A MATHEMATICA PACKAGE FOR COMPUTING ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF P-FINITE RECURRENCE EQUATIONS. 1. The Problem

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1 A MATHEMATICA PACKAGE FOR COMPUTING ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF P-FINITE RECURRENCE EQUATIONS MANUEL KAUERS Abstract. We describe a simple package for computig a fudametal system of certai formal series solutios up to a prescribed order of a give P-fiite recurrece equatio. These solutios ca be viewed as describig the asymptotic behavior of sequeces satisfyig the recurrece. 1. The Problem The Mathematica code described below solves the followig problem: Give a liear recurrece equatio of order r with polyomial coefficiets (also kow as a P-fiite recurrece) p 0 ()a + p 1 ()a p r ()a +r = 0 Fid r liearly idepedet solutios of the form γ e u(1/r) ρ α( ( 0 + β 01 + β 0 + β ) + ( 0 + β 11 + β 1 + β ) log() + + ( 1 + β k 11 + β k 1 + β k ) log() k 1) where αγρβ ij are costats rsk are positive itegers ad u is a polyomial. It is well kow ad ot difficult to compute this data [6] ad besides our implemetatio described below there are several others which do the same job [7 1]. The Mathematica code I programmed a couple of years ago was origially iteded for private use oly but over the time I have also give copies to other users ad this ofte raised the questio o how to use the package ad how exactly to iterpret its output. The purpose of this techical report is to aswer these questios. A priori the output is oly correct i some formal algebraic sese. But accordig to Birkhoff ad Trjitzisky [] it is also correct aalytically i the sese that every sequece (a ) satisfyig the iput recurrece has a liear combiatio of the output series as its asymptotic Supported by the Austria FWF grat Y464-N18. 1

2 MANUEL KAUERS expasio. This meas i particular that we ca obtai asymptotic estimates of the form a = c γ e u(1/r) ρ α( ( 0 + β β 0N N + O( 1 N+1 )) + + ( 1 + β k β k 1N N + O ( 1 N+1 )) log() k 1 ) ( ) for ay prescribed N ad some costat c which ca be determied umerically to high accuracy. The also a c γ e u(1/r) ρ α log() k 1 ( ) a where a b meas asymptotic equivalece i the sese that lim b = 1. Ulike the algebraic correctess the aalytic correctess is ot easy to show. It should be remarked that the argumets give by Birkhoff ad Trjitzisky are log ad complicated ad that some people hesitate to believe i them. O cocrete examples however I have ever observed ay mismatch betwee the actual asymptotic behavior of a solutio (a ) ad the formal expasios. A alterative approach whose uderlyig aalytic theory is more widely accepted but which would require more programmig effort is to go via the geeratig fuctio A(z) := =0 a z of the sequece (a ) uder cosideratio. Flajolet ad Sedgewick [3] give a comprehesive accout o the correspodece betwee the asymptotic behavior of A(z) ear its sigularities closest to the origi ad the asymptotic behavior of the sequece (a ). Salvy s Maple package gdev [5] follows this approach.. The Package The package is available for dowload at the URL It was writte ad tested for Mathematica 6 ad 7. Two commads are provided by the package. The first Asymptotics takes a recurrece as iput ad returs the domiat term of all its (formal) asymptotic solutios. I[1]:= << Asymptotics.m Asymptotics Package by Mauel Kauers c RISC Liz V 0.3 ( ) I[]:= Asymptotics[( + 1)f[ + ] (3 + )f[ + 1] + ( + )f[] f[]] { (3 5 ) (1+ ( ) 5)/ (1 5)/ Out[]= For processig this output further it ca at times be more coveiet to get it ot as a expressio. For this case we offer the possibility to receive the output i a iteral format: I[3]:= Asymptotics[(+1)f[+] (3+)f[+1]+(+)f[] f[] Retur iteral ] { { { Out[3]= { 1+ 5 { { 1 5 {1

3 ASYMPTOTIC EXPANSIONS 3 I this format a solutio () γ exp(µ1 1/r + µ /r + + µ r 1 1 1/r )ρ α e ( (0 β β 0N N + O( 1 )) N ( 1 + β k β k 1N N + O ( 1 )) ) log() k 1 N+1 is represeted i the form { γ {µ1... µ r 1 ρα {0β {1β k Here is a example with a logarithmic term: I[4]:= Asymptotics[( + )f[ + ] ( + 3)f[ + 1] + ( + 1)f[] f[]] { 1 Out[4]= 1 + log() I[5]:= Asymptotics[(+)f[+] (+3)f[+1]+(+1)f[] f[] Retur iteral ] Out[5]= {{0 1 { 0 {1 {0 1 { 0 {0 1 {1 0 By default the commad determies oly the domiat terms of the solutio. If more terms of the expasio are desired they ca be requested by explicitly specifyig the order. I[6]:= Asymptotics[( + 1)f[ + ] (3 + )f[ + 1] + ( + )f[] f[]order 3] { (3 5 ) (1+ ( 5)/ ) 3 ( ) (1 ( ) 5)/ I[7]:= Asymptotics[( + 1)f[ + ] (3 + )f[ + 1] + ( + )f[] f[]order 5] { (3 5 ) (1+ ( 5)/ ) 5 ( ) (1 ( ) 5)/ Out[6]= Out[7]= Ofte oly the higher order terms of oe of the solutios are of iterest or oe computes with big effort the first terms of a expasio ad realizes oly afterwards that some additioal terms are eeded. For these cases istead of wastig computatio time ito the computatio of uiterestig data or ito the recomputatio of data which is already kow there is the secod commad of the package FurtherTerms which takes as iput oe trucated series solutio i the iteral format ad completes it to a trucated series solutio of the specified order. I[8]:= sols = Asymptotics[( + 1)f[ + ] (3 + )f[ + 1] + ( + )f[] f[] Order 3 Retur iteral ] { { Out[8]= { 1+ 5 { { { 1 5 { I[9]:= FurtherTerms[( + 1)f[ + ] (3 + )f[ + 1] + ( + )f[] f[]sols[[]]5] { { 1 5 { Out[9]=

4 4 MANUEL KAUERS I[]:= FurtherTerms[( + 1)f[ + ] (3 + )f[ + 1] + ( + )f[] f[]%15] { { 1 5 { Out[]= The Multiplicative Costat The sequece solutios of a P-fiite recurrece equatio p 0 ()a + p 1 ()a p r ()a +r = 0 form a vector space of fiite dimesio. A particular solutio (a ) ca therefore be characterized uiquely by a fiite umber of iitial values a 0 a 1... a N. Give such a particular solutio we may woder about its asymptotic behavior. Clearly if we kow r liearly idepedet asymptotic solutios (b (1) )... (b (r) ) (e.g. from the Asymptotics commad) the we have a c 1 b (1) + + c r b (r) ( ) for certai costats c 1... c r. I geeral these costats caot be computed i closed form but it is possible to obtai very accurate umerical approximatios to them. Here is how A Sigle Domiat Term. First assume for simplicity that the asymptotic solutios (b (1) )... (b (r) ) are such that oe of them asymptotically domiates all the others b say lim (1) = 0 for all k Æ ad i =... r. The the terms (b () )... (b (r) ) are too k b (i) small to cotribute to the asymptotics ad we have i fact a c 1 b (1) ( ) i.e. there is oly a sigle costat c 1 to be determied. Also if fier asymptotic estimates with higher order terms are cosidered oly those comig from (b (1) ) will play a role. I this case the computatio of c 1 is easy. Sice we have lim approximatios for c 1 by computig the quotiet b (1) b (1) a = c 1 we ca obtai decet /a for some large idex. The higher the idex ad the more terms of the asymptotic expasio (b (1) ) are take ito accout the more digits of the quotiet b (1) /a will agree with the digits of the actual costat c 1. I the followig typical example sessio we defie the sequece (a ) by two iitial values ad a recurrece of secod order. This allows to compute a for every specific idex (e.g. for = 50). Next we determie the domiat terms of the two asymptotic solutios of the recurrece. The secod domiates the first because the basis of its expoetial term is greater tha i the first. Next we determie 15 terms of the asymptotic expasio. The

5 ASYMPTOTIC EXPANSIONS 5 approximate value of the costat is fially obtaied by computig the quotiet b[]/a[] for large idices. I[11]:= a[0] = 1; a[1] = ; a[ Iteger] := a[] = ((3( )+)a[ 1] a[ ])/( 1); I[1]:= a[50] Out[1]= I[13]:= Asymptotics[( + 1)f[ + ] (3 + )f[ + 1] + ( + )f[] f[]] { (3 5 ) (1+ ( ) Out[13]= 5)/ (1 5)/ I[14]:= {N[(3 5)/]N[(3 + 5)/] Out[14]= { I[15]:= terms = Last[FurtherTerms[( + 1)f[ + ] (3 + )f[ + 1] + ( + )f[] f[] { { 1 5 {115]]; I[16]:= b[ Iteger] := ( 3+ ) 5 (1 5)/ Sum[terms[[k + 1]] k {k015]; I[17]:= $RecursioLimit = 5 ; I[18]:= N[b[]/a[] / ] Out[18]= I[19]:= N[b[]/a[] / ] Out[19]= Which of these digits ca we trust? It is istructive to compute the quotiet for several idices ad see which digits remai fixed. The calculatio above strogly suggests that at least the first 30 digits are OK. The covergece is very quick because 15 terms of the expasio were take ito accout. This meas the error decays to zero i speed O( 16 ). More terms will lead to eve faster covergece. The figure below illustrates how for the preset example the umber of correct decimal digits (vertical axis) grows with (horizotal axis) whe 15 terms of the expasio are used (left) ad whe 30 terms of the expasio are used (right) As ca be see from this example doublig the umber of terms i the expasio teds to double the the accuracy of the estimate while doublig the evaluatio idex will usually ot double the accuracy. 3.. Several Domiat Terms. Whe o sigle term domiates all others the several costats have to be determied. Two terms are of the same growth for example whe the expoetial parts have the same absolute value (like ad ( ) ) or whe they differ by a polyomial multiple (like ad 3 ; although the latter grows more quickly tha the former both terms have to be take ito accout because the higher order terms of their

6 6 MANUEL KAUERS expasios may iterfere with each other). Suppose that is such that (b (1) )... (b (m) such that lim b (1) k b (i) a c 1 b (1) ) are such that b(i) b (j) + + c rb (r) ( ) = O( k ) for all 1 ij m ad some k ad = 0 for i = m r ad all k. The the first m terms cotribute sigificatly to the asymptotic behavior of (a ) ad the cotributio of the remaiig terms ca be eglected: a c 1 b (1) + + c m b (m) ( ) We have to determie approximatios for the costats c 1... c m. This ca be doe by solvig a suitable system of liear equatios as illustrated i the followig typical example sessio. I this example there are two domiat terms: the secod ad the third. For both of them we compute the first 15 terms of the expasio. The we set up a liear system for the desired costats c c 3 ad solve it. I[0]:= ClearAll[a b]; I[1]:= a[0] = ; a[1] = 1; a[] = 7; I[]:= a[ Iteger] := a[] = (( )a[ 3 + ] + ( )a[ + ] + ( )a[ 1 + ])/( ); I[3]:= a[] Out[3]= I[4]:= rec = 4(1+ )f[]+( 3+0 )f[1+] 3( 3+11 )f[+]+( 1+18 )f[3+]; I[5]:= Asymptotics[rec f[]] Out[5]= { ( 1 ) ( 3 ) (3 71)/6 ( 3 ) (3+ 71)/6 I[6]:= terms = Last[FurtherTerms[rec f[] {0 3 { {115]]; I[7]:= b[ Iteger] := ( 3 ) (3 71)/6 Sum[terms[[k + 1]] k {k015]; I[8]:= terms3 = Last[FurtherTerms[rec f[] {0 3 { {115]]; I[9]:= b3[ Iteger] := ( 3 ) (3+ 71)/6 Sum[terms3[[k + 1]] k {k0 15]; I[30]:= $RecursioLimit = 5 ; I[31]:= A = N[{b[]/a[]b3[]/a[] /. {{ 00 { 00050]; I[3]:= LiearSolve[A {1 1] Out[3]= { I[33]:= A = N[{b[]/a[]b3[]/a[] /. {{ 000 { ]; I[34]:= LiearSolve[A {1 1] Out[34]= { The umber of correct digits is ow less tha before because some accuracy is lost durig the liear system solvig. But still the umber of correct digits ca be icreased by takig ito accout more terms of the expasio. The figure blow shows for this example the umber of correct digits i depedece of the evaluatio idex ( + 00) whe 15 (left) or 30 (right) terms of the expasios are used.

7 ASYMPTOTIC EXPANSIONS Whe you eed more terms tha you ca compute. The best way of gettig a accurate estimate for the multiplicative costat(s) is to use as may terms of the expasio as possible. Sometimes oe would like to use eve more terms tha ca be computed explicitly with reasoable effort. For this situatio there exists a simple way to use higher order terms without eve kowig them explicitly. This is kow as Richardso s covergece acceleratio [4]. The idea is that if ( a c 1 + β k k + β ) k+1 k+1 + ( ) the ad therefore ( a c 1 + β k k k + β ) k+1 k+1 k+1 + k a a k 1 ( c ) k + O( (k+1) ) ( ) ( ). This meas that ( k a a ) coverges to the same limit as (a k 1 ) but oe order of magitude faster. Of course the scheme ca be iterated such as to elimiate several terms at oce. Here is some Mathematica code for doig this. I[35]:= Richardso[expr k Iteger] := Together[ k (expr /. ) expr ]; I[36]:= Richardso[expr {k0 Iteger k1 Iteger] := Fold[Richardso[#1 #]& expr Rage[k0 k1]] k 1 I the followig example we redo the calculatio of Sectio 3.1 by computig oly 1 terms of the expasio explicitly ad elimiatig three more terms with the commad just defied. I[37]:= ClearAll[a b]; I[38]:= a[0] = 1; a[1] = ; a[ Iteger] := a[] = ((3( )+)a[ 1] a[ ])/( 1); I[39]:= terms = Last[FurtherTerms[( + 1)f[ + ] (3 + )f[ + 1] + ( + )f[] f[] { { 1 5 {11]]; I[40]:= b[ Iteger] := ( 3+ ) 5 (1 5)/ Sum[terms[[k + 1]] k {k01]; I[41]:= $RecursioLimit = 5 ; I[4]:= u[ Iteger] = Richardso[b[]/a[] {13 15]; I[43]:= N[u[500] 50] Out[43]= Possible Issues There seems to be a bug related to the costructio of logarithmic terms. Whe a result is retured it seems correct but i some istaces the computatio aborts with a error whe it should ot. This bug will be fixed i a future versio.

8 8 MANUEL KAUERS I some examples Mathematica has trouble hadlig algebraic umbers. Typically these troubles become more likely for higher order terms. I such situatios it ca help to compute oly low order terms first the to rephrase all Root expressios i terms of AlgebraicNumber expressios of a commo umber field ad the to apply FurtherTerms to this. This is ot oly more stable but also more efficiet. Please report other problems to mkauers@risc.jku.at Refereces [1] Cyril Baderier Felix Cher ad Hsie-Kuei Hwag. Asymptotics of D-fiite sequeces. i preparatio. [] G.D. Birkhoff ad W.J. Trjitzisky. Aalytic theory of sigular differece equatios. Acta Mathematica 60: [3] Philippe Flajolet ad Robert Sedgewick. Aalytic Combiatorics. Cambridge Uiversity Press 009. [4] Lewis Fry Richardso. The approximate arithmetic solutio by fiite differece of physical problems ivolvig differetial equatios with a applicatio to the stress i a masory dam. Philosophical Trasactios of the Royal Society Lodo Series A : [5] Bruo Salvy. Examples of automatic asymptotic expasios. SIGSAM Bulleti 5() [6] Jet Wimp ad Doro Zeilberger. Resurrectig the asymptotics of liear recurreces. Joural of Mathematical Aalysis ad Applicatios 111: [7] Doro Zeilberger. Asyrec: A Maple package for computig the asymptotics of solutios of liear recurrece equatios with polyomial coefficiets. The Persoal Joural of Shalsoh B. Ekhad ad Doro Zeilberger 008. Mauel Kauers Research Istitute for Symbolic Computatio J. Kepler Uiversity Liz Austria address: mkauers@risc.ui-liz.ac.at