Bisection Method Notes

# Bisection Method Notes - Chapter 03.03 Bisection Method of...

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Chapter 03.03 Bisection Method of Solving a Nonlinear Equation After reading this chapter, you should be able to: 1. follow the algorithm of the bisection method of solving a nonlinear equation, 2. use the bisection method to solve examples of finding roots of a nonlinear equation, and 3. enumerate the advantages and disadvantages of the bisection method. What is the bisection method and what is it based on? One of the first numerical methods developed to find the root of a nonlinear equation 0 ) ( = x f was the bisection method (also called binary-search method). The method is based on the following theorem. Theorem An equation 0 ) ( = x f , where ) ( x f is a real continuous function, has at least one root between x and u x if 0 ) ( ) ( < u x f x f (See Figure 1). Note that if 0 ) ( ) ( u x f x f , there may or may not be any root between x and u x (Figures 2 and 3). If 0 ) ( ) ( < u x f x f , then there may be more than one root between x and u x (Figure 4). So the theorem only guarantees one root between x and u x . Bisection method Since the method is based on finding the root between two points, the method falls under the category of bracketing methods. Since the root is bracketed between two points, x and u x , one can find the mid- point, m x between x and u x . This gives us two new intervals 1. x and m x , and 2. m x and u x . 03.03.1

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03.03.2 Chapter 03.03 Figure 1 At least one root exists between the two points if the function is real, continuous, and changes sign. Figure 2 If the function ) ( x f does not change sign between the two points, roots of the equation 0 ) ( = x f may still exist between the two points. f ( x ) x x u x f ( x ) x x u x
Bisection Method 03.03.3 Figure 3 If the function ) ( x f does not change sign between two points, there may not be any roots for the equation 0 ) ( = x f between the two points. Figure 4 If the function ) ( x f changes sign between the two points, more than one root for the equation 0 ) ( = x f may exist between the two points. Is the root now between x and m x or between m x and u x ? Well, one can find the sign of ) ( ) ( m x f x f , and if 0 ) ( ) ( < m x f x f then the new bracket is between x and m x , otherwise, it is between m x and u x . So, you can see that you are literally halving the interval. As one repeats this process, the width of the interval [ ] u x x , becomes smaller and smaller, and you can zero in to the root of the equation

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## This note was uploaded on 05/01/2011 for the course CHBE 2120 taught by Professor Gallivan during the Spring '07 term at Georgia Institute of Technology.

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Bisection Method Notes - Chapter 03.03 Bisection Method of...

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