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The golden section search is a technique for finding the extremum (minimum or maximum) of a strictly unimodal function by successively narrowing the range of values inside which the extremum is known to exist. The technique derives its name from the fact that the algorithm maintains the function values for triples of points whose distances form a golden ratio. The algorithm is the limit of Fibonacci search (also described below) for a large number of function evaluations. Fibonacci search and Golden section search were discovered by Kiefer (1953). (see also Avriel and Wilde (1966)).The diagram above illustrates a single step in the technique for finding a minimum. The functional values of are on the vertical axis, and the horizontal axis is the x parameter. The value of has already been evaluated at the three points:, , and. Since is smaller than either or, it is clear that a minimum lies inside the interval from to (since f is unimodal).

approximation, golden ratio, mathematics