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Bisection Algorithm and Programming

Procedures (These are the general instructions I must follow):

You may use calculators and Computer Algebra programs when doing calculations. The only high level commands you can use are plotting commands. Never use commands like FindRoot, Solve, Taylor, etc, on any exam or quiz.

Problem is attached: Interested in addressing things such as rounding error, stopping, number of iterations, slow convergence, root solutions.

Math 609D Quiz 3 February 5, 2008
(20 points) Let f(x) be the function defined by
f(x) = (ln x)5
− 5.5  (ln x)3 + 10.0  ln x −
ln x
for 0 < x < 1 and 1 < x < 1.
Use only the Bisection Method to try to approximate all the roots of f(x) for x > 0 with error
less than 10&#8722;6. For each root found, give the starting interval [a, b] and the number of iterations
needed to approximate that root. Carefully explain any difficulties you encounter when trying to
find the roots of f(x).
You must do this problem with the following restrictions.
1. The function f(x) is to treated as a "black box." You give it a value of x and it returns the
value of the function at x.
2. You may not perform any algebraic operations on f(x) nor differentiate it. If you are using a
computer algebra system you may not use any commands like solve, fsolve, findroot, etc.
3. You may assume that f(x) is continuous for 0 < x < 1 and 1 < x < 1.
Since plotting only involves evaluation of f(x) you may plot the function over any interval in its
domain. Graphs of functions can be misleading, so don't make any definitive statements about the
roots based only on a graph. You will lose points if you make any statements about the roots of
f(x) which can't be verified under the conditions given above.


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Solution Summary

Bisection algorithm and programming are investigated.