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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