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 . Minimization Using Derivatives Suppose that is unimodal over and has a unique minimum at . Also, assume that is defined at all points in . Let the starting value lie in . If , then the minimum point p lies to the right of . If , then the minimum point p lies to the left of . Our
Introductory Methods of Numerical Analysis