An approach for finding the minimum of in a given interval is to evaluate the function many times and search for a local minimum. To reduce the number of function evaluations it is important to have a good strategy for determining where is to be evaluated. Two efficient bracketing methods are the golden ratio and Fibonacci searches. To use either bracketing method for finding the minimum of , a special condition must be met to ensure that there is a proper minimum in the given interval. The function is unimodal on , if there exists a unique number such that is decreasing on , and is increasing on . In the golden ratio search two function evaluations are made at the first iteration and then only one function evaluation is made for each subsequent iteration. The value of remains constant on each subinterval and the search is terminated at the subinterval, provided that or where are the predefined tolerances. The Fibo
Introductory Methods of Numerical Analysis