Integrable lattices and convergence acceleration algorithms

PhysicsLettersAl79(1993) 111—115

PHYSICS LETTERS A

North-Holland

Integrable lattices and convergence acceleration algorithms V. Papageorgiou LPN, Université Paris VII, Tour 24-14, 5èmeétage, 75251 Paris, France and Department ofMathematics and Computer Science, C/arkson University, Potsdam, NY 13699-5815, USA

B. Grammaticos LPN, UniversitéParis VII, Tour24-14, SEme étage. 75251 Paris, France

and A. Ramani CPT. Ecole Polytechnique, CNRS, UPR 14, 91128 Palaiseau, France Received 3 February 1993; revised manuscript received 15 April 1993; accepted for publication 25 May 1993 Communicated by A.P. Fordy

We show that a well-known convergence acceleration scheme, the e-algorithm, when viewed as a two-variable difference equation, is nothing but the discrete Korteweg—de Vries lattice equation. The complete integrability of the latter confers to the ealgorithm iliteresting properties among which the singularity confinement is outstanding. In fact, this property is used in order to derive the generalizations of the &accelerator leading to the most general form of the p-algorithm. A new acceleration algorithm based on the modified Korteweg—de Vries lattice equation is also derived.

Discrete systems play a fundamental role in numerical analysis. Integration schemes for ordinary or partial differential equations can be greatly improved if they conserve exactly the quantities that are conserved by the initial (continuous) systems. The literature on this topic is important and new schemes are regularly produced incorporating these complete or partial integrability features [1,2]. Thus the integrable lattices we are referring to in the title of this paper are usually understood as the discrete analogs of integrable (multidimensional) evolution equations. Thanks to the recent progress in this domain (see ref. [3], and references therein), the discrete forms of many of the “classical” integrable partial differential equations are known today. Here we shall not deal with this aspect of integrable discrete systems. Our study is motivated by a numerical analysis problem but not by integrability per se. The issue here concerns the numerical schemes that can be used in order to accelerate the convergence of a sequence [4—6]. Let us first consider the problem of convergence acceleration. The typical situation is that one is given a sequence of numbers x~(usually the finite sums of a series up to a given order: x,~= ~, o am) and wishes to obtain an estimate of their value when n—~co(which, in the example, would correspond to the sum of the infinite series). The idea behind the acceleration method is to guess a (simple) form for the terms of the sequence that would lead to an exact result in a given number of acceleration steps. There is, of course, a considerable freedom in the number of steps and thus the implied ansatz is usually not too restrictive. Let us illustrate this point in the case of the Aitken—Shanks algorithm [7,8], where, given a sequence ofx~,we construct x’~through ~

0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

=

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~ —x

—

~

PHYSICS LETTERSA

2 August 1993

—x~)(x~—x~_1)

x,~fI

If we ask that the algorithm converges in one step this means that all the x’,, are equal to the sought limit a of x~.Introducing u,, = x~ a we rewrite (1) as u,,+1 u,,_1 = u the solution of which is f3~. Thus the Aitken— Shanks algorithm converges in one step when x,, is of the form a + ft2.”. More complicated forms (essentially, sums of powers) are obtained when one asks for convergence in a higher number of steps [8,9]. When the form of the terms ofthe sequence is not the one leading to an exact result the acceleration method is still useful: it amounts to successively stripping the sequence ofthe terms forwhich exact convergence is attained. However there is no a priori guarantee that the method will lead to any useful result: a given accelerator cannot accelerate the convergence of any sequence. Nonaccelerable families of sequences are known to exist [101. An observation of form (1) leads naturally to a question that introduces the main theme ofthis paper: given the rational form of the algorithm we may wonder what happens whenever the denominator vanishes accidentally. The obvious answer is that x’~diverges and that this divergence propagates indefinitely with the iterations of the mapping. This is, clearly, an undesirable feature for a numerical scheme. The next question is, thus, whether one can derive accelerator schemes for which the (spontaneously appearing) singularities are confined to a few iteration steps. Singularity confinement has recently been proposed [11] as an integrability criterion for discrete systems. The method has already led to a wealth of results and is becoming firmly established. Thus the singularityconfining acceleration schemes should in principle be integrable and one would except to be able to use the standard integrability tools in this case. This paper does not intend to address the question of “integrable” accelerator schemes in its full generality. Rather, we will focus on one such scheme, chosen as a paradigm of this class. We will show how the integrability requirement leads naturally to the generalisations of the scheme and we derive conditions for exact (finitenumber of steps) convergence. The accelerator we are going to study is the so-called c-algorithm [12,131. In order to implement it for the acceleration of the convergence of x,, we need two successive iterations, say x~’ and x~from which we cornputex~ as —

~,

‘~.

(2)

1

k k xn+I —xn_I

The initial data are the x~while the x°~ are set to zero. Iterating (2) we obtain a convergence to the result ~ 4,) on odd-numbered (in k) steps, while the even-numbered ones grow indefinitely and are just intermediate quantities in the calculation. These intermediaries play an important role as far as the singularity confinement/integrability properties of the algorithm are concerned. We can do away with them by replacing the even-numbered zerox~’ and derive an algorithm on thex~/’and odd-numbered iterationswhen only. we In proceed practice 2iterations =0 and a by given from which we compute x~’. However wecompute start from x~we mustreset x~tozero. The resulting algorithmcorresponds to iterating Aitken’s expression, to x~”~2 2k±I_ —

2k—I..L.

i 2k—i 2k—I’.I 2k—I 2k—Is ~2 ~Xn J~~Xn ~Xn_2 ) 2k—I I 2k—I 2k—I xn+2 — xn -T-xn_

(3

2

for the n of a given parity. Alternatively, we can eliminate consistently the intermediate quantities (without any assumption as to their values). We obtain in this case the cross-rule of ref. [91, 2_x~ x~_x~_2 x~÷—x~ ~ =0. (4) x~ We remark here that the cross-rule necessitates two sequences 4, and 4, for its initialization. So, in practice, one starts from a given 4,, computes 4, through Aitken’s scheme (3) and then uses rule (4). 112

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Schemes (3) and (4) are not equivalent either with respect to their acceleration properties or as far as singularity confinement is concerned. In order to assess the latter statement we start applying the singularity confinement test on the c-algorithm. We assume that at a certain iteration we have x~1=4,_i + ö. We find then that x~ diverges as 1 /ö, and computing the next iteration we find k+2_ k ~j.ç2 x~ k+I k 1—x,,_1 ~ ,, x~_1—x~_1 ~ This precise relation leads to a cancellation of the divergent terms in ~ , which is then finite: the singularity is indeed confined. Aitken’s algorithm, on the contrary, does not confine. Indeed, let us assume that in eq. 2/2d+.... (1), the Iterating relation ~ This leads and to a so divergentx, ofsingularity the form x’n=(xn+i —x~_1) once we find x’~= (x~+ 2/4ö+... on. Thus the never vanishes. The interpretation of 1 x,,_1 ) is clear: the c-algorithm is integrable while Aitken’s is not. this result from our point of view In fact, the singularity confinement criterion can be used in order to derive the generalisations of the c-algorithm, the so-called p-algorithms [13]. We start from the expression 2\

—

j~j~

—

XflXfl+

k

4,

(5)

k

xn+l —xn_l

and look for the possible forms of 4, that lead to confined singularities. As previously we start with x~÷ 1

=

x~1+ ö in which case x~’ diverges as 4,15. At the next iteration we find ~

k+1 =x~_1 +(i_ ~!!±!)5+O(52)

k+I

x~ =x~1+

Z~_1

5+0(52)

3 finite, is The condition for singularity confinement, i.e. x~ k+2 k+l k+I k.....~ z,,

~

(6)

.

The general solution of the latter is 4, =f( n + k) + g( n k). This expression gives the most general form of the p-algorithm. In order to compare with existing results, we rewrite (5) in the form most often encountered ~ln a numerical analysis setting —

XrnXrn+i+

k

xm+i

(7)

k

Xm

with m = (n k+ 1). This means that the general solution for z compatible with integrability is just F( in + k) + G ( m). Indeed, all the known cases fall within this class. In ref. [9], Brezinski gives ~

—

4=F(m),

4,,=F(m+k),

4

=

4,,=F(m+k)—F(m),

obviously included in our general form. So in this case the simple singularity confinement requirement suffices in order to derive the complete result. One interesting question concerns the finite-step exact convergence using the general algorithm (5) together with (6). The answer is not known in the general case (6), so let us work it out for an exact convergence in two steps. This means that starting from x2 = 0 and 4, being given, we compute 4, and 4, and demand that the latter be already the sought-out limit a = ~ 4,. Given the translation invariance of (5) we can translate the value of a to zero, which simplifies considerably the calculations. We find thus readily k+I Zp~ k

xn

k Z~i

—

k

k

xn÷2—xn

k

k Zn_i k

xn—xn_ 2

Next we put UmX~/X~_2and we end up with

Um+i

=x~,÷2/x~ and taking into account the fact that 4,=F(m+k)+G(m) 113

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Um+

1

PHYSICS LETTERS A

[F(m+l)+G(m+l)]um—G(m+l)+G(m) [F(m+l)—F(m+2)]um+F(m+2)+G(m)

2 August 1993

9

( )

Thus, for a two-step convergence, u must be a solution of a homographic mapping (which is the discrete form of a Riccati equation [14]). As we have previously explained the singularity confinement property is related to integrability. Thus the c-algorithm is expected to be integrable. And in fact it is! The reason is simple: this algorithm, considered as a two-variable difference equation, is precisely the (discrete) lattice potential KdV equation. The latter has been studied extensively and its integrability is firmly established [15,161. In the case of the accelerator algorithm we are in presence of a special type of solutions of the KdV lattice, namely solutions involving singular columns. In fig. 1 we give this solution schematically where S is understood as a small parameter that goes to zero. The zeros to the left of the rightmost column of infinities are 0(52) leading further left to a column of terms that diverge as 1/52 and so on. The presence of these divergences is not in contradiction with the singularity confinement requirement. The latter concerns only movable singularities while the divergent columns in fig. 1 are fixed singularities. From the above analysis, one may tend to consider integrable lattice equations as candidates for efficient acceleration schemes. This is, however, not true in general. We have investigated this question in the case of the modified lattice KdV equation. Its discrete form has been given in ref. [16] and writes simply k k+I Xfl

k PXn_l k

k—I Xn+l —x~ k —

—x~_1

PXn+l

The parameter p can be taken constant, but the most general integrable nonautonomous form is known [17], corresponding to p =f( k+ n )g( k n) where fand g are free functions. A study of the convergence acceleration properties of (10) (starting either from 4, = 1 or 4, = (— 1 )n, and a given x,’,) has shown that the mKdV acts on certain sequences rather as an anti-accelerator, i.e. it amplifies oscillations around the limit value leading to a loss of convergence for any p ~ 1. The value p = 1, on the other hand, is special in the sense that one must first iterate (10) and then take p = 1. It turns out in this case mKdV can act as an accelerator. However, taking this limit may be rather delicate in practice, so one must look for simplifications. One possibility is to eliminate the intermediate, even-numbered, steps, by putting x~= 1, and derive an algorithm a la Aitken on the oddnumbered steps only. We find readily —

0 0

÷00

(÷~5)

00

00

o

i/zt+Xi

0

-00

(-6)

-1/2t±Xz

o

~

o (+6)

00 i/z6+x3

0 114

(— 6)

Fig. 1. Structure of the singular lattice-KdV sponds to the acceleration scheme (see text). solution that corre-

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2k+I_ P1

—

i 2k—i 2k—I 2k—I ~~fl+I X,,_i fl 2k—i 2k—I j_

Xp~

PHYSICS LETTERS A 2k—u

X,,

2k—li

1X,,_1 mX~

j.. j_

2k—i

2k—I

i,X,,.~ 1Xn_i 2k—I

2k—I

i~X~÷u -rXn_i

.j.

2k—I \

Xn

i

2k—i

j

2 August 1993

~ll

JXn

This algorithm does indeed accelerate as can be seen also in the fact that it possesses solutions corresponding to convergence in a finite number of steps. If we ask that all ~ have converged to some value a we find that x~—’ must be of the form — (l_fi2?l)(l_flAPI_I) Xn_a(l+P2fl)(l+flAflI).

On the other hand algorithm (11) is not singularity confining and thus certainly not integrable. We can now ask whether the equivalent of the cross rule would exist in the mKdV case (p = 1) as well. Its form can be obtained by iterating the mKdV algorithm and then taking the p—il limit. It turns out that it is identical to the cross-rule (4)! Still, its implementation here is different: starting from 4, we compute 4, using scheme (11) and not Aitken’s scheme (3). Thus, we obtain a new scheme that is both integrable and accelerating. To conclude, we can state, once more, that discrete integrability has encountered another domain of applicability, that of convergence acceleration schemes. The singularity confinement property that allows a numerical scheme to deal with near divergences is intimely related to integrability. This sheds light on the acceleration process and makes possible a novel interpretation of these algorithms. V. Papageorgiou wishes to acknowledge the hospitality of the Laboratoire de Physique Nucléaire de l’Université Paris VII. He is also indebted to professor C. Guilpin for bringing the c-algorithm to his attention.

References [1] [2] [3] [4]

T.R. Taha and M.J. Ablowitz, J. Comput. Phys. 55 (1984)192,203. B.M. Herbst and M.J. Ablowitz, Phys. Rev. Lett. 62 (1989) 2065. F.W. Nijhoff, V.G. Papageorgiou and H.W. Capel, in: Lecture notes in mathematics, Vol. 1510 (Springer, Berlin, 1991) p. 312. C.M. Bender and S.A. Orszag, in: Advanced mathematical methods for scientists and engineers (McGraw-Hill, New York, 1978)

Ch. 8. [5] C. Brezinski, Lecture notes in mathematics, Vol. 584, Accélération de Ia convergence and analyse numérique (Springer, Berlin, 1977). [6] J. Wimp, Sequence transformations and their applications (Academic Press, New York, 1981). [7] A.C. Aitken, Proc. R. Soc. (Edinburgh) 46 (1926) 289. [8] D. Shanks, J. Math. Phys. 34(1955)1. [9] C. Brezinski, Algorithmes d’accélération de la convergence (Technip, Paris, 1977). [10] J.-P. Delahaye, SCM, Vol. 11, Sequence transformations (Springer, Berlin, 1980). [11] B. Grammatiéos, A. Ramani and V.0. Papageorgiou, Phys. Rev. Len. 67 (1991) 1825. [12] P. Wynn, Math. Tables Aids Computing 10 (1956) 91. [13] C. Brezinski, Numer. Math. 35 (1980) 175. [14] A. Ramani, B. Grammaticos and 0. Karra, Physics A 181(1992) 115. [15] R. Hirota, J. 1~hys.Soc. Japan 43 (1977) 1424, 2079. [16] H.W. Capel, P.W. Nijhoff and V.G. Papageorgiou, Phys. Lett. A 155 (1991) 337. [17] J. Satsuma, B. Grammaticos and A. Ramani, Phys. Lett., in press.

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