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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 6 Roots of Equations Bracketing Methods

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Last Time - Accuracy and Precision Accuracy Precision

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Last Time - 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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Last Time - Roundoff Errors A= d 2

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Last Time - Error Definition E t =true value - approximation True Error t = (E t /True Value)100% Relative True Error

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Last Time - Error Definition Approximate Relative Error Iteration Relative Error

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Last Time - 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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Last Time - Taylor’s Theorem Error of Order (x i+1 – x i ) n+1

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Last Time - Numerical Differentiation Forward DifferenceBackward DifferenceCentral Difference

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The Problem Analytic Solution

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The Problem To design the parachute: v=10 m/s t=3 sec m=64 kg g=9.81 c=? CANNOT rearrange to solve for c

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The Problem Define Function c must satisfy c is the ROOT of the equation

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Objectives Master methods to compute roots of equations Assess reliability of each method Choose best method for a specific problem

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Classification Methods BracketingOpen Graphical Bisection Method False Position Fixed Point Iteration Newton-Raphson Secand

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Graphical Methods c f(c) v=10 m/s t=3 sec m=65 kg g=9.81

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Graphical Methods No Roots Even Number of Roots Lower and Upper Bounds of interval yield values of same sign

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Graphical Methods Lower and Upper Bounds of interval yield values of opposite sign Odd number of Roots

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Bisection Method Choose Lower, x l and Upper x u guesses that bracket the root xlxl xuxu

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Bisection Method Calculate New Estimate x r and f(x r ) xlxl xuxu x r =0.5(x l +x u )

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Bisection Method Check Convergence Root = If Error

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Bisection Method Define New Interval that Brackets the Root Check sign of f(x l )*f(x r ) and f(x u )*f(x r ) xlxl xuxu Previous Guess xuxu

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Bisection Method Repeat until convergence xlxl xuxu Previous Guess x r =0.5(x l +x u )

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Bisection - Flowchart Loop x old =x x=(x l +x u )/2 Error=100*abs(x-x old )/x Sign=f(x l )*f(x r ) Sign x u =xx l =xError=0 Error<E all ROOT=x FALSE <0>0

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