Search Fibonacci

The optimum seeking methods which have been found to be particularly useful are the modified Fibonacci search (search by golden section) for one-dimensional searches and the Hooke-Jeeves search for multi-dimensional searches. Beveridge and Schechter (8) give a complete description of these searches. 

A one-dimensional search optimization technique, such as the Fibonacci search, is employed to minimize Equation 8-113. A computer program (PROG81) was developed to estimate the equivalent number of ideal tanks N for the given effluent tracer response versus time data. Additionally, the program calculates the mean residence time, variance, dimensionless variance, dispersion number, and the Peclet number. 

The sequence having been named after Leonardo Fibonacci (1180-1225), a pioneer in the study of infinite series, the technique about to be described is called the Fibonacci search method. 

The last cycle of the method will be a dichotomous search, which we know is minimax for two experiments. Keifer (Kl) and Johnson (J2) show that the Fibonacci search is in fact minimax among all sequential techniques. In order to reduce the interval of uncertainty to less than 1% it only takes 11 Fibonacci experiments, three less than for a sequential dichotomous search. The advantage increases with the number of experiments. 

It should be remembered that all of these methods are very conservative, since they are all based on the assumption that nothing is known about the function y except that it is unimodal. If, as is often the case with physical systems, the function is known to be smooth and continuous, the engineer may wish to fit a curve to his points and estimate the maximum by ordinary differentiation. When doing this, however, it is worthwhile to locate the points according to the Fibonacci sequence so as to be able to shift to a Fibonacci search if the function does not behave according to preliminary estimates. 

How this optimization is accomplished is not pertinent to the present development. The operator may work by Fibonacci search, differential calculus, or even brute-force evaluation of all possible decisions all that is important is the resultant finding of the optimal yield Y and the associated decision D. 

Federal environment regulations, 75-78 Feed-tray location in distillation towers, lo Fiberglass reinforced plastics (FRP), 436-437 Fibonacci search, 407 Fifo method for materials accounting, 148... 

G. A. Omura, Modified Fibonacci search. J Clin Oncol 21 3177 (2003)... 

Santora, N.J. and Auyang, K. (1975) Non-computer approach to structure-activity study. An expanded Fibonacci search applied to structurally diverse types of compounds. J. Med Chem. 18 959-963. 

Other line search methods that involve only function evaluations, that is, no derivative calculations, are the dichotomous search, the Fibonacci search (Kiefer 1957), and the quadratic fit line search. The Fibonacci search is the most efficient derivative-free line search technique in the sense that it requires the fewest function evaluations to attain a prescribed degree of accuracy. The quadratic fit method... 

The determination of the minimum of the objective function by changing the values of Ani can be performed iteratively or with the help of appropriate solvers (e.g., one-dimensional search routines, such as the method of the golden section or the Fibonacci search [11]). 

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