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The Bisection Method

The document describes the bisection method for finding roots of equations. It provides background on the method, outlines its implementation which involves iteratively applying it over an interval to find a root, and discusses some limitations. It then presents the bisection theorem which proves the method converges to a root if the function has opposite signs at the endpoints. Several examples applying the method to find roots of cubic equations are also given.

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92 views5 pages

The Bisection Method

The document describes the bisection method for finding roots of equations. It provides background on the method, outlines its implementation which involves iteratively applying it over an interval to find a root, and discusses some limitations. It then presents the bisection theorem which proves the method converges to a root if the function has opposite signs at the endpoints. Several examples applying the method to find roots of cubic equations are also given.

Uploaded by

Anonymous UyGHBJU
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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The Bisection Method

Background. The bisection method is one of the bracketing methods for finding
roots of equations.
Implementation. Given a function f(x) and an interval which might contain a
root, perform a predetermined number of iterations using the bisection method.
Limitations. Investigate the result of applying the bisection method over an
interval where there is a discontinuity. Apply the bisection method for a function
using an interval where there are distinct roots. Apply the bisection method over a
"large" interval.

Theorem (Bisection Theorem). Assume that and that there exists a


number such that .
If have opposite signs, and represents the sequence of
midpoints generated by the bisection process, then

for ,

and the sequence converges to the zero .

That is, .

Proof Bisection Method Bisection Method

Computer Programs Bisection Method Bisection Method

Mathematica Subroutine (Bisection Method).


Example 1. Find all the real solutions to the cubic equation .
Solution 1.

Example 2. Use the cubic equation in Example 1 and perform the


following call to the bisection method.

Bisection[0,1,30];

Solution 2.

Reduce the volume of printout.

After you have debugged you program and it is working properly, delete the
unnecessary print statements.

Concise Program for the Bisection Method


Now test the example to see if it still works. Use the last case in Example 1 given
above and compare with the previous results.

Reducing the Computational Load for the Bisection Method

The following program uses fewer computations in the bisection method and is the
traditional way to do it. Can you determine how many fewer functional
evaluations are used ?
Various Scenarios and Animations for the Bisection Method.
Example 3. Convergence Find the solution to the cubic equation
. Use the starting interval .
Solution 3.

Example 4. Not a root located Find the solution to the equation . Use
the starting interval .

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