Homework 4: Optimization

Due March 1.

1. Write a program to find the minimum of a function of one variable using the Golden Search method.

   As with the root finding homework, write your code in a flexible manner so that you can use it later for other problems, and choose your stopping criteria carefully.

2. Test your implementation on the following problem:

   The rotation curves (that is, rotation velocity vs. distance from the center) for galaxies are observed to rise linearly close to the center, and to be constant far from the center. A possible (but dynamically, not well motivated) function which can be fit to such a rotation curve is;
   

   $$v_{model}(r) = v_{inf} (1 - e^{-r/r_0}),$$
   
   
   where $v_{inf}$ is the assympotic velocity and $r_0$ is a characteristic radius.

   Using the Golden Search method, and assuming that $v_{inf}$ is 100 km/s, find the $r_0$ that gives the best fit of the above formulae to the following ``data'':

   $r_{obs}$ (kiloparsecs)	$v_{obs}$(km/s)
   1	41.96
   2	59.64
   3	77.96
   4	67.57
   5	77.33
   6	90.83
   7	96.2
   8	89.68
   9	96.53
   10	81.7

   (This data will be available on the web site as
   http://www.astro.washington.edu/astro497d/rot.dat.) As a criteria for goodness of fit, use the standard least squares formulae:
   

   $$E = \sum_{i = 1}^{N_{data}} (v_{obs} - v_{model}(r_{obs}))^2$$
   
   

   Plot the ``trajectory'' of the Golden Search iterations by displaying each iterate on a plot of $r_0$ vs. $E$ and connecting successive points.

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