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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 5 Approximations, Errors and The Taylor Series

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Objectives Distinguish between algorithm errors and roundoff errors. Introduce Taylor Theorem Calculate the numerical error of a finite difference formula for derivatives

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Significant Figures

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Designate the reliability of a numerical value The significant digits of a number are those that can be used with confidence

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Accuracy and Precision Accuracy Precision

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Truncation Errors vivi titi t i+1 v i+1 True Slope Approximate Slope Truncation errors due to using approximation in place of exact solution

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Roundoff Errors A= d 2

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Roundoff Errors A1= d 1 2 A2= d 2 2 d1=1.0d2=1.000001 A1=3.14A2=3.14000628

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Error Definition True Value = Approximation + Error Numerical Error E t =true value - approximation Does not account for order of magnitude

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Error Definition True Value = Approximation + Error Relative Error = True Error/True Value t = (True Error/True Value)100%

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Approximate Error Numerical Methods: the true value is not known apriori

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Approximate Error Numerical Methods: approximate procedures Mclaurin Series Exact

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Approximate Error

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The Taylor Series Predict value of a function at one point in terms of the function value and its derivatives at another point

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Taylor’s Theorem

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Error of Order (x i+1 – x i ) n+1

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Numerical Differentiation

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First Divided Difference

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Forward Difference vivi titi t i+1 v i+1 True Slope Approximate Slope

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Backward Difference

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Central Difference Forward Backward Central

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Homework Problems 4.5 and 4.6 Due Wednesday Sept 10

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