lecture_28 - Introduction to Scientific Computing Volodymyr...

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National Center for Supercomputing Applications University of Illinois at Urbana-Champaign Introduction to Scientific Computing Volodymyr Kindratenko
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This lecture is entirely based on the first chapter of Scientific Computing: An Introductory Survey by Michael T. Heath http://www.cse.illinois.edu/heath/scicomp/
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Lecture Topics Scientific Computing Approximations in scientific computing Computer Arithmetic Survey of scientific computing methods
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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 Time, distance, speed, … are continuous in nature Considers effects of approximations Data is not precise, digital computers perform only approximate calculations Why scientific computing? Simulation of natural phenomena Virtual prototyping of engineering designs
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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 Knowing how accurate the solution is becomes important
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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
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Sources of Approximation Before computation Modeling Some features may be simplified or omitted Empirical measurements Lab instruments have finite precision Previous computations May be only approximate During computation Truncation or discretization Some features may be omitted or simplified Rounding Computer representation of real numbers is inexact Accuracy of final result reflects all these
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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 (~6370 km) is based on empirical measurements and previous computations Value for requires truncating infinite process Values for input data and results of arithmetic operations are rounded in computer The accuracy of the computation depends on all these approximations
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Some errors are due to the input data , some are due to the computational process Typical problem: compute value of function f : for given argument x = true value of input f(x) = desired result x = approximate (inexact) input f = approximate function actually computed Total error: f ( x ) − f(x) =
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This note was uploaded on 08/24/2011 for the course ECE 190 taught by Professor Hutchinson during the Spring '08 term at University of Illinois, Urbana Champaign.

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lecture_28 - Introduction to Scientific Computing Volodymyr...

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