One- and multidimensional Fibonacci search very easy!

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1 One and ultidiensional ibonacci search One and ultidiensional ibonacci search very easy!. Content. Introduction / Preliinary rearks...page. Short descrition of the ibonacci nubers...page 3. Descrition of a sile algorith...page 4. ibonacci search for discrete functions...page 4 5. Progra for the ndiensional ibonacci search...page 5 6. Derivation of the ibonacci forula...page 9. Introduction / Preliinary rearks About the ibonacci nubers, there are any aers in the literature, therefore, discusses the features in this article only briefly. In the literature and on the Internet. any treatises can be found. Many descritions of search using ibonacci nubers assue that starting fro an interval [a, b] within which the iniu or axiu of a function to be deterined, will be deterined according to certain rules with given search stes ore and ore saller intervals in where the iniu or axiu is sought. This rocess is of course correct, but the derived algorith is rather colex and for a ultidiensional search highly iractical. As a byroduct of y diloa thesis (about 35 years ago) that dealt with tieotial control, I had designed an algorith, which extreely silified the ibonacci search and easily alicable to ultidiensional robles ade. ro age 5 you will find a Python rogra for the n diensional ibonacci search. Recently added was a treatise on the ibonacci search for discrete functions (age 4).. Short descrition of the ibonacci nubers or ibonacci nubers is the following foration rule: = = = + for > This results in the infinite sequence of nubers:,,,,3,5,8,3,,.. The individual values can not only recursive, but also directly calculate with the forula = 5 éæ êç + ê ëè 5 ö ø æ ç è 5 ö ø ù Eq. () 3. Descrition of a sile algorith At the start of the ibonacci search it has to be set with how any stes the Miniu / Maxiu of a function y = f (x) in an interval x є [a, b] shall be found. Is the nuber of search stes, at the end of the calculations, it reains an uncertainty = + Thus ust be chosen so that the residual error is corresondingly sall. Eq. () Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page of

2 One and ultidiensional ibonacci search It is iortant that in the onediensional search that the function in the interval ust be uniodal. In the ultidiensional search the exained area ust be convex. That eans, in the interval ay exist only one iniu / axiu. Otherwise the search will not be successful. As noted in ite, the usual algorith functions such that eversaller intervals are fored. After the th ste is then the solution found. With the algorith described here, the ethod is extreely sile. Basis for the silified algorith was the idea to avoid the deterination of the various intervals. There were the distances between the various exained oints deterined. As a result was found a very sile relationshi. Consider the interval to be exained [a, b]. The nuber of search stes is. urtherore, we assue that a iniu should be searched. In this interval, the first two oints to be exained are x = a + + The distances of x to the left interval boundary and x to the right interval boundary are equal. In our considerations, we start the calculation fro the left interval boundary a. The distance between the first Search oint and a is v = + or the distance of the second Search oint fro the first follows v and ( + ) = = + = = The oint with the saller function value is taken as a starting oint for the next search ste. Other considerations, which will not be discussed in detail here showed the distance of the third Point fro the revious search to x a v 3 = + and so on u to the th search ste v = = + + Generally follows the ste width of the kth search ste to vk = + k + Eq. (3) This is the basis of a flow chart for the onediensional ibonacci search algorith like shown on the next age. It is so sile, so that the extension to higher diensions is not a roble. Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page of

3 One and ultidiensional ibonacci search Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page 3 of

4 One and ultidiensional ibonacci search 4. ibonacci search for discrete functions Is not given a function y = f (x), but it is only y in the for of a finite sequence of nubers as they can for exale be stored in an array, then the algorith like described before ust be odified soewhat. Suose in an array Ay of the length l is the sequence consisting of l discrete values stored. ro this values the iniu or axiu value shall be deterined. It is understood iediately that in this case, only integer increents ay be used. ro Eq. (3) shows that this is true, if the interval length (ba) is equal +. The following three features are to be considerd:. The length l of the array Ay is not the difference between the first and last index, but it is larger by one.. ro Eq. () results fro ba = + an uncertainty interval of. 3 In the ibonacci search the function values at the interval ends are not considered. When the search is discrete but this is necessary. This follows + = l + = Þ l = + or the otial calculation of the iniu / axiu of such a sequence arise now following rules:. The length of the array or the nuber of discrete values lus ust be a ibonacci nuber.. After deterination of the index of the ibonacci nuber can then be deterined by stes the iniu / axiu. 3. To consider all of the values, at the beginning of the sequence a duy is to be added. The value is uniortant, since it is only needed as a starting oint and does not enter into the calculation. If no duy added, the length of the first search ste ust be reduced by Sale Given is the following sequence: {,3,5,6,8,9,,3,5,7,9,8} There are values. Added by is 3 and this is the ibonacci nuber 7. Consequently, the nuber of search stes is = 7 = 5. With the added duy follows the sequence {,,3,5,6,8,9,,3,5,7,9,8} The order of search lengths is: 5, 4, 3,, (5,3,,,). At the iniu search sequentially following the above sequence nubers would be queried: 8>3>5>3> This follows the iniu value of. At axiu, it would be the following search order: 8>3>7>9>8. This follows the axiu value to 9. Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page 4 of

5 One and ultidiensional ibonacci search 5. Progra for the ndiensional ibonacci search A Python scrit that the iniu of a function y = f (x,, x n) = f (x) calculates is to find on the following ages. The nuber of indeendent variables is arbitrary (n ). This was ade ossible by recursively calling the search function. The indeendent variables are the n coonents of the vector x, x[],..., x [n]. Basis of the rogra, the slightly odified algorith for the onediensional search (see age 3). or the correct oeration of the rogra, the following entries are required:. In the array are for the n variables x[],..., x [n] entered the required nuber of search stes.. In the array intervall are for the n variables x[],..., x [n] entered the search interval boundaries. 3. In the function definition def UNCTION (x): the function y = f(x) is calculated and returned. To the axiu of a function y = f (x) to deterine with this rogra, there are two ossibilities:. In all cases, where y and y be coared, the coarison oerator '<' will relaced by '>'. Or even siler. Set function y = f(x). The rogra can be easily orted to any other rograing languages. It can be downloaded fro htt:// Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page 5 of

6 One and ultidiensional ibonacci search #!/usr/bin/env ython # Progra for an ndiensional search of the iniu of a function # by eans of the ibonacci nubers. # The rogra is written as an ython file(version.5.4). # It can be easily orted to any other rograing language. #****************************************************************************************** # Version. / revised version # Date: 6.. # Author: KlausEckart Schul / Berlin # The rogra can be used as required, odified and assed on, # however, the reference to the author aintained. # or any alication the author assues no liability. #****************************************************************************************** # Very sile exale for n=5diensional # Data entry =[4,6,8,,] # Nuber of search stes for each of the n=5 intervals. intervall=[[,8], # n=5 search intervals [8,], [,8], [,8], [,8] ] # Definition of the function y=f(x,...,xn) =f(x) def UNKTION(x): return (x[]4)**+(x[]+4)**+(x[]4)**+(x[3]4)**+(x[4]4)** # #===== Recursive rocedure for the ndiensional ibonacci search =====# def fibo_search(di): global k,x,x,y,x,y,s,,es, k[di]= while k[di] < [di]: if di==len(): if k[di]==: x[di]=x[di]+[[di]k[di]]*es[di] x[di]=x[di] y[di]=unktion(x) s[di]= else: x[di]=x[di]+[[di]k[di]]*es[di]*s[di] y[di]=unktion(x) if y[di]<y[di]: x[di]=x[di] y[di]=y[di] else: s[di]=s[di]*() else: if k[di]==: x[di]=x[di]+[[di]k[di]]*es[di] x[di]=x[di] s[di]= fibo_search(di+) y[di]=y[di+] else: x[di]=x[di]+[[di]k[di]]*es[di]*s[di] Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page 6 of

7 One and ultidiensional ibonacci search fibo_search(di+) y[di]=y[di+] if y[di]<y[di]: x[di]=x[di] y[di]=y[di] else: s[di]=s[di]*() k[di]=k[di]+ ################################################################################ #+++++ Initialiation of the variables # Deterination of the array for the ibonacci nubers '' _ax=ax() =[]*(_ax+3) []=. []=. for i in range(,_ax+3): [i]=[i]+[i] # Deterination of 'es[]=interval_length/ib[+]' for all diensions # es[] is the ax. ossible absolute error for each diension es=([]*len()) for i in range(,len()): es[i]=(intervall[i][]intervall[i][])/[[i]+] # Arrays 's' for the search direction, and 'k' for the control variables s=[]*len() k=[]*len() x=[.]*len() x=[.]*len() x=[.]*len() y=[.]*len() y=[.]*len() for i in range(,len()): x[i]=intervall[i][] # Start of the Progra fibo_search() # Result Outut rint x # ndiensional array with the n solutions rint y[] # Result y=f(x) Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page 7 of

8 One and ultidiensional ibonacci search Very sile ndiensional search with ibonacci nubers (n, this eans di ) Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page 8 of

9 One and ultidiensional ibonacci search Aendix 6. Derivation of the ibonacci forula Introductory Rearks or the Derivation of forula a tool is used, as is often the case in atheatics. The original function fro the original area (tie area) is first transfored into the iage area, where it is rocessed with sile tools and then transfored back into the original area. or exale, the Lalace transfor is a very suitable tool for solving differential equations. With it a tie function f(t) will assigned with the socalled onesided Lalace transforation by eans of the Lalace integral a colex function of frequency. The Lalace transfor of derivatives can be deterined as follows: Or generally { } Iortant is the tie shifting function to right. Using Lalace transfor a differentiation in the original area shall be a ultilication in the iage area. or the reverse transforation back to the original field can be used the socalled Residue forula. Here, n is the order of the ole at = ν. ò t L f ( t) = ( ) = f ( t) e dt it = + j ' () { } { } { } L f ( t) = L f ( t) = L f ( t) f () { } { } n ( n) n n k ( k) å L f ( t) = L f ( t) f () t { } { } k= L f ( t t ) = e L f ( t) ( t > ) d f t L e e ( n )! d n t t n { } å { } å n { } ( ) = ( ) = Res ( ) = li ( ) ( ) = If f(t) not a continous function but a sequence [x n], then can be used the discrete LalaceTransforation (L*) to transfor with the relation * * { n} L ( ) å n n= L x = X = x e nt If in this forula will set e T =, then it coes to socalled transfor, which we will use for our case. This roduces the following transforation and inverse transforation rules: * n Z { xn} = X ( ) = å xn n= x n In this case is the order of the ole at = k. d = åres éx = X = ë k k { k } * n * n ( ) ù å li ( ) ( ) k k ( )! d Again, the tie shifting is iortant, esecially for the solution of our roble. It is k { } = { } Z x Z x and nk n ( ) é Z { x } éz { x } x x... x ù Z { x } x ë k k k k i n+ k = ë n k = ê n å i i= Such as the Lalace transfor is suitable for solving differential equations, the transfor can be used to solve difference equations. ù Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page 9 of

10 One and ultidiensional ibonacci search Calculation or the ibonacci nubers is (look age) = + for > and the initial conditions = und = To can use the initial conditions, the nd tie shift ust be alied. It follows (using the variable x) x + = x + x + with x = and x = The alication of the transforation yields This follows * * * é ëx ( ) x x ù = X ( ) + é ëx ( ) x ù é ë ( ) ù = ( ) + é ë ( )ù * * * X X X * X ( ) = 5 with, ( ) ( )( ) * X ( ) = = ± = ± 5 Now the reverse transforation takes lace by eans of Residue forula. Since and are sile oles, is for both cases =. * d x = åres éx ( ) = li X ( ) ( ) = ë k k k= k k= * { k } x = li X ( ) ( ) ( )( ) ( ) ( )( ) ( ) li { k } * ù å k k ( )! d x = li + x å ( ) ( ) é + 5 ù é 5 ù ê = + = = ë ê ë ( + 5) ( 5) x éæ+ 5 ö æ 5 ö ù = ê 5 êç ç è ø è ø ë Literature Gerhard Wunsch: Systeanalyse Bd., Lineare Systee, VEB Verlag Technik Berlin,969 Dil.Ing (TU) KlausEckart Schul / Deceber 9 Birnbauring 64 Addition Aug. D359 Berlin Gerany Page of