mws_gen_nle_ppt_bisection(1)

mws_gen_nle_ppt_bisection(1) - Bisection Method Major: All...

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10/8/2010 http://numericalmethods.eng.usf.edu 1 Bisection Method Major: All Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates
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Bisection Method http://numericalmethods.eng.usf.edu
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http://numericalmethods.eng.usf.edu 3 Basis of Bisection Method Theorem x f(x) x u x An equation f(x)=0, where f(x) is a real continuous function, has at least one root between x l and x u if f(x l ) f(x u ) < 0. Figure 1 At least one root exists between the two points if the function is real, continuous, and changes sign.
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x f(x) x u x http://numericalmethods.eng.usf.edu 4 Basis of Bisection Method Figure 2 If function does not change sign between two points, roots of the equation may still exist between the two points. ( ) x f ( ) 0 = x f
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x f(x) x u x http://numericalmethods.eng.usf.edu 5 Basis of Bisection Method Figure 3 If the function does not change sign between two points, there may not be any roots for the equation between the two points. x f(x) x u x ( ) x f ( ) 0 = x f
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x f(x) x u x http://numericalmethods.eng.usf.edu 6 Basis of Bisection Method Figure 4 If the function changes sign between two points, more than one root for the equation may exist between the two points. ( ) x f ( ) 0 = x f
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http://numericalmethods.eng.usf.edu 7 Algorithm for Bisection Method
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http://numericalmethods.eng.usf.edu 8 Step 1 Choose x and x u as two guesses for the root such that f(x ) f(x u ) < 0, or in other words, f(x) changes sign between x and x u . This was demonstrated in Figure 1. x f(x) x u x Figure 1
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x f(x) x u x x m http://numericalmethods.eng.usf.edu 9 Step 2 Estimate the root, x m of the equation f (x) = 0 as the mid point between x and x u as x x m = x u + 2 Figure 5 Estimate of x m
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10 Step 3 Now check the following a) If , then the root lies between x and x m ; then x = x ; x u = x m . b) If , then the root lies between x m and x u ; then x = x m ; x u = x u . c) If ; then the root is x m. Stop the algorithm if this is true. ( ) ( ) 0 < m
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.

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mws_gen_nle_ppt_bisection(1) - Bisection Method Major: All...

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