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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 7 Roots of Equations Bracketing Methods

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

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

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

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

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

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

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

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Last Time Bisection Method Check Convergence Root = If Error

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Objectives Master methods to compute roots of equations Assess reliability of each method Choose best method for a specific problem REGULA FALSI Method (False Position)

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False Position Method xlxl xuxu x r =0.5(x l +x u ) Recall Bisection Method No consideration on function values

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False Position Method f(x l ) f(x u ) xlxl xuxu xrxr NEW ESTIMATE

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False Position Method f(x l ) f(x u ) xlxl xuxu xrxr

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False Position Method f(x l ) f(x u ) xlxl xuxu xrxr From Similar Triangles

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False Position Method

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Add and subtract New Estimate

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Loop x old =x r Error=100*abs(x-x old )/x r Sign=f(x l )*f(x r ) Sign x u =x r f u =f(x u ) x l =x r f l =f(x l ) Error=0 Error<E all ROOT=x r FALSE <0>0 f u =f(x u ), f l =f(x l )

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False Position Typically Faster Convergence than Bisection

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False Position Not Efficient in this Case

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