Question

Consider the function f(x): = xcos x+eˣ/² over the interval x ∊ [-2, 1]. Write and execute MATLAB scripts/functions that help answer the parts of this question. You may use either of the codes given in the notes: if you use a functions as is, then leave the authorship intact; if you modify the function, then add that you modified; if you rewrite the functions anew, then record that you are the author.

(a) Use MATLAB to plot the function over the interval: why is the function unimodal?
(b) Add code to execute a Dichotomous Search for the minimum to find it to an error ≤ 0.05. Record the x-values computed in the iterations. I
(c) Summarise what the search finds, and the number of functions evaluations.
(d) Now use the same Dichotomous Search for the same function to the same error but over the bigger interval [-2,8]. Record your results. In this case the computed 'minimiser' is wrong: why? Give reasons.

Answer 1

To answer the question, we plot the given function and conduct a Dichotomous Search for the minimum. We summarize the search results and explain the inaccuracies encountered when extending the interval.

Step-by-step explanation:

To answer the given question, we need to plot the function f(x) = xcos(x) + e^(x/2) over the interval [-2, 1] using MATLAB. This can be done by defining the function and using the 'plot' function in MATLAB. The graph will show the function's behavior over the given interval.

To execute a Dichotomous Search for the minimum of the function with an error ≤ 0.05, we can define a function for the search algorithm and use it to iteratively find the minimum. We can record the x-values computed in each iteration to track the progress of the search.

After conducting the search, we can summarize what the search finds as the minimum value of the function and the x-value at which it occurs. The number of function evaluations can also be recorded as the number of iterations performed in the search.

Now, if we conduct the same Dichotomous Search for the same function but over the bigger interval [-2, 8], we may encounter inaccuracies in the computed 'minimizer'. This is because the function has multiple local minima within the interval, which can cause the search algorithm to converge to a different minimum than the global minimum.