Exercises 31.4 Homework
1.
Parabolic interpolation is also useful when applied just once, without iteration. For example, after evaluating \(f\) on a grid \(x_1, \dots, x_n\) and choosing \(k\) with smallest \(f(x_k)\) (i.e., brute force minimization) we can get a more precise location of the minimum with parabolic interpolation through the points \((x_j, f(x_j))\text{,}\) \(j=k-1, k, k+1\text{.}\)
Write a script that applies the above idea to the function \(f(x) = \sin (x)\) on the interval \([0, 7]\text{.}\) Use \(n=100\) points for brute force minimization, locating a grid point \(x_k\) with the smallest function value \(f(x_k)\text{.}\) Then find a more precise point of minimum \(x^*\) using parabolic interpolation once. Since the exact point of minimum is \(3\pi/2\text{,}\) the script should display the differences \(|x_k - 3\pi/2|\) and \(|x^* - 3\pi/2|\) to illustrate the improvement achieved with parabolic interpolation.
There should be no loops in the script.