# bisection - BisectionMethod Theorem...

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Bisection Method

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http://numericalmethods.eng.usf.edu 2 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.
x f(x) x u x                                             http://numericalmethods.eng.usf.edu 3 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.  ( 29 x f ( 29 0 = x f

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x f(x) x u x                                             http://numericalmethods.eng.usf.edu 4 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 ( 29 x f ( 29 0 = x f
x f(x) x u x                                             http://numericalmethods.eng.usf.edu 5 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. ( 29 x f ( 29 0 = x f

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http://numericalmethods.eng.usf.edu 6 Algorithm for Bisection Method
http://numericalmethods.eng.usf.edu 7 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 8 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
http://numericalmethods.eng.usf.edu 9 Step 3 Now check the following a) If                     , then the root lies between x  and x m then x  = x  ; x u  = x m . a)

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bisection - BisectionMethod Theorem...

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