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Rootfinding and Optimization : Newton's, Secant, Bisection, False Position Methods

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3. (Rootfinding and Optimization)
(a) Suppose that f is differentiable on [a, b]. Discuss how you might use a rootfinding method to identify a local extremum of f inside [a, b].
(b) Let f(x) = logx ? cosx. Prove that f has a unique maximum in the interval [3,4]. (NB: log means natural logarithm.)
(c) Approximate this local maximum using six iterations of the enclosure methods (Bisection and False Position) with starting interval [3, 4].
(d) Approximate this local maximum using six iterations of the two fixed-point methods (Secant and Newton). For Newton's Method, use = 3. For the Secant Method, use P0 = 4 and p' = 3.
(e) What is your best estimate for p. the location of the maximum?
(f) Provide the following two tables, comparing the four algorithms. The headin for the two tables should be the following.
[TABLE]

(g) Plot the absolute error for all four methods on the same graph.
(h) What happens if you attempt to approximate the maximum by starting Newton's Method with p = 5?

Please see the attached file for the fully formatted problems.

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Solution Summary

Roots are found and optimized using Newton's, Secant, Bisection, False Position Methods. Methods are compared. The solution is detailed and well presented.

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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