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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,      .

Computer Programs  Bisection Method  Bisection Method

Mathematica Subroutine (Bisection Method).

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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

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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.

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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 ?

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Various Scenarios and Animations for the Bisection Method.

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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  .
Solution 4.

Animations (Bisection Method  Bisection Method).  Internet hyperlinks to animations.

Old Lab Project (Bisection Method  Bisection Method).  Internet hyperlinks to an old lab project.

Bisection Method Bisection Method  Internet hyperlinks to web sites and a bibliography of articles.

(c) John H. Mathews 2004