Optimization in One Dimension

It is hopefully not hard to see why this mathematical problem (or closely related problems) are important in DFT calculations. The problem of finding the lattice constant for an fee metal that we looked at in Chapter 2 can be cast as a minimization problem. If we do not know the precise arrangement 

For now we will consider the original one-dimensional problem stated above Find a local minimum of /(x). If we can find a value of x where f (x) = df/dx = 0, then this point defines either a local maximum or a minimum of the function. Since it is easy to distinguish a maximum from a minimum (either by looking at the second derivative or more simply by evaluating the function at some nearby points), we can redefine our problem as find a value of x where f(x) = 0. We will look at two numerical methods to solve this problem. (There are many other possible methods, but these two will illustrate some of the key concepts that show up in essentially all possible methods.) 

As a simple example, we can apply the bisection method to find a minimum 

If we use the bisection method starting with xi = 1.8 and x2 = 3, then we generate the following series of approximations to the solution x 2.4, 

These are getting closer to the actual solution as we continue to repeat the calculation, although after applying 

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