Function Fibonacci

The Simplex algorithm and that of Powell s are examples of derivative-free methods (Edgar and Himmelblau, 1988 Seber and Wild, 1989, Powell, 1965). In this chapter only two algorithms will be presented (1) the LJ optimization procedure and (2) the simplex method. The well known golden section and Fibonacci methods for minimizing a function along a line will not be presented. Kowalik and Osborne (1968) and Press et al. (1992) among others discuss these methods in detail. 

Indirect methods solve the necessary conditions for an optimum (looking at the shape of the function) directly via iteration. Region elimination techniques such as Fibonacci and Golden Section searches use function evaluations only to delete a portion of the independent variable range at each iteration. Interpolation techniques use polynomial fitting (quadratic or cubic oftentimes) to predict the location of the optimum. 

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. 

The symmetry of quasicrystals can be represented by introducing a different perturbation function, which is based on the Fibonacci numbers. An infinite Fibonacci sequence is derived fi-om two numbers, 0 and 1, and is formed according to the following rule ... 

We observe (Figure 5) the expected linear increase of the FLI with time for all the orbits except the periodic one. For such orbit the FLI after a transitory linear increase becomes constant. Our aim is to explain such a behavior but before we study the relation between the FLI values and the order of periodic orbits. At this purpose we have computed the FLI as a function of time for a particular set of periodic orbits, the Fibonacci sequence. 

To avoid any numerical precision problems that may arise with the Mulcrone formulation, R. Biyani has suggested a formulation involving only integer calculations. In particular, we can use a recursive function that computes the sex, s, of the xth person in the yth year using a previously generated sequence of the number of persons taken in each year (the Fibonacci sequence). The recursive relationship is ... 

Therefore, /(x) serves as a generating function for the Fibonacci numbers ... 

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... 

Unfortunately, some of these advantages may also turn out be disadvantages at times too. The Fibonacci method minimizes the maximum final interval of uncertainty only if the required function evaluations are accomplished. 

Many functions have interesting, highly exploitable features (e.g., continuity and derivability). Specifically, many can be well approximated by means of quadratic functions as their minimum is approached. Conversely, the Fibonacci method does not discriminate between functions and takes all of them in the same way the worst one. 

We might also think about coupling the Fibonacci method with other algorithms that can exploit the function s features Fibonacci is a slow but sure method and... 

It is also possible to modify the Fibonacci method to obtain a series that minimizes the maximum final interval of uncertainty when a point is already positioned in the starting interval. In this case, the final interval is not univocally determined by the number of points, but different widths can be obtained depending on the position of the first point and the function to be minimized. 

Another method based on function comparison is the golden section method. The golden section method was proposed before the Fibonacci method. The golden section also exploits the position of the point still inside the new interval of uncertainty, in a sequential search. 

Comparison methods such as Fibonacci s method and golden section search exploit function unimodality within a specific interval of uncertainty with the aim... 

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]). 

Various optimum search methods exist for the minimization of objective functions, which can be used for the estimation of kinetic constants [3], for example, the Fibonacci method, the golden section method, the Newton-Raphson method, the Levenberg-Marquardt method, and the simplex method. Recently, even genetic algorithms have been... 

Fig. 10 A plot of 1 -modular Farey sequences as a function of the natural numbers defines a set of infinite festoons that resembles the arrangement of nuclides in Figs. 5 and 7. The segment, obtained as a subset defined by limiting Fibonacci fractions that converge from 1 to r and subject to the condition A(mod4) = 0 —> 3, corresponds to the observed field of nuclide stability...

Figure 12.2. The Fourier Transform/ p) of the Fibonacci sequence as given by Eq. (12.9), with w = 1 and the 3-function represented by a window of width 0.02 (cf. Eq. (G.53)).

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