Exact Horadam Numbers with a Chebyshevish Accent by Clifford A. Reiter

Transcription

1 Exact Horadam Numbers with a Chebyshevish Accet by Clifford A. Reiter (reiterc@lafayette.edu) A recet paper by Joseph De Kerf illustrated the use of Biet type formulas for computig Horadam umbers [1]. This was a effective demostratio of the power of those formulas ad the grace of APL for implemetatio. That article spurred my iterest i the Horadam umbers a log eglected iterest. Oe of my earlier related iterests was i computig very large Fiboacci umbers [6]. That work takes advatage of idetities ivolvig Fiboacci umbers but does ot directly use the Biet formula. Sice the Biet formulas are most aturally implemeted with floatig poit computatios, the computatios are rapid, but of fiite precisio, o most systems. Thus, my goal i this ote is to describe a very differet way of computig Horadam umbers ad to implemet those ideas i J where exact ratioal computatios are available. Moreover, we will see several additioal examples of the pervasive ature of the Horadam umbers icludig a glace at Chebyshev polyomials. The recursive algorithm that we will use also illustrates a powerful J programmig style. While matrix costructios of the Fiboacci umbers are well kow [2], the correspodig matrix costructios for Horadam umbers are ot early as well kow, but they do appear [9]. Matrix View of Horadam Numbers. Followig the otatio i [1], we will defie the Horadam sequece by H p, H q, H sh rh However, the otatio for these umbers varies greatly. For example, Horadam [3] ad Rabiowitz [5] use a otatio where the recursio cotais a mius sig ad i may cotexts it would be more coveiet to begi the sequeces with H 0. Notice that H as defied above depeds upo p, q, r, ad s i additio to. Whe we eed to emphasize some or all of that rs depedecy, we ca write pq H or pq H for H. We rewrite the above recursio i matrix form as Iteratig this -1 times, we see H H H 0 1 r s H 1 1.

2 2 H H 0 1 r s 1 1 H1. H Cosiderig the iitial coditio, H p 1, H q 0, we see Likewise, whe H p 0, H q 1, we see 1 2 Puttig these together, we have Sice pq H p H q H 1 H H r s H H r s H H 0 1 H H. r s ( 10 ) ( 01 ), we ca compute geeral Horadam umbers by computig powers of the matrix W 0 1. I particular, r s pq H is the dot product of the vector p, q with the first row of W 1. Moreover, the matrix idetities W W m m ad W W W give rise to powerful m m idetities that allow oe to skip may itermediate steps i computig Horadam umbers. Ideed, i the Fiboacci case, idetities arisig from those matrix relatios give methods for rapid computatio of very large Fiboacci umbers [6]. The idetities for the Horadam umbers may be writte explicitly ad implemeted (a worthwhile diversio see Appedix B). However, we will use the graceful array capabilities of J to utilize the matrix relatios directly. I particular, we will recursively compute eve powers of W by W W odd powers of W usig W ad compute the 2 2 WW. I the ext sectio we will implemet these ideas

3 3 Horadam Numbers i J We first defie fuctios that give the 0 th ad 1 st powers of W. I all the fuctios we defie for givig powers of W, we will let the left argumet be the list r,s ad the right argumet be the ultimately desired power eve though some of that iformatio is uused for these low powers. w0=:(=i.2)"_ 2 3 w th power of W w1=:0 1&,:@x:@[ ]W=.2 3 w x=:+/. * 1 st power of W Matrix product W x W W Next we cosider the geeral cases for computig oegative powers of W. There are 4 cases accordig to whether the power is 0, 1, a geeral eve power, or a geeral odd power. The fuctio case distiguishes betwee these cases by addig the power mod 2 to twice the result of the Boolea test: is the power greater tha 1?. case=: 2& + +:@(1&<) (,:case)i we=:[ x~@:w -:@] wo=:w1 x [ w <:@] We see how the four cases deped o the power Eve powers of W : the square of the half power Odd powers of W : W times the previous power w=: w0`w1`we`wo@.(case@]) Ageda o the four cases gives geeral powers 2 3 w 2 W w 50 W

4 4 Recall that the Horadam umbers are the dot product of the list p,q with the first row of W 1. We will follow [1] ad take the left argumet of our Horadam fuctio to be the list p,q,r,s. Thus 2&{.@[ gives p,q ad _2&{.@[ gives r,s. The defiitio of horadam may be read as p,q dot product with the first row of W 1. We apply rak 1 0 sice we expect the left argumet to be a vector ad the right argumet to be the umber of the desired term i the sequece. horadam=:(2&{.@[ x _2&{.@[ {.@w <:@])" horadam The followig exact itegers are cosistet with the correct etries i the table i the appedix of [1].,.2 3 _3 3 horadam _ sxfmt 2 3 _3 3 horadam Short exact format _ (60) (1) (60) The fuctio sxfmt is a utility for displayig the begiig ad ed of exact itegers. It is coveiet to use it whe the umber of digits is very large. These short exact format expressios also iclude the umber of decimal digits of the exact iteger i paretheses. The defiitios of sxfmt (short exact format) ad a related fuctio, sxmfmt, (short exact matrix format) are give i Appedix A ad are available o-lie [7]. We ca also use the horadam fuctio for oiteger p,q,r,s ad maitai exact computatios sice exact ratioal arithmetic is built-ito J. The experimets below are agai cosistet with the correct etries i the appropriate table i [1]. x: _ _ _3.3 Chage decimal to exact ratioal _3r5 49r50 _1089r400 _33r10

5 5 float=: x:^:_1 float x: _ _ _3.3 Chage back to decimal _ _ _3.3 float (x: _ _ _3.30) horadam e19 _ e20 Some Special Horadam Sequeces De Kerf [1] oted some special cases of Horadam umbers ad other special cases are oted i [3,4]. We cosider these to demostrate the pervasiveess of the Horadam umbers. 1 3 _1 2 horadam >:i.10 Arthimetic progressio startig _1 2 horadam >:i.10 Arthimetic progressio startig horadam >:i.10 Geometric progressio horadam >:i.10 Fiboacci umbers horadam >:i.10 Lucas umbers horadam >:i.10 Pell umbers horadam >:i.10 Pell-Lucas umbers horadam >:i.10 Jacobsthal umbers horadam >:i.10 Jacobsthal-Lucas umbers I fact, because of the two step recursive defiitio of the Horadam sequece, may other classical sequeces ad fuctios may be obtaied from the Horadam umbers. For example, the Chebyshev Polyomials [8] are recursively defied by

6 6 2 T( x) 2xT 1( x) T 2 ( x) where T 0 ( x) 1, T( 1 x) x, T 2 ( x) 2x 1. Thus, the value of T( x ) is the th Horadam umber with p,q,r,s equal to x, 2x 2 1, 1, 2x. chebyshev=: (, <:@+:@*:, _1:, +:)@]"0 horadam [ 2 chebyshev 0.9 T(.) load 'plot' $Y=: chebyshev"0 1 X=:_1++:(i.%<:) plot X;Y Figure 1 shows the resultig plot of five Chebyshev polyomials. As oe desceds just to the left of x=1, the 1 st through the 5 th Chebyshev polyomials are crossed i order. Do you otice properties of these polyomials? Several ca be observed. Figure 1. Five Chebyshev Polyomials

7 7 Timig ad Efficiecy The efficiecy of this matrix approach to computig Horadam umbers is of iterest. First, we ca temporarily modify our fuctio w so that we ca see which powers were recursively called. I particular, we chage w as follows. wrsc=:(1!:2)&2 w=: w0`w1`we`wo@.(case@wrsc@]) Write to scree fuctio Modified w horadam Sice there ca ever be two or more odd powers i a row, o more tha 2log 2 ( ) recursive steps are required. Our approach could be improved i a umber of ways. We oly used the top row of the highest power of W that we compute; so we ought to be able to speed thigs up by ot computig the bottom row. Some improvemet ca also be expected if we use 21 W ( W W ) W i order to compute the odd powers of W. However, more dramatic improvemets (betwee a factor of 2 ad 4) ca be obtaied by implemetig vector/scalar formulas istead of usig matrices. See Appedix B for a brief descriptio ad implemetatio. Table 1 shows the required computer time to compute various size Fiboacci umbers usig horadam o a 400MHz Petium based computer. The 5 millisecods eeded to exactly compute the 209 digit Fiboacci umber F 1000 is ot as fast as the floatig poit computatios doe i [1], but eve accoutig for the differeces i processor speed, is remarkably close ad very respectable. We see that computatios i J with thousads or tes of thousads of digits are routie while calculatios producig a couple millio digits take sigificat amouts of time. We

8 8 did ot directly measure the space required for these computatios, but the J sessio as a whole required less tha 80M of memory. F 10 time (sec) (209) (2090) (20899) (208988) ( ) Table 1. Time required to compute some Fiboacci Numbers. Appedix A. Defiitio Summary We give here a summary of our J fuctios. Readers may obtai the script horadam.ijs from [7]. x=:+/. * case=: 2& + +:@(1&<) w0=:(=i.2)"_ w1=:0 1&,:@x:@[ we=:[ x~@:w -:@] wo=:w1 x [ w <:@] w=: w0`w1`we`wo@.(case@]) horadam=:(2&{.@[ x _2&{.@[ {.@w <:@])"1 0 chebyshev=: (, <:@+:@*:, _1:, +:)@]"0 horadam [ wrsc=:(1!:2)&2 sxfmt=:(10&{.,'...('"_,":@(#-'_'&=@{.),')...'"_, _10&{.)@":"0 sxmfmt=: ([,' Appedix B. Direct Implemetatio Usig Idetities I squarig a 2 by 2 matrix we ordiarily compute 8 products. We will see the special ature of the Horadam umbers is such that we ca do this with 3 products. I practice, this leads to almost a 8r3 fold improvemet i efficiecy. We will ot give detailed derivatios of the requisite idetities, but will explai briefly how they may be obtaied. For coveiece, we defie K 0, K 1, K sk rk

9 9 Thus, K 1 1/ r is a reasoable extesio ad i geeral K 1 01 H ad it is t too hard to show by iductio o that rk We ca verify that 2 10 H. Therefore pq H prk 2 qk 1. W rk 1 K rk sk rk 1 usig the matrix power expasio of W i terms of the H s that we saw ad the traslatig its etries ito K s usig the above relatios. ad hece Squarig the right side of W W 2 2 ad equatig the etries i the first rows yields: rk r K rk ad K 2rK K sk K rk K ad K K ( 2rK sk ) which goes from the pair ( K2 1, K2) to ( K1, K) usig just three big multiplicatios. We implemet this as follows. horadamb=:4 : 0"1 0 'p q r s'=.x: x. k0=.((%r),0)"_ k1=.0 1x"_ ke=.((r,1)&x@:*:, {: * ((+:r),s)&x)@:k@-: ko=.(0 1,:r,s)&x@k@<: k=.k0`k1`ke`ko@.case ((p*r),q) x k <:y. ) Lastly, we remark that det( W) r ad hece ( r) det( W ) rk ( sk rk ) rk That idetity ca be used to modify the formula for K 2 1 ito a formula usig just oe big multiplicatio if oe assumes that ( ) r 1 is ot a big computatio relative to K. The ( K2 1, K2) may be computed from ( K1, K) usig just two big multiplicatios. We leave the implemetatio of this approach as a exercise.

10 10 Refereces [1] Joseph De Kerf, From Fiboacci to Horadam, Vector 15 3 (1999) [2] Verer E. Hoggart, Jr., Fiboacci ad Lucas Numbers, The Fiboacci Associatio, Sata Clara, [3] A. F. Horadam, Special Properties of the Sequece W a b p q (, ;, ), The Fiboacci Quarterly, 5 5 (1967) [4] A. F. Horadam, Jacobsthal Represetatio Numbers, The Fiboacci Quarterly, 34 1 (1996) [5] Staley Rabiowitz, Algorithmic Maipulatio of Secod Order Liear Recurreces, The Fiboacci Quarterly, 37 2 (1999) [6] Clifford A. Reiter, Fast Fiboacci Numbers, The Mathematica Joural, 2 3 (1992) [7] Clifford A. Reiter, Horadam Numbers, [8] Theodore J. Rivli, The Chebyshev Polyomials, Joh Wiley & Sos, New York, [9] Marcellus E. Waddill, Matrices ad Geeralized Fiboacci Sequeces, The Fiboacci Quarterly, 12 4 (1974) Clifford A. Reiter Departmet of Mathematics Lafayette College Easto, PA USA