the approximations obtained from the d(1) -, (d=d )d(1) -, and d(2) -transformations, respectively.

(The inÿnite series diverge.)

™(0)

™ ™ ™ ™

(0) (0)

n Rn |SRn ’ S| |Sn ’ S| |S Rn ’ S| |S n ’ S| |Bn ’ S|

0 1 2:46D + 00 1:46D + 00 3:92D + 00 3:92D + 00 3:92D + 00

2 3 3:74D + 00 1:28D ’ 01 2:80D + 00 1:65D ’ 01 5:33D ’ 01

4 5 4:69D + 00 1:01D ’ 03 1:39D + 00 4:64D ’ 04 9:62D + 00

6 9 6:17D + 00 4:71D ’ 06 1:55D + 00 9:73D ’ 06 2:50D + 00

8 14 7:62D + 00 2:32D ’ 07 5:13D + 00 8:13D ’ 08 1:05D + 00

10 21 9:27D + 00 2:24D ’ 09 9:90D + 00 4:19D ’ 10 2:01D ’ 01

12 32 1:14D + 01 8:85D ’ 12 1:69D + 01 5:88D ’ 12 4:02D ’ 02

14 47 1:38D + 01 1:33D ’ 14 2:56D + 01 1:71D ’ 14 1:79D ’ 03

16 69 1:67D + 01 2:51D ’ 18 3:74D + 01 8:66D ’ 18 1:41D ’ 05

18 100 2:00D + 01 2:74D ’ 20 5:23D + 01 2:88D ’ 20 1:50D ’ 07

20 146 2:42D + 01 2:76D ’ 23 7:23D + 01 4:34D ’ 23 1:91D ’ 09

22 212 2:92D + 01 6:72D ’ 27 9:79D + 01 3:13D ’ 26 2:21D ’ 11

24 307 3:51D + 01 6:38D ’ 27 1:31D + 02 1:54D ’ 26 1:41D ’ 13

Since

∞

’

n

Bi n’i

Sn’1 ∼ ( + 1) ’ as n ’ ∞;

i

i=0

and since B0 = 1; B1 = ’ 1 , while B2i = 0; B2i+1 = 0; i = 1; 2; : : : ; we have that with the exception of

2

™

ÿ2i+1 ; i = 1; 2; : : : ; all the other ÿi are nonzero, and that exactly the same applies to the ÿi . (Here Bi

are the Bernoulli numbers and should not be confused with Bnj) .) Consequently, (3.19) of Theorem

(

3.4 holds with = there. Thus, whether limm’∞ Sm exists or not, as j ’ ∞; |Sn j+1) ’ S|=|Sn j) ’ S|

( (

is O( ’1 ) for n = 0, O( ’2 ) for n = 1; O( ’3 ) for n = 2, and O( ’(2i+1) ) for both n = 2i ’ 1 and

™( j+1) ™ ™( j) ™

™

n=2i, with i =2; 3; : : : : Similarly, whether limm’∞ S m exists or not, as j ’ ∞; |S n ’ S|=|S n ’ S|

is O( ’1 ) for n = 0; O( ’2 ) for n = 1; O( ’3 ) for n = 2, and O( ’(2i+1) ) for both n = 2i ’ 1 and

2i, with i = 2; 3; : : : :

™ ™

As for the approximations Bnj) to S obtained from the d(2) -transformation on {S m }, Theorem 2:2

(

™ ™

in [11] implies that, as j ’ ∞; |Bnj+1) ’ S|=|Bnj) ’ S| is O( ’1 ) for n = 0; O( ’2 ) for n = 2; O( ’3 )

( (

for n = 4, and O( ’(2i+1) ) for both n = 2(2i ’ 1) and n = 4i, with i = 2; 3; : : : :

The numerical results of Tables 2 and 3 pertain to Process II and show clearly that our approach

to the computation of derivatives of limits is a very e ective one.

Example 5.2. Consider the summation of the inÿnite series ∞ vk ,

k=0 ™ where vm = bm ( )m =m! and

m’1 ∞ Á’i

( )m = i=0 ( + i); and bm ∼ i=0 „i m as m ’ ∞. By the fact that ( )m = ( + m)= ( )

and by formula 6:1:47 in [1] we have that ( )m =m! ∼ ∞ i m ’i’1 as m ’ ∞. Consequently,

i=0

vm ∼ ∞ ‚i mÁ+ ’1’i as m ’ ∞, so that the d(1) -transformation can be applied successfully to sum

i=0

272 A. Sidi / Journal of Computational and Applied Mathematics 122 (2000) 251“273

Table 4

Numerical results on Process II for F( ; 1 ; 3 ; 1) in Example 5.2, where F(a; b; c; z)is the Gauss

22

hypergeometric function, with = 0:5. The d(1) - and (d=d )d(1) -transformations on {Sm } and

™ ™

{S m } and the d(2) -transformation on {S m } are implemented with = 1:2 in (4.4). Here Sn j) , (

™( j)

S n , and Bn j) are the approximations obtained from the d(1) -, (d=d )d(1) -, and d(2) -transformations,

(

respectively. (The inÿnite series converge.)

(0)

™ ™ ™ ™ ™

(0) (0)

n Rn |SRn ’ S| |Sn ’ S| |S Rn ’ S| |S n ’ S| |Bn ’ S|

0 1 5:71D ’ 01 1:57D + 00 2:18D + 00 2:18D + 00 2:18D + 00

2 3 3:29D ’ 01 4:70D ’ 02 1:64D + 00 2:18D ’ 01 6:79D ’ 01

4 5 2:54D ’ 01 4:06D ’ 05 1:41D + 00 4:06D ’ 04 1:51D ’ 01

6 9 1:89D ’ 01 1:69D ’ 06 1:16D + 00 1:22D ’ 05 4:59D ’ 02

8 14 1:51D ’ 01 1:95D ’ 08 9:96D ’ 01 1:39D ’ 07 3:76D ’ 03

10 21 1:23D ’ 01 1:11D ’ 10 8:63D ’ 01 7:94D ’ 10 1:47D ’ 04

12 32 9:99D ’ 02 3:11D ’ 13 7:41D ’ 01 2:20D ’ 12 4:42D ’ 06

14 47 8:24D ’ 02 3:99D ’ 16 6:43D ’ 01 2:61D ’ 15 9:14D ’ 08

16 69 6:80D ’ 02 1:20D ’ 19 5:57D ’ 01 1:41D ’ 19 1:26D ’ 09

18 100 5:64D ’ 02 2:04D ’ 22 4:84D ’ 01 2:38D ’ 21 1:57D ’ 11

20 146 4:67D ’ 02 2:03D ’ 25 4:18D ’ 01 2:03D ’ 24 2:21D ’ 13

22 212 3:88D ’ 02 6:77D ’ 29 3:61D ’ 01 7:81D ’ 28 1:86D ’ 15

24 307 3:22D ’ 02 2:41D ’ 29 3:12D ’ 01 1:51D ’ 28 1:13D ’ 18

∞

vk , as described in the previous section. Now vm = vm [ m’1 1=( + i)], and m’1 1=( + i) ∼

™

k=0 i=0 i=0

log m+ k=0 ei m as m ’ ∞. Therefore, we can apply the (d=d )d -transformation to sum ∞ vk

∞ ’i (1)

k=0 ™

™m can be obtained by term-by-term di erentiation by

provided that the asymptotic expansion of S

Theorem 4.1. (We have not shown that this last condition is satisÿed).

We have applied the (d=d )d(1) -transformation with bm = 1=(2m + 1). With this bm the series

∞ ∞ ∞ 13

k=0 vk and √ vk both converge. Actually we have k=0 vk = F( ; 2 ; 2 ; 1). By formula 15.1.20 in

k=0 ™

[1], ∞ vk√ =( =2)( (1’ )= (3=2’ ))=S. Di erentiating both sides with respect to , we obtain

k=0

™

∞

=2)( (1 ’ )= (3=2 ’ )){ ( 3 ’ ) ’ (1 ’ )} = S, where (z) = (d=d z) (z)= (z).

k=0 vk = (

™ 2

™

Letting now = 1 throughout, we get S = =2 and S = log 2, the latter following from formulas

2

6:3:2 and 6:3:3 in [1].

In our computations we picked the Rl as in the ÿrst example. We also applied the d(2) -transformation

™

directly to {S m } with the same Rl ™s.

Table 4 contains numerical results pertaining to Process II.

Acknowledgements

This research was supported in part by the Fund for the Promotion of Research at the Technion.

References

[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Vol. 55 of National Bureau of Standards,

Applied Mathematics Series, Government Printing O ce, Washington, DC, 1964.

A. Sidi / Journal of Computational and Applied Mathematics 122 (2000) 251“273 273

[2] R. Bulirsch, J. Stoer, Fehlerabschatzungen und extrapolation mit rationalen Funktionen bei Verfahren vom

Richardson-Typus, Numer. Math. 6 (1964) 413“427.

[3] P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1984. 2nd Edition.

[4] W.F. Ford, A. Sidi, An algorithm for a generalization of the Richardson extrapolation process, SIAM J. Numer.

Anal. 24 (1987) 1212“1232.

[5] D. Levin, Development of nonlinear transformations for improving convergence of sequences, Int. J. Comput. Math.

Series B 3 (1973) 371“388.

[6] D. Levin, A. Sidi, Two new classes of nonlinear transformations for accelerating the convergence of inÿnite integrals

and series, Appl. Math. Comput. 9 (1981) 175“215.

[7] A. Sidi, Some properties of a generalization of the Richardson extrapolation process, J. Inst. Math. Appl. 24 (1979)

327“346.

[8] A. Sidi, An algorithm for a special case of a generalization of the Richardson extrapolation process, Numer. Math.

38 (1982) 299“307.

[9] A. Sidi, Generalizations of Richardson extrapolation with applications to numerical integration, in: H. Brass,

G. Hammerlin (Eds.), Numerical Integration III, ISNM, Vol. 85, Birkhauser, Basel, Switzerland, 1988, pp. 237“250.

[10] A. Sidi, Convergence analysis for a generalized Richardson extrapolation process with an application to the

d(1) -transformation on convergent and divergent logarithmic sequences, Math. Comput. 64 (1995) 1627“1657.

[11] A. Sidi, Further results on convergence and stability of a generalization of the Richardson extrapolation process,

BIT 36 (1996) 143“157.

[12] A. Sidi, A complete convergence and stability theory for a generalized Richardson extrapolation process, SIAM J.

Numer. Anal. 34 (1997) 1761“1778.

[13] A. Sidi, Further convergence and stability results for the generalized Richardson extrapolation process GREP(1)

with an application to the D(1) -transformation for inÿnite integrals, J. Comput. Appl. Math. 112 (1999) 269“290.

[14] A. Sidi, Extrapolation methods and derivatives of limits of sequences, Math. Comput. 69 (2000) 305“323.

[15] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer, New York, 1980.

Journal of Computational and Applied Mathematics 122 (2000) 275“295

www.elsevier.nl/locate/cam

Matrix Hermite“PadÃ problem and dynamical systems

e

Vladimir Sorokina;— , Jeannette Van Iseghemb

a

Mechanics-Mathematics Department, Moscow State University, Moscow, Russia

b

UFR de MathÃ matiques, UniversitÃ de Lille, 59655 Villeneuve d™Ascq cedex, France

e e

Received 8 June 1999; received in revised form 3 November 1999

Abstract

The solution of a discrete dynamical system is studied. To do so spectral properties of the band operator, with inter-

mediate zero diagonals, are investigated. The method of genetic sums for the moments of the Weyl function is used to

ÿnd the continued fraction associated to this Weyl function. It is possible to follow the inverse spectral method to solve

dynamical systems deÿned by a Lax pair. c 2000 Elsevier Science B.V. All rights reserved.

MSC: 41A21; 47A10; 47B99; 40A15; 11J70

Keywords: Hermite-PadÃ approximation; Matrix continued fraction; Lax pair; Resolvent function; Weyl function

e

1. Introduction

Let us consider the inÿnite system (E) of di erential equations, for n = 0; 1; : : : ,

bn+3 = bn+3 (Un+3 ’ Un’2 );

(E)

Un+3 = bn+6 (bn+9 + bn+7 + bn+5 ) + bn+4 (bn+7 + bn+5 ) + bn+2 bn+5 ;

where the solutions bn+3 (t) are unknown real functions of t, t ∈ [0; +∞[ with conditions bn+3 (t) = 0,

n ¡ 0. These conditions are boundary conditions with respect to n for system (E). We formulate the

Cauchy problem for system (E), i.e., with initial conditions

bn+3 (0); n¿0:

Why are we interested in such dynamical system? Bogoyavlensky [4,5] has given a classiÿcation

of all dynamical systems which are a discrete generalization of the KdV equation; they depend on

two parameters p and q. He showed that such systems have interesting applications in hamiltonian

mechanics.

—

Corresponding author.

E-mail addresses: vnsormm@nw.math.msu.su (V. Sorokin), jvaniseg@ano.univ-lille1.fr (J.Van Iseghem).

0377-0427/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 0 4 2 7 ( 0 0 ) 0 0 3 5 4 - X

276 V. Sorokin, J. Van Iseghem / Journal of Computational and Applied Mathematics 122 (2000) 275“295

Our system is such a dynamical system for p = 3, q = 2. This case includes the main ideas of

the general case. In Section 7, we give di erent transformations of system (E). For p = q = 1, we

get the Langmuir chain [9], for p = 1 and q any integer or q = 1 and p any integer, the systems

have been studied, respectively, by Sorokin [11] and by Aptekarev [1]:

q q

p = 1; a n = an an+k ’ an’k ;

k=1 k=1

p p

q = 1; an = a n an+k ’ an’k :

k=1 k=1

In order to be explicit, we will often restrict to the case p = 3, q = 2, the important assumption

being that p and q are relatively prime. Studies of the same kind, with no references to dynamical

systems, but concerned with the ÿnite-dimensional approximations of the resolvent of an inÿnite

three diagonal band matrix can be found in [3, 2]. What we call in our notation the matrix A(3)

has three diagonals (i.e., p = q = 1), but nonbounded or even complex data are considered. We have

not considered nonbounded operators in this paper.

The ÿrst natural question for a Cauchy problem is the existence and uniqueness of the solution.

A classical theory of ODE does not give the answer because system (E) is inÿnite.

We will prove two results through this paper.

Theorem A. If all bn+3 (0) are positive; bounded

0 ¡ bn+3 (0)6M; n¿0;

then the Cauchy problem for system (E) has a unique solution deÿned on [0; +∞[.

As usual, it is possible to prove only the local existence and uniqueness of the solution of discrete

dynamical systems [8]. As a remark, in the case where all bn+3 (0) are not positive, there exists

examples where the solution is not unique, so the condition is not only su cient but also necessary.

In physical applications, the bn+3 are exponents of physical data, so are positive.

If the (bn+3 (0))n¿0 are not bounded, then the solution does not extend beyond some interval [0; t1 ].

In this paper, we do not consider this case, which would be possible following the same scheme,

but the theory of operators, in case of an unbounded, nonsymmetric operator, is to be used, and yet

there do not exist su cient results.

The next question is how to ÿnd the solution of system (E). Tools are known, it is the inverse

spectral problem. In this paper, we give the solution in classical form, in terms of continued fraction,

as for the Langmuir chain. We get (the matrix of constants will be deÿned in the text (7) and

i; j

i; j

the notation F=z means that each component is f =z )

Theorem B. The solution of system (E) is

1 d (x; t) I3 C1 C2