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Unformatted text preview: Lecture 13 – October 7, 2010 Agenda: • Solving Nonlinear functions: f ( x ) = 0 (root finding) Closed (Bracketing) Methods o Bisection (talked about this last time) o False Position Open Methods (nonbracketing) o NewtonRaphson o Secant • Discuss HW 3 (due 10/15) Root Finding: False Position • False Position is a method for finding a root ( x r ) when the function has little curvature within the bracket • Example: find the solution to x 3 = 2 ( x r = 2 1/3 1.3) • Root finding equation: f ( x ) = x 3 – 2 Root Finding: False Position • Choose x l and x u such that f l f u < 0 (good bracket) • Say x l = 0.5 and x u = 1.75 • The search is fixed between the two limits • Define a line between our chosen points: • The root of the above occurs when l ( x guess ) = 0 l u l u l l x x f f x x f x l Root Finding: False Position l u l u l l l u l u l l f f x x f x x x x f f x x f x guess guess guess l • Solve for x guess (or x guess – x l ): • This is the first estimate of the true root ( x r ) • From here we evaluate f ( x guess ) and repeat if needed Root Finding: False Position • In the first step, x r and x g will likely be too far apart, so we need to form a new bracket • Test to see if x g should be the new x l or x u based on whether f l f g < 0 or f g f u < 0 • In the example, x g should be the new x l • The procedure is analogous to the bisection method Root Finding: False Position • Again, we draw a line from ( x l , f l ) to ( x u , f u ) • Solve l ( x guess ) = 0 • Check to see if x guess should be the new x l or x u • Continue until the absolute approximate error is within a certain tolerance: tol iter guess 1 iter guess iter guess appox x x x Root Finding: False Position • Overview of False Position algorithm: Choose an appropriate bracket ( x l , x u ) Choose a tolerance (tol) for the error Initialize error to something > tol...
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This note was uploaded on 09/21/2011 for the course CH E 310 taught by Professor Staff during the Spring '08 term at Iowa State.
 Spring '08
 Staff

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