CALCULATING FIBONACCI VECTORS

THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet of Mathematics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: ovak@ithacaedu Draft of 008 May 30 1

INTRODUCTION The aim of this chapter is to fid a formula to calculate ay Fiboacci vector directly, ot recursively, geeralizig Biet s formula for the origial Fiboacci sequece We begi with the recursive defiitio of Fiboacci vector sequeces Next, we discuss the roots of the Fiboacci characteristic polyomial ad costruct the root vector ad root matrix The, we cosider liear combiatios of the powers of the root vector Evetually, we fid the particular combiatio that represets Fiboacci vectors We show that i the case of two-dimesioal vectors, this represetatio of the compoets of the vectors is idetical to Biet s formula for calculatig Fiboacci umbers We also illustrate the use of the Fiboacci vector calculatio formula i the case of three-dimesioal vectors THE RECURSIVE DEFINITION Let be the dimesio of a vector space Let F represet a sequece of vectors i this - dimesioal space, defied as follows Such a sequece of vectors is called a Fiboacci vector sequece ad each vector i the sequece is called a Fiboacci vector Let F k represet the Fiboacci vector of idex k, the vector that is displaced by k steps i the sequece from the seed vector F 0 To move oe step from a give vector F k to the ext vector F k + 1 i the sequece is accomplished by applyig the -dimesioal Fiboacci vector sequece augmetatio matrix Q F T k + 1 Q F T k 1 The superscript T attached to the symbol of a vector or matrix idicates the traspose of that etity Because we usually defie vectors without such a superscript to represet row vectors, for example F k, the symbol of a vector with a T superscript represets a colum vector, for example F T k I this recursive defiitio of a Fiboacci vector sequece, the augmetatio matrix Q trasforms the give vector ito the ext vector 0 0 0 1 0 0 1 1 Q 0 1 1 1 1 1 1 1 The elemets of the augmetatio matrix are represeted by Q i, j, where 1 i is the row idex ad 1 j is the colum idex 0 i + j Q i, j 1 i + j > The effect or the applicatio of the augmetatio matrix is to make the ith compoet of the ext vector equal to the sum of the last i + 1 compoets of the give vector Note that, because the augmetatio 3

matrix is a symmetric matrix, Q T Q, we ca write equatio 1 without superscript T s F k + 1 F k Q 4 The two equatios 1 ad 4 are equivalet defiitios of a Fiboacci vector sequece They are described as recursive defiitios, because they are used repeatedly to relate vectors that are separated by more tha oe step i the sequece The recursive defiitio of the sequece is ot eough, however We also eed to specify a startig poit, the seed vector of the sequece For a Fiboacci vector sequece, we specify the followig seed vector F 0 F 0 0 0 0 1 5 All but the last compoet of the seed vector are 0s; the last compoet is a 1 0 1 j < F 0, j 1 j 6 Note that we could have specified a differet seed vector, either trivially by specifyig a differet vector i the same sequece, or sigificatly by specifyig a vector ot i this sequece ad geeratig a o-fiboacci vector sequece But we shall restrict our attetio here to Fiboacci sequeces grow from the specified seed vectors The set of the first Fiboacci vectors F k for 0 k 1 forms the the seed matrix F 0 1 F 0 F 0 F 1 1 F 1 F 0 F 0 Q F 0 Q 1 The kth row of the seed matrix is the Fiboacci vector of idex k 1, F k 1 7 F 0 1 k F k 1 F 0 Q k 1 8 THE CHARACTERISTIC POLYNOMIAL The characteristic polyomial of degree i the variable x is D x, D x Q xi 9 where the pair of vertical lies extracts the determiat of the eclosed matrix ad I is the by idetity matrix It may be expressed explicitly as a polyomial, a power series of degree i the variable x D x d jx j 10 j0 3

I aother chapter, we determie the values of the coefficiets d j This polyomial has roots x i for 1 i The polyomial may also be writte i terms of these roots D x 1 i1 x x i 11 The roots of the characteristic polyomial are real ad distict, so that they may be ordered We will place the roots i ascedig order, so that i < j x i < x j The ordered set of roots is the root vector x x x 1 x x 1 It is coveiet to have a otatio for a vector, each compoet of which is a certai power of the same compoet of aother vector Therefore, defie the jth power of a vector, v j, as the vector the ith compoet of which is the jth power of the ith compoet of the vector v v j i vi j 13 Thus, for example, we will write x j i for the ith compoet of the jth power of the vector x, istead of its equivalet expressio x i j for the jth power of the ith compoet of the vector x Defie the root matrix X as the matrix whose ith row, X i, is x i 1, the i 1th power of the root vector x X i x i 1 14 Thus, the elemet X i, j of the root matrix X i its ith row ad jth colum is the i 1th power of the jth root of the characteristic polyomial of degree X i, j x i 1 j 15 Displayed i more detail, the root matrix X appears as follows x 0 1 1 1 1 x 1 x x 1 x x X x 1 x 1 x 1 1 x 1 x 1 16 Here, the symbol 1 represets the -dimesioal row vector, all of whose elemets are 1s The defiig property of the roots of the characteristic polyomial is that, if the variable x is set equal to the value of oe of its roots x i, the characteristic polyomial evaluates to zero D x i d jx j i 0 17 j0 4

Let us form arbitrary liear combiatios A k of the kth powers of the roots x i of the characteristic polyomial A k x k a x k ia i 18 Thik of this as the ier scalar or dot product of x k, the kth power of the -dimesioal root vector x, with a arbitrary -dimesioal colum vector a Such arbitrary combiatios satisfy the followig relatio d ja j 0 19 j0 j0 i1 i0 j0 i0 j0 The proof of this relatio 19 follows straightforwardly from the defiitio of the arbitrary combiatios 18 ad the defiig property of the roots of the characteristic polyomial 17 by rearragemet d ja j d j x j ia i a i d jx j i a id x i 0 0 i1 THE GENERALIZATION OF BINET S FORMULA Now, let us form special liear combiatios of the roots of the characteristic polyomial to represet the compoets of Fiboacci vectors Form the jth compoet of the kth Fiboacci vector, F k, j as the ier product of x k with a differet -dimesioal colum vector f, j for each compoet F k, j x k f, j x k if i, j 1 The symbol f represets a matrix, the Fiboacci root power combiatio matrix; f i, j, the elemet i its ith row ad jth colum; f, j, its jth colum vector; ad f i,, its ith row vector The kth Fiboacci vector is represeted by F k F k x k f i1 x k if i, We do ot kow the value of the combiatio matrix f, but if we could figure it out, the we would have a procedure for calculatig ay Fiboacci vector, without havig to calculate all the vectors i betwee i1 Let us represet aew the seed matrix F 0 1, which we defied i 7, by substitutig ito each of its rows this represetatio of each of the first Fiboacci vectors as combiatios of powers of the root vector F 0 F 0 F 1 1 F 1 x 0 f x 1 f x 1 f 1 x x 1 f 3 5

I equatio 3, you see that the seed matrix is the product of what you recogize as the root matrix X ad the combiatio matrix f F 0 1 X f 4 We ca fid the combiatio matrix f by multiplyig this equatio by the iverse of the root matrix X 1 f X 1 F 0 1 5 Now, substitute this ito equatio to fid the kth Fiboacci vector Fially, we fid that the kth Fiboacci vector is the product of the kth power of the root vector x k, the iverse of the root matrix X 1 ad the seed matrix F 0 1, i that order F k x k X 1 F 0 1 Here at last is the Fiboacci vector calculatio formula, which it was the aim of this chapter to fid To costruct the seed matrix, you have to calculate the first 1 vectors after the seed vector, which is give Ad to costruct the root vector ad the root matrix, you have to determie the roots x i for 1 i of the characteristic polyomial D x But havig doe so, it is ot too difficult to apply the Fiboacci vector calculatio formula of equatio 6 to calculate ay vector you wat The recursive defiitio of Fiboacci vector sequeces, equatio 1 or 4, provides a procedure to calculate ay Fiboacci vector, but i order to do so, you must calculate all the vectors betwee the seed vector ad the oe that you wat, oe after the other What we have foud here i equatio 6 is a formula for calculatig ay Fiboacci vector, without the eed to calculate all the vectors i betwee It is certaily a more complicated process, but if your aim is to calculate a Fiboacci vector F k of high idex k, it is surely less tedious The formula i equatio 6 is the geeralizatio to Fiboacci vectors of Biet s formula for calculatig the umbers of the origial Fiboacci sequece, for, as you will see i the ext sectio, our formula for reduces to Biet s formula for the case 6 Recall that the compoets of the vectors of the two-dimesioal Fiboacci vector sequece F are the umbers of the origial Fiboacci sequece F BINET S FORMULA Cosider the case The characteristic polyomial is D x 1 x + x The root vector x is x 1, x, but let us write it as x 1, x for brevity i this case x 1, x 1 φ, φ 1 5, 1 + 5 1 cos α, cos α 7 where α π 5 is the miimal agle betwee the edges sides or diagoals of a regular petago 6

Now, let us apply the geeral formula 6 for calculatig Fiboacci vectors i dimesios to calculate the kth Fiboacci vector i two dimesios 1 F k x k 1, x k 1 1 0 1 1 φ k, φ k 1 1 x 1 x 1 1 1 φ φ φ k 1 φ k, φ k 1 1 φ k 1 + φ k 1 φ k 0 1 1 1 1 where F k, F k 1 + F k F k xk x k 1 φk 1 φ k x x 1 φ 1 φ 1 F k, F k + 1 1+ k 5 5 1 k 5 cos α k 1 cos α k 4 cos α 1 8 9 Observe that this is Biet s formula for calculatig the kth Fiboacci umber Thus, we see that the geeral formula 6 for calculatig Fiboacci vectors i dimesios is the geeralizatio of Biet s formula from two to > higher dimesioal Fiboacci vectors A COMPUTER PROGRAM TO CALCULATE FIBONACCI VECTORS To calculate -dimesioal Fiboacci vectors usig the Biet formula is possible with pecil ad paper, but it ca be doe more readily with a had-held calculator To calculate higher dimesioal Fiboacci vectors, however, you really eed more complex techology, a computer ad sophisticated mathematical software The followig Mathematica program calculates the kth Fiboacci vector i dimesios usig the geeralized Biet formula 3 qm Table[If[i+j<,0,1],{i,1,},{j,1,}] sv Table[If[i<,0,1],{i,1,}] sf[v_] : qm v sm NestList[sf,sv,-1] pc Table[-1^k+Floor[-k/]*Biomial[-k+Floor[k/],-k],{k,0,}] cp Sum[pc[[k+1]]*x^k,{k,0,}] xr Flatte[NSolve[cp0,x]] xv Table[xr[[i,]],{i,1,}] xm Table[xv^i,{i,0,-1}] fv[k_] : xv^k Iverse[xm] sm k 0 Prit[fv[k]] 7

I this program, the symbol represets the dimesio, qm represets the augmetatio matrix Q, sv, the seed vector F 0, sm, the seed matrix F 0 1, pc, the vector of characteristic polyomial coefficiets d, cp, the characteristic polyomial D x, xv, the root vector x, xm, the root matrix X, ad fv[k], the kth Fiboacci vector F k Specific values 3 ad 0 are assiged i this program to the variables ad k, respectively, but you are free to assig them ay values you choose Almost every lie i this program implemets a equatio i this chapter For example, the secod lie implemets the defiitio of the augmetatio matrix Q, equatio 3 The third lie from the bottom is the climax of the program, as it implemets the geeralized Biet formula, equatio 6, to calculate a Fiboacci vector F k for arbitrary ad k There is, however, oe lie that implemets a equatio that is ot i this chapter, the sixth lie, which implemets the defiitio of the characteristic polyomial coefficiets d k by a equatio i the chapter o the characteristic polyomial CASE 3 Now, cosider the case 3 The characteristic polyomial is D 3 x 1 + x + x x 3 The root vector x 3 is x 1, x, x 3 3, but, as before, let us write it as x 1, x, x 3 for brevity x 1, x, x 3 1 7 cosβ1 β 3, 1 7 cosβ1 + β 3, 3 3 3 1 7 cosβ3 30 0801937735, +0554958131, +46979604 31 There are iterestig relatios amog the roots β 1 ta 1 3 π 3 β 3 ta 1 7 3 3 33 x 1 1 1 x 34 x 1 + x + x 3 35 CONCLUSION I this chapter, we achieved the goal which it was our aim to reach We foud a formula, equatio 6, for calculatig Fiboacci vectors, which we showed is the geeralizatio of Biet s formula, equatio 9, for calculatig Fiboacci umbers The importace of this formula is that it allows you to calculate a desired Fiboacci vector F k of high idex k > without havig to calculate all the vectors betwee it ad the seed vector F 0 The sigificace of this fidig is that demostrates oce agai that Fiboacci vector sequeces geeralize the origial Fiboacci sequece i the most atural ad powerful maer The most useful techique we have employed i these ivestigatios is the cosistet use of vector ad matrix 8

otatio The most amazig observatio we have made here is that, although Biet s formula ivolves 5, the square root of 5, ad its higher dimesioal geeralizatios ivolve complicated combiatios of the roots of other umbers, the output of these formulas is always itegral 9