Scientific Computing Approximations Computer Arithmetic Introduction Computational Problems General Strategy Scientific Computing What is scientific computing ? Design and analysis of algorithms for numerically solving mathematical problems in science and engineering Traditionally called numerical analysis Distinguishing features of scientific computing Deals with continuous quantities Considers effects of approximations Why scientific computing ? Simulation of natural phenomena Virtual prototyping of engineering designs Michael T. Heath Scientific Computing 3 / 46 Scientific Computing Approximations Computer Arithmetic Introduction Computational Problems General Strategy Well-Posed Problems Problem is well-posed if solution exists is unique depends continuously on problem data Otherwise, problem is ill-posed Even if problem is well posed, solution may still be sensitive to input data Computational algorithm should not make sensitivity worse Michael T. Heath Scientific Computing 4 / 46 Scientific Computing Approximations Computer Arithmetic Introduction Computational Problems General Strategy General Strategy Replace difficult problem by easier one having same or closely related solution infinite → finite differential → algebraic nonlinear → linear complicated → simple Solution obtained may only approximate that of original problem Michael T. Heath Scientific Computing 5 / 46 Scientific 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 final result reflects all these Uncertainty in input may be amplified by problem Perturbations during computation may be amplified by algorithm Michael T. Heath Scientific Computing 6 / 46 Scientific Computing Approximations Computer Arithmetic Sources of Approximation Error Analysis Sensitivity and Conditioning Example: Approximations Computing surface area of Earth using formula A = 4 πr 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 π requires truncating infinite process Values for input data and results of arithmetic operations are rounded in computer Michael T. Heath Scientific Computing 7 / 46 Scientific Computing...
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chap01_8up - Scientific Computing Approximations Computer...

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