Minimum of functions in one dimension

Similarly to the most robust methods of solving nonlinear equations, we start with bracketing. Assume that the interval [xy, X ] contains a single minimum point r, i.e., the function f is decreasing up to r and increasing afterwards. Then the function is said to be unimodal on the interval [xy, Xy], This property is exploited in cut-off methods, purported to reduce the length of the interval which will, however, include the minimum point in all iterations. 

The idea we use is similar to bisection, but now we need to evaluate the function at two inner points x and x2 of the interval, where XL xi x2 XU f( i) f(x2), then the minimum point is in the interval [xy, x2], since we assumed that the function is decreasing up to the minimum point, see Fig. 2.b.a. Similarly, f(x ) f(x2) implies that the minimum 

We select the internal points x and x2 with the same spacing from either end, as shown in Fig. 2.7, where i denotes the ratio of the longer segment to the total length of the uncertainty interval, i.e., 

The efficiency of the golden section stems from the special value of the ratio i. We require the ratio of the larger of the two segments to the total length of the interval be the same as the ratio of the smaller to the larger segment, 

To show why this famous ratio X is good for us, assume that f(x ) f(x2) as shown in Fig. 2.8, and hence we cut off the interval [xL, x ). Then the ratio of the remaining two segments is given by 

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