Method of summation of some slowly convergent series

Applied Mathematics and Computation 215 (2009) 1622–1645

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Method of summation of some slowly convergent series Paweł Woz´ny, Rafał Nowak * Institute of Computer Science, University of Wrocław, ul. Joliot-Curie 15, 50-383 Wrocław, Poland

a r t i c l e

i n f o

Keywords: Convergence acceleration Extrapolation Levin-type method Nonlinear sequence transformation Iterative methods Power series Orthogonal series Hypergeometric series Basic hypergeometric series

a b s t r a c t A new method of summation of slowly convergent series is proposed. It may be successfully applied to the summation of generalized and basic hypergeometric series, as well as some classical orthogonal polynomial series expansions. In some special cases, our algorithm is equivalent to Wynn’s epsilon algorithm, Weniger transformation [E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10 (1989) 189–371] or the technique recently introˇ ízˇek et al. [J. C ˇ ízˇek, J. Zamastil, L. Skála, New summation technique for rapidly duced by C divergent perturbation series. Hydrogen atom in magnetic field, Journal of Mathematical Physics 44 (3) (2003) 962–968]. In the case of trigonometric series, our method is very similar to the Homeier’s H transformation, while in the case of orthogonal series — to the K transformation. Two iterated methods related to the proposed method are considered. Some theoretical results and several illustrative numerical examples are given. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Many fields of numerical mathematics come up against the convergence problem. In the case of slow convergence, many methods of acceleration were proposed. But still the development of the new algorithms plays an important role. In this paper, a new method of summation of slowly convergent series is introduced, and its efficiency is examined by many numerical examples. The method is a Levin-type sequence transformation; its relations to the classic methods are analysed. For the purpose of this paper, let us consider an infinite series

s¼

1 X

an

n¼0

Pn1 P ð1Þ with terms an , partial sums sn ¼ j¼0 aj , and remainders rn ¼ 1 annihilate the j¼n aj . Let the linear difference operator L ð1Þ remainder r n , i.e. L ðrn Þ ¼ 0. Throughout the paper, we assume that all difference operators act on the variable n. Observe that

s¼

Lð1Þ ðsn Þ Lð1Þ ð1Þ

provided Lð1Þ ð1Þ–0. In general, it is difficult to obtain explicit form of the annihilator Lð1Þ . Let the linear difference operator LðmÞ ; m 2 N, be an approximation of Lð1Þ in the sense that

(

ðmÞ

LðmÞ ðr n Þ ¼ 0; LðmÞ ð1Þ – 0;

* Corresponding author. E-mail addresses: [email protected] (P. Woz´ny), [email protected] (R. Nowak). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.07.016

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1623

ðmÞ

where r n :¼ rn  r nþm . Then, we can expect that LðmÞ ðsn Þ=LðmÞ ð1Þ gives an approximation of s, of accuracy growing when m ! 1. In this paper, we consider the Q transformation, where QðmÞ : fsn g#fQðmÞ n g; m 2 N, is defined by

QðmÞ :¼ n

LðmÞ ðsn Þ : LðmÞ ð1Þ

Since LðmÞ is a linear difference operator, Q is a Levin-type sequence transformation; see, e.g., [1], [3, Section 2.7], [4] or [5]. Different Levin-type transformations, and hence different methods of convergence acceleration, were studied by Weniger [1], Homeier [4], Brezinski and Matos [6], and Matos [7]. In Section 2, we give a method of obtaining the annihilators LðmÞ , satisfying LðmÞ ¼ PðmÞ Lðm1Þ for some linear difference operators PðmÞ ; m 2 N. Thus we will consider

LðmÞ ¼ PðmÞ Pðm1Þ    Pð1Þ and we will say that Q is determined by operators PðmÞ . Next, we will make use of the fact that numerators and denominators on the right side of (2.3) satisfy the recurrence relation

X nfðmÞg ¼ fPgfðmÞg ðX fðm—1Þg Þ n ðmÞ to propose an efficient algorithm of computing QðmÞ ; m 2 N, provided n . One could obtain explicit forms of the operators P the analytical form of terms an is known. In the sequel, we give some applications of the Q transformation. In Section 3, we give explicit forms of the operators PðmÞ in the case of generalized hypergeometric series, while in Section 4 we give analogous results for basic hypergeometric series. In Section 5, we show that our method may be successfully applied to the summation of some classical orthogonal polynomial series, too. Two iterated methods, related to the Q transformation, are considered in Section 6. In the sequel, we will use the following notation:

 shift operator: Exn ¼ xnþ1 ; more generally Ek xn ¼ xnþk ; k 2 Z;  identity operator: I :¼ E0 ;  linear difference operator:

K¼

v X

kj ðnÞEj

ðku ðnÞX0; kv ðnÞX0Þ of order ord K :¼ v  u;

ð1:1Þ

j¼u

 forward difference operator: D :¼ E  I. All computations and numerical tests are made in Maplee11 system, using 128-digit arithmetic. Following Paszkowski [8], we measure the accuracy of approximation r by

r    accðrÞ :¼ log10   1; s i.e., by the number of exact significant decimal digits in approximation

r of the sum s.

2. The method Consider the series

s¼

1 X

an

ð2:1Þ

n¼0

Pn1 P with terms an , partial sums sn ¼ j¼0 aj , and remainders rn ¼ 1 j¼n aj . In Section 2.1, we propose a method of obtaining the ðmÞ annihilators L ; m 2 N, satisfying

LðmÞ ðrðmÞ n Þ ¼ 0;

ð2:2Þ

where

rðmÞ ¼ rn  r nþm ¼ an þ anþ1 þ    þ anþm1 : n We consider the transformation QðmÞ : fsn g#fQðmÞ n g, defined by

QðmÞ :¼ n

LðmÞ ðsn Þ ; LðmÞ ð1Þ

m 2 N:

In Section 2.2, we describe an algorithm of computing QðmÞ n .

ð2:3Þ

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1624

2.1. Constructing the annihilators LðmÞ According to (2.2) with m ¼ 1, the operator Lð1Þ should annihilate the term an . A possible choice is the first-order operator

  1 1 1 Lð1Þ :¼ D I ¼ E  I: an anþ1 an

ð2:4Þ

The operators Lð2Þ ; Lð3Þ ; . . . are constructed recursively by

LðmÞ ¼ PðmÞ Lðm1Þ ;

m P 2:

ð2:5Þ ðmÞ

ðm1Þ

We give the details of a typical step of this construction. Since r n ¼ r n þ anþm1 and by the fact that Lðm1Þ annihilates the ðm1Þ , we are looking for a linear difference operator PðmÞ satisfying term rn

LðmÞ ðanþm1 Þ ¼ PðmÞ Lðm1Þ ðanþm1 Þ ¼ 0:

ð2:6Þ

Let us put

e L ðm1Þ :¼ Em1 Lð1Þ Emþ1

ð2:7Þ

and observe that e L ðm1Þ annihilates the term anþm1 . Now, assume that operators P

ðmÞ

ðmÞ

and R

are such that

L ðm1Þ : PðmÞ Lðm1Þ ¼ RðmÞ e

ð2:8Þ

We obtain ðmÞ ðm1Þ LðmÞ ðr ðmÞ ðrn Þ þ LðmÞ ðanþm1 Þ ¼ PðmÞ Lðm1Þ ðr nðm1Þ Þ þ RðmÞ e L ðm1Þ ðanþm1 Þ ¼ PðmÞ ð0Þ þ RðmÞ ð0Þ ¼ 0; n Þ ¼ L

and thus (2.2) holds. Note that, in view of (2.5), the operators LðmÞ are of the form (we assume Pð1Þ :¼ Lð1Þ )

LðmÞ ¼ PðmÞ Pðm1Þ    Pð1Þ :

ð2:9Þ

Now we explain how to obtain operators P ator Lð1Þ is of the form

Lð1Þ ¼

‘ X

ð1Þ

ð1Þ

ki ðnÞEi ;

ðmÞ

ðmÞ

;R

satisfying (2.8). Without loss of generality we can assume that oper-

ð1Þ

k0 ðnÞX0; k‘ ðnÞX0:

ð2:10Þ

i¼0

Thus operator (2.7) can be written as follows:

e L ðm1Þ ¼

‘ X

ð1Þ

ki ðn þ m  1ÞEi ;

m P 2:

ð2:11Þ

i¼0

Assume that Lðm1Þ has the form

Lðm1Þ ¼

ðm1Þ‘ X

ðm1Þ

kj

ðnÞEj ;

m P 2:

ð2:12Þ

j¼0

Let us suppose that condition (2.8) can be satisfied by the operators of the form

PðmÞ ¼

‘ X

i pðmÞ RðmÞ ¼ i ðnÞE ;

ðm1Þ‘ X

i¼0

j qðmÞ j ðnÞE ;

ð2:13Þ

j¼0

where fpi ðnÞg‘i¼0 ; fqj ðnÞgm‘‘ j¼0 are unknown functions. By performing multiplication of the operators in (2.8), we obtain ðmÞ

‘ m‘‘ X X i¼0

ðmÞ

ðm1Þ

kj

ðmÞ

ðn þ iÞ pi ðnÞEiþj ¼

m‘‘ X

‘ X

j¼0

i¼0

j¼0

ð1Þ

ðmÞ

ki ðn þ m  1 þ jÞ qj ðnÞEiþj :

ð2:14Þ

Equating the variable coefficients of Ek on both sides of the equation, we obtain the following system of m‘ þ 1 equations:

X

ðm1Þ

kj

ðmÞ

ðn þ iÞ pi ðnÞ ¼

iþj¼k

X

ð1Þ

ðmÞ

ki ðn þ m  1 þ jÞ qj ðnÞ;

which are satisfied by a set of m‘ þ 2 unknown functions corresponding to k ¼ 0; 1; 2, are as follows: ðm1Þ

k ¼ 0; 1; . . . ; m‘

ðmÞ pðmÞ i ðnÞ and qj ðnÞ. The beginning three equations of the system,

ðmÞ ð1Þ ðmÞ ðnÞ 0 ðnÞ ¼ k0 ðn þ m  1Þ 0 ðnÞ; ðm1Þ ðmÞ ðm1Þ ðmÞ ð1Þ ðmÞ ð1Þ ðn þ 1Þ 1 ðnÞ þ k1 ðnÞ 0 ðnÞ ¼ k0 ðn þ mÞ 1 ðnÞ þ k1 ðn þ m  k0 ðm1Þ ðmÞ ðm1Þ ðmÞ ðm1Þ ðmÞ ðn þ 2Þ 2 ðnÞ þ k1 ðn þ 1Þ 1 ðnÞ þ k2 ðnÞ 0 ðnÞ k0 ð1Þ ðmÞ ð1Þ ðmÞ ð1Þ ðmÞ ¼ k0 ðn þ m þ 1Þ 2 ðnÞ þ k1 ðn þ mÞ 1 ðnÞ þ k2 ðn þ m  1Þ 0 ðnÞ:

k0

ð2:15Þ

iþj¼k

p

q

p p

p

q

p

q

p

q

q

ðmÞ

1Þ q0 ðnÞ;

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1625

Since the one unknown is superabundant, so it can be chosen arbitrarily. Then the next unknowns are determined uniquely. Observe that such calculations can be made using computer algebra system like Maplee or Mathematicaâ . Observe that left-hand side of (2.14) is actually an explicit form of LðmÞ (cf. (2.5)), which can be rearranged in the form (2.12) with m  1 replaced by m. Notice that such construction of operators LðmÞ depends on the choice of the first operator Lð1Þ , as well as of the operators ðmÞ and RðmÞ , satisfying (2.8) (cf. (2.13)). In most cases, solving (2.15) gives coefficients of the linear difference operators P PðmÞ ; RðmÞ of the form (2.13). Also, we have

ord PðmÞ ¼ ord e L ðm1Þ ¼ ord Lð1Þ ¼ ‘;

and ord RðmÞ ¼ ðm  1Þ‘:

ð2:16Þ

As we shall see later, the complexity and efficiency of our algorithm depend on the order of Lð1Þ , thus — on the orders of PðmÞ ; m 2 N. Choosing higher order operator Lð1Þ may result in a better acceleration of the convergence of series. Since the operators LðmÞ are of the form (2.9), we have

ord LðmÞ ¼ m‘:

ð2:17Þ ð1Þ

In Section 2.3, we will show that if ord L ¼ 1, then our algorithm is equivalent to Wynn’s epsilon algorithm. In the case of generalized hypergeometric series, we propose to use higher order operator Lð1Þ (see Section 3). Our method can then be ˇ ízˇek et al. (see [2], or [9]). In the case of basic identified as a special case of the G transformation recently proposed by C hypergeometric series, considered in Section 4, our transformation seems to be a new method of summation. In the case of orthogonal polynomial series with terms an ¼ bn W n ðxÞ, we propose to annihilate the term an by using the second-order linear recurrence equation, satisfied by polynomials W n ðxÞ (see Section 5 and (6.8)). Such idea is similar to the one proposed by Homeier in derivation of H; I, or K transformation; see [4,10–12]. Notice that annihilators LðmÞ of the form (2.9) were studied, among others, by Brezinski and Redivo-Zaglia [13], or Matos [7], in derivation of some Levin-type sequence transformations. Example 2.1. Consider the series 1 X

2 ln 2 ¼

an

with an :¼

n¼0

1 : ðn þ 1Þ 2n

ð2:18Þ

One can check that the operator

Lð1Þ :¼ ðn þ 1ÞI  ð2n þ 4ÞE

ð2:19Þ

satisfies Lð1Þ ðan Þ ¼ 0. We show in details the construction of operators PðmÞ for m ¼ 2; 3. We get

e L ð1Þ ¼ ðn þ 2ÞI  ð2n þ 6ÞE;

e L ð2Þ ¼ ðn þ 3ÞI  ð2n þ 8ÞE;

respectively (cf. (2.11)). Since Lð1Þ has the form (2.10) with ‘ ¼ 1, the assumption (2.8) (for m ¼ 2) can be satisfied by operators ð2Þ

ð2Þ

ð2Þ

Pð2Þ ¼ p0 ðnÞI þ p1 ðnÞE;

ð2Þ

Rð2Þ ¼ q0 ðnÞI þ q1 ðnÞE ð2Þ

(cf. (2.13)). Remark that one of the unknowns can be chosen arbitrarily, so we put q0 ðnÞ ¼ n þ 1, and obtain ð2Þ

ð2Þ

ð2Þ

ð2Þ

Pð2Þ Lð1Þ ¼ ðn þ 1Þ p0 ðnÞ I þ ½ðn þ 2Þ p1 ðnÞ  ð2n þ 4Þ p0 ðnÞ E  ð2n þ 6Þ p1 ðnÞ E2 ; ð2Þ e ð1Þ

ð2Þ 1 ðnÞ

¼ ðn þ 2Þðn þ 1ÞI þ ½ðn þ 3Þ q

R L

ð2Þ 2 1 ðnÞ E :

 ð2n þ 6Þðn þ 1ÞE  ð2n þ 8Þ q

Equating the coefficients of I; E; E2 , we compute

pð2Þ pð2Þ qð2Þ 0 ðnÞ ¼ n þ 2; 1 ðnÞ ¼ ð2n þ 8Þ; 1 ðnÞ ¼ ð2n þ 6Þ: Hence

Pð2Þ ¼ ðn þ 2ÞI  ð2n þ 8ÞE; and L

ð2Þ

ð2Þ ð1Þ

¼P L ð3Þ

P

ð2:20Þ 2

¼ ðn þ 1Þðn þ 2ÞI  4 ðn þ 2Þðn þ 3ÞE þ 4 ðn þ 3Þðn þ 4ÞE . Now we will find the operators

ð3Þ 0 ðnÞI

¼p

Rð2Þ ¼ ðn þ 1ÞI  ð2n þ 6ÞE;

ð3Þ

ð3Þ

ð3Þ

ð3Þ

Rð3Þ ¼ q0 ðnÞI þ q1 ðnÞE þ q2 ðnÞE2 ;

þ p1 ðnÞE;

ð3Þ

satisfying (2.8) for m ¼ 3. We put q0 ðnÞ ¼ ðn þ 1Þðn þ 2Þ and obtain ð3Þ

ð3Þ

ð3Þ

Pð3Þ Lð2Þ ¼ ðn þ 1Þðn þ 2Þ p0 ðnÞI  ðn þ 2Þðn þ 3Þð4 p0 ðnÞ  p1 ðnÞÞE   ð3Þ ð3Þ ð3Þ  4ðn þ 3Þðn þ 4Þ p1 ðnÞ  p0 ðnÞ E2 þ 4ðn þ 4Þðn þ 5Þ p1 ðnÞE3 ;

  ð3Þ ð3Þ ð3Þ ð3Þ L ð2Þ ¼ ðn þ 1Þðn þ 2Þðn þ 3ÞI  ðn þ 4Þð2ðn þ 1Þðn þ 2Þ  q1 ðnÞÞE þ ðn þ 5Þ q2 ðnÞ  2 q1 ðnÞ E2  2ðn þ 6Þ q2 ðnÞE3 : Rð3Þ e

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1626

Using computer algebra system like Maplee or Mathematicaâ , one can compute

pð3Þ pð3Þ 0 ðnÞ ¼ n þ 3; 1 ¼ ð2n þ 12Þ; ð3Þ q1 ðnÞ ¼ ð2n þ 4Þð2n þ 8Þ; qð3Þ 2 ðnÞ ¼ ð2n þ 8Þð2n þ 10Þ: Hence we obtain

Pð3Þ ¼ ðn þ 3ÞI  ð2n þ 12ÞE;

Rð3Þ ¼ ðn þ 1Þðn þ 2ÞI  ð2n þ 4Þð2n þ 8ÞE þ ð2n þ 8Þð2n þ 10ÞE2 :

ð2:21Þ

2.2. Algorithm Now, we present an efficient algorithm of performing the Q transformation, determined by the operators PðmÞ , hence by the annihilators LðmÞ obtained by the method of Section 2.1. Let us denote the numerators and denominators of the right side ðmÞ :¼ LðmÞ ðsn Þ and DðmÞ :¼ LðmÞ ð1Þ (notice the dependence on n, since LðmÞ annihilates rn ), respectively. By (2.9), of (2.3) by N ðmÞ n n we have the following recurrence relations:

NðmÞ ¼ PðmÞ ðNðm1Þ Þ; n n

DðmÞ ¼ PðmÞ ðDðm1Þ Þ: n n

Thus, the algorithm of computing QðmÞ does not need the operators LðmÞ explicitly; only PðmÞ are needed. We assume the inin tialization Pð1Þ :¼ Lð1Þ . P1 ðmÞ Algorithm 1. Consider the series ; m 2 N, are such that j¼0 aj with partial sums sn . If the operators P ðmÞ ðmÞ ðmÞ ðmÞ ðm1Þ ð1Þ :¼ P P    P annihilate r n , then array Qn can be computed in the following recurrent way: L

Nð0Þ n ¼ sn ;

Dð0Þ n ¼ 1;

N ðmÞ ¼ PðmÞ ðNðm1Þ Þ; n n QðmÞ :¼ n

NðmÞ n DðmÞ n

DðmÞ ¼ PðmÞ ðDnðm1Þ Þ; m P 1; n

:

In the following example, we are able to obtain the operators PðmÞ explicitly. Example 2.2. Consider the series (2.18) once more. We prove that continuing the construction described in Example 2.1 gives the operators PðmÞ of the form

PðmÞ :¼ ðn þ mÞI  ð2n þ 4mÞE;

m 2 N;

ð2:22Þ

(cf. (2.19)–(2.21)). Letting LðmÞ :¼ PðmÞ Pðm1Þ    Pð1Þ , one can check that

LðmÞ ðanþk Þ ¼

m! ðkÞm 2

nþk

ðn þ k þ 1Þmþ1

;

k 2 N0 ; m 2 N;

ð2:23Þ

where ðzÞk is the Pochhammer symbol defined for any complex number z by

ðzÞ0 :¼ 1;

ðzÞk :¼

k1 Y

ðz þ jÞ; k 2 N:

ð2:24Þ

j¼0

The proof of (2.23) is by induction on m 2 N. For any k 2 N0 we have

Lð1Þ ðanþk Þ ¼ ððn þ 1ÞI  ð2n þ 4ÞEÞðanþk Þ ¼ ðn þ 1Þanþk  ð2n þ 4Þanþkþ1 ¼

ðn þ 1Þðn þ k þ 2Þ  ðn þ 2Þðn þ k þ 1Þ 2nþk ðn þ k þ 1Þ2

;

and (2.23) holds for m ¼ 1. Let us suppose that it holds for some natural number m 2 N. Then for any k 2 N0 we obtain

Lðmþ1Þ ðanþk Þ ¼ Pðmþ1Þ LðmÞ ðanþk Þ ¼ Pðmþ1Þ ¼

!

m! ðkÞm 2nþk ðn þ k þ 1Þmþ1

¼

m! ðkÞm ðn þ m þ 1Þ 2nþk ðn þ k þ 1Þmþ1



m! ðkÞm ðn þ 2m þ 2Þ 2nþk ðn þ k þ 2Þmþ1

m! ðkÞm

2

nþk

½ðn þ m þ 1Þðn þ k þ m þ 2Þ  ðn þ 2m þ 2Þðn þ k þ 1ÞÞ : ðn þ k þ 1Þmþ2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ðmþ1ÞðmkÞ

This completes the proof. Since ðkÞm ¼ 0 for k < m, one can get that LðmÞ ðanþk Þ ¼ 0 for all k ¼ 0; 1; . . . ; m  1, and consequently that (2.2) holds. ðmÞ Hence, computing of Qn can be executed by Algorithm 1, i.e., in the following recurrent way:

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645 n1 X

Nð0Þ n :¼ sn ¼

aj ;

1627

Dð0Þ n :¼ 1;

j¼0 ðm1Þ

:¼ PðmÞ ðNðm1Þ Þ ¼ ðn þ mÞNnðm1Þ  ð2n þ 4mÞNnþ1 ; NðmÞ n n

m P 1;

ðm1Þ 4mÞDnþ1 ;

m P 1;

DðmÞ n

ðmÞ

:¼ P

:¼ QðmÞ n

ðDðm1Þ Þ n

NðmÞ n DðmÞ n

¼ ðn þ

mÞDðm1Þ n

 ð2n þ

:

Given the partial sums s1 ; s2 ; . . . ; s5 , one can obtain the following array of quantities QðmÞ (underlined digits are exact): n ð0Þ

ð1Þ

ð2Þ

ð3Þ

ð0Þ

ð1Þ

ð2Þ

ð3Þ

ð0Þ

ð1Þ

ð2Þ

ð0Þ

ð1Þ

ð4Þ

Q1 ¼ 1:000 Q1 ¼ 1:37500 Q1 ¼ 1:38596491 Q1 ¼ 1:38628472 Q1 ¼ 1:38629408 Q2 ¼ 1:250 Q2 ¼ 1:38333 Q2 ¼ 1:38621795 Q2 ¼ 1:38629227 Q3 ¼ 1:333 Q3 ¼ 1:38542 Q3 ¼ 1:38627451 Q4 ¼ 1:365 Q4 ¼ 1:38601 ð0Þ Q5

¼ 1:377 ð4Þ

Let us observe that Q1 is the following combination of the first 5 partial sums: ð4Þ

Q1 ¼

5 40 280 2240 672 s1  s2 þ s3  s4 þ s5 : 501 167 167 501 167 ð0Þ

Thus our method gives five exact digits more than partial sum s5 does (cf. Q5 ). Using Wynn’s e-algorithm (see, e.g., [3, Section2.3]) and Aitken’s iterated D2 process (see, e.g., [1, (5.1–15)] or [5, pp. 149–152]) one obtains ð1Þ eð1Þ A2 ¼ 1:38611111: 4 ¼ 1:38596491;

2.3. Relation to the Wynn’s epsilon algorithm Wynn’s e-algorithm (see, e.g., [3, Section2.3] or [1, Section4]) is defined by

eðnÞ eðnÞ 1 ¼ 0; 0 ¼ sn ; ðnÞ ðnþ1Þ 1 ekþ1 ¼ ek1 þ ½ekðnþ1Þ  eðnÞ n; k 2 N0 k  ; and solves the system of equations

sr ¼ ^s þ

m 1 X

cj Dsrþj ;

r ¼ n; n þ 1; . . . ; n þ m;

ð2:25Þ

j¼0 ðnÞ

being a model of Shanks transformation (see [14], or [13, p. 78]), with unknown ^s and coefficients cj , in the sense that ^s ¼ e2m . Consider the series (2.1) and assume that Lð1Þ is of the form

Lð1Þ ¼ k0 ðnÞI þ k1 ðnÞE: ð2Þ

ð3Þ

ð2:26Þ ðmÞ

ðmÞ

Let us denote by L ; L ; . . . ; L the operators obtained by the method of Section 2.1. By (2.17), we get ord L ¼ m, and thus one can solve the system of Eqs. (2.25) by the following trick. Applying the operator LðmÞ to both sides of the first equation of (2.25), we obtain ðmÞ

L

ðsn Þ ¼ L

ðmÞ

ð^sÞ þ LðmÞ

m1 X

! cj Dsnþj :

j¼0

One can prove that the last term in the right-hand side vanishes as a consequence of (2.2). Thus, we get

^s ¼

LðmÞ ðsn Þ : LðmÞ ð1Þ

Thus we obtain the following relation of the proposed method to Wynn’s epsilon algorithm: ðnÞ

QðmÞ ¼ e2m : n

ð2:27Þ

Example 2.3. Let us consider the series (2.18). It is quite fast convergent, since:

accðs5 Þ ¼ 2:2;

accðs15 Þ ¼ 5:6;

accðs25 Þ ¼ 8:8;

accðs35 Þ ¼ 11:9; ð1Þ

accðs45 Þ ¼ 15:1: ð1Þ

However, let us use only the partial sums s1 ; s2 ; . . . ; s15 . Observe that L ¼ P is, in view of (2.22), of the form (2.26). The ð1Þ ð1Þ ð1Þ first row of the array obtained by e-algorithm contains the quantities e0 ; e1 ; . . . ; e14 . Algorithm 1 yields the quantities

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1628

Table 1 Values of accðQðmÞ n Þ. Results obtained by e-algorithm (above the stepping line) are marked in red.

ð0Þ

ð1Þ

ð14Þ

ð14Þ

ð1Þ

Q1 ; Q1 ; . . . ; Q1 (cf. Table 1), which is more than e-algorithm gives, since Q1 ¼ e28 . The reason is that we have the formulae (2.22), which are available thanks to the knowledge of analytical form of the terms an , as described in Section 2.1. Wynn’s method does not need such information.

3. Generalized hypergeometric series Now, we present the application of the proposed method to the case of generalized hypergeometric series

 !

pþ1 Fp

a0 ; a1 ; . . . ; ap 

x b1 ; b2 ; . . . ; bp 

:¼

1 X ða0 Þn ða1 Þn ða2 Þn    ðap Þn n x ðb1 Þn ðb2 Þn    ðbp Þn n! n¼0

(see, e.g., [15, Section2.1]), where ðzÞn is the Pochhammer symbol (see (2.24)). 3.1. Main results Without loss of generality we can assume that a0 :¼ 1, since one can write

 !

pþ1 Fp

a0 ; a1 ; . . . ; ap 

x b1 ; b2 ; . . . ; bp 

¼ pþ2 Fpþ1

 ! 1; a0 ; a1 ; . . . ; ap  x : 1; b1 ; b2 ; . . . ; bp 

In Theorem 2, we present the explicit form of operators PðmÞ which are the most important input in Algorithm 1 of computing the proposed transformation in the case of considered series. We will show that our method of summation can be ˇ ízˇek et al. , recently introduced in [2], and analysed by Weniger identified as some special variant of the transformation of C in [9]. Numerical examples show that if we choose the second or higher order operator Pð1Þ , then our method is not equivalent to e-algorithm. Given a vector c ¼ ðc1 ; c2 ; . . . ; cp Þ, we will use the notation

Ac þ B :¼ ðAc1 þ B; Ac2 þ B; . . . ; Acp þ BÞ;

A; B 2 C;

and

hcin :¼

p Y

ðcj Þn ;

hci :¼ hci1 ;

n 2 N0 :

ð3:1Þ

j¼1

Hence, we write the considered series as

pþ1 Fp

 ! 1 X 1; a1 ; a2 ; . . . ; ap  an x ¼ b1 ; b2 ; . . . ; bp  n¼0

with an :¼

hain n ða1 Þn ða2 Þn    ðap Þn n x ¼ x : hbin ðb1 Þn ðb2 Þn    ðbp Þn

ð3:2Þ

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1629

Theorem 2. Let  us define 

hbin n x I ; hain  p  X mp :¼ ðDj hb þ n þ mðp þ 1Þ  j  2iÞDpj ; j j¼0

Pð1Þ :¼ Dp

PðmÞ

ð3:3aÞ m ¼ 2; 3 . . . :

ð3:3bÞ

Then the operators LðmÞ :¼ PðmÞ Pðm1Þ    Pð1Þ can be written in the form

LðmÞ ¼ Dmp

  hbinþm1 I ; hain xn

m 2 N;

ð3:4Þ

LðmÞ ðan þ anþ1 þ    þ anþm1 Þ ¼ 0:

ð3:5Þ

and satisfy

Proof. By the following well-known property of operator D:

Dr ðfn IÞ ¼

r   X r ðDj fnþrj ÞDrj ; j j¼0

r 2 N0 ;

ð3:6Þ

(cf. [16, Theorem 1.3.2]), we obtain

Dmp ðhb þ n þ m  2iIÞ ¼ PðmÞ Dðm1Þp ;

m ¼ 2; 3; . . . ;

PðmÞ being given by (3.3b). Hence, (3.4) follows by induction on m. To prove (3.5), notice that LðmÞ ðanþk Þ ¼ 0 for k ¼ 0; 1; . . . ; m  1, since

hbinþm1  anþk ¼ ha þ nik  hb þ n þ kim1k  xk hain xn is a polynomial in n of degree ðm  1Þp. h The following two corollaries give the form of operators PðmÞ with coefficients being polynomials in n. Corollary 3 (Gauss series). Given the series 2 F1



  X 1 1; a  x ¼ an b 

with an :¼

n¼0

ðaÞn n x ; ðbÞn

let us define

b ðmÞ :¼ xðn þ m þ a  1ÞI  ðn þ 2m þ b  2ÞE; P b ðmÞ

Then the operators L

b ðmÞ L

b ðmÞ

:¼ P

b ðm1Þ

P

b ð1Þ

P

m 2 N:

satisfy

ðan þ anþ1 þ    þ anþm1 Þ ¼ 0:

ð3:7Þ

b ðmÞ in the sense that Proof. One can check that the operators PðmÞ , given in Theorem 2 (in the case of p ¼ 1), are related to P

b ðkÞ ¼ PðkÞ Qðk1Þ ; QðkÞ P

k 2 N;

where

Qð0Þ :¼ I;

QðkÞ :¼

ðbÞnþk1 I; k 2 N: ðaÞnþk xnþk

Thus LðmÞ ¼ QðmÞ b L ðmÞ , m 2 N, and consequently (3.7) holds. h Corollary 4 (Series 3 F 2 ). Given the series 3 F2



  X 1 1; a1 ; a2  x ¼ an  b ;b 1

2

n¼0

with an :¼

ða1 Þn ða2 Þn n x ; ðb1 Þn ðb2 Þn

let us define

b ðmÞ :¼ x2 hn þ 2m þ a  2i I  2xhn þ 2m þ a  1i½hn þ 2m þ b  2i P 2  mðm  1Þ E þ hn þ m þ b  1ihn þ 3m þ b  2i E2

ð3:8Þ

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1630

b ðmÞ P b ð1Þ satisfy b ðm1Þ    P with a ¼ ða1 ; a2 Þ; b ¼ ðb1 ; b2 Þ. Then the operators b L ðmÞ :¼ P

b L ðmÞ ðan þ anþ1 þ    þ anþm1 Þ ¼ 0:

ð3:9Þ

Proof. The proof follows very closely the previous proof, since the operators PðmÞ , given in Theorem 2 (in the case of p ¼ 2), b ðmÞ in the sense that are related to P

b ðkÞ ¼ PðkÞ Qðk1Þ ; QðkÞ P

k 2 N;

where

Qð0Þ :¼ I;

QðkÞ :¼

ðb1 Þnþk1 ðb2 Þnþk1 I; ða1 Þnþ2k ða2 Þnþ2k xnþ2k

k 2 N:

Thus LðmÞ ¼ QðmÞ b L ðmÞ ; m 2 N, and hence (3.9) holds. h Example 3.1. Consider the series 3 F2

 ! 1 X 1; 12 ; 12  ð1Þn  1 ¼  2 3 3  ; 2 2 n¼0 ð2n þ 1Þ

ð3:10Þ

summing up to the Catalan’s constant G  0:9159655941772190. We compared the results obtained by:     

eðnÞ 2k – Wynn’s e-algorithm (see [1, (4.2-2)]), ðnÞ

Ak – Aitken’s iterated D2 process (see [1, (5.1-15)]), ðkÞ sn – The method proposed by Lewanowicz and Paszkowski in [17, Theorem 3.2], ðkÞ t u LðkÞ n ð1; fsn gÞ and Ln ð1; fsn gÞ – t and u variants of Levin transformation (see [1, (7.1-7), (7.3-5), (7.3-7)], Q – transformation (2.3) determined by operators (3.8). The input for all algorithms was the set of the partial sums s1 ; s2 ; . . . ; s15 of series (3.10). In Table 2, we present the values: ðkÞ

ð1Þ

ð1Þ

ð2kÞ

ð2kÞ

accðQ1 Þ; accðe2k Þ; accðAk Þ; accðt L1 ð1; fsn gÞÞ; accðu L1 ð1; fsn gÞÞ ðkÞ

for k ¼ 0; 1; . . . ; 7, respectively. In the case of method of Lewanowicz and Paszkowski, we present the antidiagonal skþ1 . Since the series (3.10) is alternating, e-algorithm and Aitken’s D2 process are quite efficient. Example 3.2. Consider the following positive series:

 !

A ¼ 2 F1

1 1 ; 4 2 1 5  4

¼ 3 F2

 ! 1 X 1; 14 ; 12  an ; 1 ¼ 5  1; 4 n¼0

1 1 an :¼ 4 n 25 n ; ð1Þn 4 n

summing up to lemniscate constant A  1:31102877714605990523. In this case, e-algorithm is ineffective; we have, for inð1Þ stance, accðe14 Þ ¼ 1:61. The convergence of the series can be successfully accelerated by the u variant of Levin transformað14Þ tion, or the method of Lewanowicz and Paszkowski (see [17, Theorem 2.2]). Namely, we have accðu L1 ð1; fsn gÞÞ ¼ 11:50 ð7Þ and accðs8 Þ ¼ 13:88, respectively. The Q transformation, determined by operators (3.8), yields similar results; we have ð7Þ accðQ1 Þ ¼ 13:03. Notice that ord Lð1Þ ¼ 2, and much better results than e-algorithm (corresponding to the choice of first-order operator Lð1Þ ; see Section 2.3) are obtained. Thus the order of operator Lð1Þ , being of choice in the Section 2.1, has a great importance indeed.

Table 2 Accuracy of approximations to the sum of the series (3.10). Comparison of the proposed summation method, e-algorithm, Aitken’s iterated D2 process, method of Lewanowicz and Paszkowski [17], t and u variants of Levin transformation. k

0

1

2

ð1Þ accð 2k Þ ð1Þ accðAk Þ ðkÞ accðskþ1 Þ ð2kÞ accðt L1 ð1; fsn gÞÞ ð2kÞ accðu L1 ð1; fsn gÞÞ ðkÞ accðQ1 Þ

1.04

2.59

4.14

e

3

4 5.67

5

7.21

8.74

6

7

10.27

11.81

1.04

2.59

4.40

6.40

8.60

13.06

11.81

14.46

1.04

3.00

4.89

6.76

8.62

10.46

12.30

14.14

1.04

4.91

5.76

11.24

12.11

14.23

16.91

20.15

1.04

2.70

5.10

8.78

10.95

13.95

16.61

18.78

1.04

3.44

6.20

9.55

12.27

14.95

18.15

21.19

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1631

3.2. Some theoretical results Theorem 5. The Q transformation, applied to the generalized hypergeometric series (3.2) with x–1, is regular, i.e.,

 ! 1; a1 ; a2 ; . . . ; ap  for all m 2 N: x b1 ; b2 ; . . . ; bp 

lim QðmÞ n ðsn Þ ¼ pþ1 Fp

n!1

Proof. Let LðmÞ ; m 2 N, denote the operators (3.4). We have

QðmÞ n

LðmÞ ðsn Þ ¼ ¼ ðmÞ L ð1Þ

hainþmp xnþmp hbinþm1

 LðmÞ ðsn Þ

hainþmp xnþmp hbinþm1

ðmÞ

L

ð1Þ

Pmp

ðmpÞ

j¼0 kj Pmp

¼

ðn; xÞ snþj

ðmpÞ

j¼0 kj

ðn; xÞ

ð3:11Þ

;

where

hainþmp xnþmp ðmpÞ kj ðn; xÞ :¼ hbinþm1

" mpj



ð1Þ

#    mp mp hbinþjþm1 ¼ ðxÞmpj ha þ n þ jimpj hb þ n þ m  1ij ; j ¼ 0;1; .. .; mp: nþj hainþj x j j ðmpÞ

Denote the denominator in the last leg of (3.11) by Mðn; xÞ. Since kj ðn; xÞ are polynomials in n of degree mp2 with leading   mp coefficient ðxÞmpj , under assumption x–1, the function Mðn; xÞ is also a polynomial in n of degree mp2 ; more j precisely 2

Mðn; xÞ ¼ ð1  xÞmp nmp þ Oðnmp

2 1

n ! 1:

Þ;

Let sn denote the partial sums of the series (3.2) and let limn!1 sn ¼ s. Then

s

QðmÞ n

LðmÞ ðsÞ LðmÞ ðsn Þ  ¼ ¼ ðmÞ L ð1Þ LðmÞ ð1Þ

hainþmp xnþmp hbinþm1

 LðmÞ ðs  sn Þ

xnþmp

hainþmp hbinþm1

 LðmÞ ð1Þ

Pmp ¼

ðmpÞ ðn; xÞ ðs  snþj Þ j¼0 kj Pmp ðmpÞ ðn; xÞ j¼0 kj

¼

ðmpÞ mp X kj ðn; xÞ

Mðn; xÞ

j¼0

ðs  snþj Þ ! 0; n!1

because ðmpÞ

kj

ðn; xÞ

Mðn; xÞ

¼ Oð1Þ;

j ¼ 0; 1; . . . ; mp; n ! 1: 

In the following theorem, we present an efficiency of our transformation in the case of generalized hypergeometric series. Theorem 6. pþ1 Fp

 ! 1; a1 ; a2 ; . . . ; ap  nþmðpþ1Þ Þ; x  QðmÞ n ðsn Þ ¼ Oðx b1 ; b2 ; . . . ; bp  ðmpÞ

Proof. Let us use the functions kj satisfying

Mðn; xÞ ¼ Oð1Þ;

x ! 0:

ðn; xÞ and Mðn; xÞ introduced in the proof of Theorem 5. Now, Mðn; xÞ is a function of x

x ! 0:

Similarly, we obtain ðmÞ

s

QðmÞ n

LðmÞ ðs  sn Þ L ¼ ¼ LðmÞ ð1Þ ¼

ðmpÞ mp X kj ðn; xÞ j¼0

Mðn; xÞ

P

m1 k¼0 anþk ðmÞ

L

þ rnþm

ð1Þ

ðmpÞ

ðmpÞ

ðn; xÞ ¼ Oðxmpj Þ;

LðmÞ ðr nþm Þ ¼ ðmÞ ¼ L ð1Þ

hainþmp xnþmp hbinþm1

 LðmÞ ðrnþm Þ

hainþmp xnþmp hbinþm1

 LðmÞ ð1Þ

r nþjþm :

This completes the proof, since functions kj

kj



r nþmþj ¼

1 X i¼0

ðn; xÞ and r nþmþj satisfy

hainþmþiþj nþmþiþj x ¼ Oðxnþmþj Þ; x ! 0:  hbinþmþiþj

Pmp ¼

ðmpÞ ðn; xÞ r nþjþm j¼0 kj

Mðn; xÞ

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1632

Remark 7. The authors were unable to show that Q transformation accelerates the convergence in the sense that

lim

n!1

Qðmþ1Þ ðsn Þ  s n ¼ 0: ðmÞ Qn ðsn Þ  s

This seems to be a very difficult problem, in general. 3.3. Relation to the Cízˇek et al. transformation By (2.3) and (3.4), the Q transformation in the case of generalized hypergeometric series can be written in the following form:

QðmÞ n

i h sn Dmp ðn þ b1 Þm1 ðn þ b2 Þm1    ðn þ bp Þm1 h i ¼ h ¼ hbi Dmp hainþm1 Dmp ðn þ b1 Þm1 ðn þ b2 Þm1    ðn þ bp Þm1 xn

Dmp

h

hbinþm1 hain xn n

sn an 1 an

i i:

ð3:12Þ

Remind that the G transformation of Cízˇek et al. is defined by m1 GðmÞ n ðfqk gk¼1 ; fsn g; f

xn gÞ :¼

Dm Dm

hQ

m1 k¼1 ðn

þ qk Þ xsnn

m1 k¼1 ðn

þ qk Þ x1n

hQ

i i;

ð3:13Þ

where fqk gm1 k¼1 is a set of parameters, and xn is a sequence of remainder estimates (see [2] or [1, (2.13)]). In special cases qk  1, or qk ¼ k (for k ¼ 1; 2; . . . ; m  1) the Levin and Weniger [1, Section8.2] transformations are obtained, respectively. Now, it is easy to observe that the transformation (3.12) is equivalent to the transformation (3.13) with some particular choice of parameters and remainder estimates. Namely, we have the following. Proposition 8. Equation 



m 1 QðmÞ ¼ Gnðm Þ ðfqk gk¼1 ; fsn g; fxn gÞ n

holds, where

m :¼ mp;  n

x :¼ ðn þ 1Þ 

m 1 fqk gk¼1

ð3:14aÞ p1

an ; ð3:14bÞ 9 8 > > = < :¼ 1; 1; . . . ; 1; b1 ; b2 ; . . . ; bp ; b1 þ 1; b2 þ 1; . . . ; bp þ 1; . . . ; b1 þ m  2; b2 þ m  2; . . . ; bp þ m  2 : ð3:14cÞ |fflfflfflfflfflffl ffl {zfflfflfflfflfflffl ffl } > > ; : p1 times

Example 3.3. For the series

 s ¼ 5 F4

  1; 1; 2; 2; 7  99  1:0376328566238592296948;  3; 5; 6; 9 100

we obtain ð5Þ

Q1  1:03763285662385922975208; ð5Þ

ð5Þ

ð20Þ

ð20Þ

which yields accðQ1 Þ ¼ 19:26. Notice that Q1 ¼ G1 ðfqk g; fsn g; fxn gÞ; see Proposition 8 ðp ¼ 4Þ. We computed G1 and compared the choices of parameters fqk g and remainder estimates fxn g given in (3.14) with the classic ones. As shown in Table 3, the best results are obtained with the parameters chosen as in (3.14), i.e., by using Q transformation (cf. (3.12)).

Table 3 ð20Þ Values of accðG1 ðfqk g19 k¼1 ; fsn g; fxn gÞÞ. qk

(3.14c) 1 (Levin L [1, (7.1-7), b ¼ 1]) k (Weniger S [1, (8.2-7), b ¼ 1]) 2

k (cf. [2, Tables 1 and 2])

xn (3.14b)

an1 (t-variant)

ðn þ 1Þan1 (u-variant)

an  aann1 an1 (v-variant)

19.26 18.44 14.20 1.51

15.46 14.46 16.04 10.60

16.78 15.83 17.33 8.39

17.67 16.69 18.24 9.73

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1633

4. Basic hypergeometric series The Q transformation may also be successfully applied to the case of the basic hypergeometric series, or q-hypergeometric series,



pþ1 /p

!

a0 ; a1 ; . . . ; ap 

q; x b1 ; b2 ; . . . ; bp 

1 X ða0 ; qÞn ða1 ; qÞn ða2 ; qÞn    ðap ; qÞn n x ðb1 ; qÞn ðb2 ; qÞn    ðbp ; qÞn ðq; qÞn n¼0

:¼

(see, e.g., [18, (1.2.22)]), where ðz; qÞk is the q-Pochhammer symbol, defined for any complex number z by

8 > <

ðz; qÞk :¼

1; k1 Q

> :

k ¼ 0;

ð1  zqj Þ; k > 0:

j¼0

Without loss of generality we may assume that a0 :¼ q, since one can write



pþ1 /p

a0 ; a1 ; . . . ; ap 

!

q; x b1 ; b2 ; . . . ; bp 

¼ pþ2 /pþ1

 ! q; a0 ; a1 ; . . . ; ap  q; x : q; b1 ; b2 ; . . . ; bp 

We append the notation (3.1) by

hc; qin :¼

p Y ðcj ; qÞn ; j¼1

and write the considered series shortly as pþ1 /p

 ! 1 X q; a1 ; . . . ; ap  an q; x ¼ b1 ; b2 ; . . . ; bp  n¼0

with an :¼

ha; qin n x : hb; qin

ð4:1Þ

Before we propose a q-analogue of Theorem 2, let us introduce the operator Dq and its powers by

D0q :¼ I; Drq :¼ ðE  qr1 IÞDr1 q ;

r 2 N:

One can check, that q-analogue of (3.6) is

Drq ðfn IÞ ¼

r  X r j¼0

where

j

ðDjq fnþrj ÞDrj q ;

r 2 N0 ;

ð4:2Þ

q

 r is the Gauss binomial coefficient, defined by j q

 r j

:¼

q

ðq; qÞr ; ðq; qÞj ðq; qÞrj

r; j 2 N0 ;

j 6 r:

Theorem 9. Let us define

 hb; qin n x I ; ha; qin  p

X mp ~ ðmpp;pjÞ ; :¼ ðDjq h1  b qnþmðpþ1Þj2 iÞD q j q j¼0

Pð1Þ :¼ Dpq PðmÞ



m ¼ 2; 3 . . . ;

where

~ ði;0Þ :¼ I; D q

~ ði;rÞ :¼ D ~ ðiþ1;r1Þ ðE  qi IÞ; D q q

Then the operators L

LðmÞ

ðmÞ

ðmÞ

:¼ P

P

ðm1Þ

  hb; qinþm1 ¼ Dmp I ; q ha; qin xn

ð1Þ

P

m 2 N;

i 2 Z; r 2 N: are of the form

ð4:3Þ

and satisfy

LðmÞ ðan þ anþ1 þ    þ anþm1 Þ ¼ 0:

ð4:4Þ

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1634

Table 4 Values of accðQðmÞ n Þ for the approximations of the sum of the series (4.5), with a ¼ 9=10; b ¼ 5=8; c ¼ 1=2 and q ¼ 1=5. nnm

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13

0.42 0.48 0.53 0.59 0.64 0.69 0.74 0.79 0.84 0.89 0.94 1.00 1.05

3.18 4.67 6.13 7.58 9.03 10.48 11.93 13.38 14.83 16.28 17.73

8.75 11.64 14.50 17.35 20.19 23.04 25.89 28.73 31.58

14.77 18.36 21.92 25.47 29.01 32.56 36.11

22.17 26.46 30.72 34.97 39.21

30.97 35.96 40.91

41.15

Proof. The proof follows very closely the proof of Theorem 2. Namely, using the the property (4.2), we obtain nþk2 Dkp iIÞ ¼ PðkÞ Dqðk1Þp ; q ðh1  b q

k ¼ 2; 3; . . . ; m  1;

and thus the proof of (4.3) follows by induction on m. Next, observe that

hb; qinþm1  anþk ¼ haqn ; qik  hbqnþk ; qim1k  xk ; ha; qin xn

k ¼ 0; 1; . . . ; m  1;

is a polynomial in qn of degree ðm  1Þp. Since operator Drq annihilates all the polynomials in qn of degree at most r  1, we have that LðmÞ ðanþk Þ ¼ 0 for all k ¼ 0; 1; . . . ; m  1. This completes the proof of (4.4). h Summarizing, in the case of q-hypergeometric series (4.1), we can compute the Q transformation by Algorithm 1 with operators PðmÞ given in Theorem 9. Such a transformation does not seem to be related to any classic method of convergence acceleration. Example 4.1. Let us consider the following q-hypergeometric series:

 2 /1

     a; b  q; a; b  q; c=ab ¼ 3 /2 q; c=ab :   c q; c

ð4:5Þ

The Heine’s q-analogue of Gauss summation formula is given by

 2 /1

  a; b  ðc; =a; qÞ1 ðc=b; qÞ1 q; c=ab ¼ ðc; qÞ ðc=ab; qÞ c  1

1

(see, e.g., [18, (1.5.1)]), where ðz; qÞ1 is the generalized q-Pochhammer symbol, defined by

ðz; qÞ1 ¼

1 Y ð1  zqj Þ: j¼0

Let us put a ¼ 9=10; b ¼ 5=8; c ¼ 1=2 and q ¼ 1=5. Then the sum s of the series equals approximately 1.6172358319254889075. Let us notice the very slow convergence of the considered series (cf. the second column of Table 4. We used only the partial sums s1 ; s2 ; . . . ; s13 ; and the operators PðmÞ ; m ¼ 1; 2; . . . ; 6; given in Theorem 9. The results obtained are presented in Table 4.

5. Orthogonal polynomial series Let us consider the series 1 X

an ;

an :¼ bn f n ;

ð5:1Þ

n¼0

where fn satisfies a linear difference equation of the second order

jn f n þ kn f nþ1 þ ln f nþ2 ¼ 0; n 2 N0 :

ð5:2Þ

It is known that Eq. (5.2) is satisfied, if fn ¼ W n ðxÞ, where W n ðxÞ are orthogonal polynomials in an interval I with respect to a weight function xðxÞ. More specifically, if W n is

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1635

(I) the nth Chebyshev polynomial of the first kind T n ðI ¼ ½1; 1; xðxÞ ¼ ð1  x2 Þ1=2 Þ or of the second kind U n ðI ¼ ½1; 1; xðxÞ ¼ ð1  x2 Þ1=2 Þ, then

jn ¼ 1; kn ¼ 2x; ln ¼ 1;

ð5:3Þ

(II) the nth Legendre polynomial P n ðI ¼ ½1; 1; xðxÞ  1Þ, then

jn ¼ n þ 1; kn ¼ ð2n þ 3Þx; ln ¼ n þ 2; (III) the nth Laguerre polynomial

LnðaÞ ðI

ð5:4Þ

¼ ½0; 1Þ; xðxÞ ¼

xa ex ; a

> 1), then

jn ¼ n þ 1 þ a; kn ¼ ð2n þ 3 þ a  xÞ; ln ¼ n þ 2; x2

(IV) the nth Hermite polynomial Hn ðI ¼ ð1; 1Þ; xðxÞ ¼ e

ð5:5Þ Þ, then.

jn ¼ n þ 1; kn ¼ x; ln ¼ 1=2;

ð5:6Þ

see, e.g., [8, p. 36], or [19, Sections1.13, 1.11, 1.8.3, 1.8.2]. The case of Fourier series

s ¼ A0 =2 þ

1 X ðAn cos na þ Bn sin naÞ

ð5:7Þ

n¼1

corresponds to the case I, since the functions fn ¼ sin na; cos na satisfy the Eq. (5.2) with

jn ¼ 1; kn ¼ 2 cos a; ln ¼ 1:

ð5:8Þ

The summation of such series was studied, among others, by Homeier [10]. He proposed the H transformation, defined as follows: 1 Nð0Þ sn =xn ; n ¼ ðn þ bÞ

1 Dð0Þ n ¼ ðn þ bÞ =xn ; ðk1Þ

ðk1Þ

ðk1Þ NðkÞ  2 cos a ðn þ k þ bÞNnþ1 þ ðn þ 2k þ bÞN nþ2 ; n ¼ ðn þ bÞN n

DðkÞ n

¼ ðn þ

bÞDðk1Þ n

 2 cos a ðn þ k þ

ðk1Þ bÞDnþ1

þ ðn þ 2k þ

ðk1Þ bÞDnþ2 ;

ð5:9Þ

ðkÞ ðkÞ HðkÞ n ða; b; fsn g; fxn gÞ ¼ N n =Dn ;

see, e.g., [4, (155)]. The transformation of the Fourier series (5.7) to accelerate their convergence, especially near the singularities, was proposed by Oleksy [20]. The hierarchically consistent iteration of Hð1Þ transformation was studied by Homeier [11] in derivation of his I transformation. More generally, in the case of orthogonal series (5.1), hierarchically consistent iteration of the transformation

s0n :¼

jn xsnn þ kn xsnþ1 þ ln xsnþ2 nþ1 nþ2 1 1 jn x1n þ kn xnþ1 þ ln xnþ2

ð5:10Þ

yields K transformation, defined as follows:

Nð0Þ Dð0Þ n ¼ sn =xn ; n ¼ 1=xn ; 1 ðk1Þ ðk1Þ ðk1Þ þ knþk1 Nnþ1 þ lnþk1 Nnþ2 Þ; N ðkÞ n ¼ ðkÞ ðjnþk1 N n dn 1 ðk1Þ ðk1Þ ðk1Þ þ knþk1 Dnþ1 þ lnþk1 Dnþ2 Þ; DðkÞ n ¼ ðkÞ ðjnþk1 Dn dn

ð5:11Þ

ðkÞ ðkÞ ðkÞ KðkÞ n ðfsn g; fxn g; fdn g; fjn ; kn ; ln gÞ ¼ N n =Dn ;

see, e.g., [12], or [4, (174)]. The transformation (5.10) is exact for the sequences of the form

sn ¼ s þ xn ðc Pn þ d Q n Þ; where c and d are arbitrary constants, while P n and Q n are two linearly independent solution of the Eq. (5.2). For instance, it is exact in the case

sn þ bn fn ¼ s: Let us notice that QðmÞ transformation is exact for any sequence sn satisfying

sn þ rðmÞ ¼ sn þ bn fn þ bnþ1 fnþ1 þ    þ bnþm1 fnþm1 ¼ s: n

ð5:12Þ

Since Homeier’s transformations are Levin-type sequence transformations, they can be performed by the Algorithm 1 with some special operators PðmÞ . Notice that such operators do not correspond to the model (5.12), since usually the ðmÞ operators LðmÞ :¼ PðmÞ Pðm1Þ    Pð1Þ do not annihilate rn ; cf. (2.2). Indeed, in the case of orthogonal series (5.1), we usually ðmÞ ðmÞ are not able to obtain explicit forms of operators P , and so the operators LðmÞ ; m 2 N, annihilating r n . In the sequel, we

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1636

consider some special cases of the series (5.1), where we found operators PðmÞ explicitly for m 2 N. Some numerical computations, comparing Q transformation with Homeier’s methods and the methods recently introduced by Paszkowski [8], are given. In the following theorem, we present an explicit form of operators PðmÞ in the case of some Hermite series. Theorem 10. Let z 2 C, and

an :¼

1 Hn ðxÞ; zn n!

n 2 N0 :

Let us define the operators

PðmÞ :¼ I  zxE þ z2 ðn þ 3m  1Þ=2E2 ;

m 2 N:

Then the operators LðmÞ :¼ PðmÞ Pðm1Þ    Pð1Þ satisfy

LðmÞ ðan þ anþ1 þ    þ anþm1 Þ ¼ 0:

ð5:13Þ

Proof. We proceed by induction on m 2 N. One can check that

Lð1Þ ðan Þ ¼ Pð1Þ



1

zn n!

Hn

 ¼

1 ððn þ 1ÞHn ðxÞ  xHnþ1 ðxÞ þ Hnþ2 ðxÞ=2Þ ¼ 0 zn ðn þ 1Þ!

(cf. (5.6)). Suppose that (5.13) holds for some m 2 N. Observe that it is enough to show that Lðmþ1Þ ðanþm Þ ¼ 0. Using the following property:

Pðkþ1Þ Ej ¼ Ej Pð1Þ þ 2ð3k  jÞEjþ2 ;

j; k 2 N0 ;

we obtain

We can do the cancellations shown above, since Pð1Þ ðan Þ ¼ 0.

h

In Theorem 12, preceded by a lemma, we present the forms of operators PðmÞ ; m 2 N, in the case of the series (5.1) with some classes of sequences bn ; fn . Lemma 11. Let us define the function

rðnÞ as follows

ðA þ 1Þn rðnÞ :¼  1; ðA þ 1  dÞn

ð5:14Þ

where A; d; 1 2 C. Further, in the case d ¼ 0 (resp. d–0), let us define the operators

b ðmÞ :¼ u ðn þ m þ AÞI þ w ðn þ 2m þ BÞE þ v ðn þ 3m þ CÞE2 ; P

ð5:15Þ

where u; w; v; B; C 2 C (resp. A ¼ B ¼ C) are chosen so that

b ð1Þ ðrðnÞ  an Þ ¼ 0: P

ð5:16Þ

Then

b ðmÞ    P b ð1Þ ½rðnÞðan þ anþ1 þ    þ anþm1 Þ ¼ 0; b ð2Þ P P

m 2 N:

ð5:17Þ

b ðkÞ defined in (5.15) satisfy Proof. One can check that operators P

b ðkþ1Þ E2k P b ðkÞ Ek1 ¼ E1k P b ðkþ1Þ Ek P b ðkÞ ; P Let us define the operator

D :¼ ð1 þ dÞðuI þ wE þ vE2 Þ:

k 2 N:

ðaÞ

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1637

Observe that

b ð1Þ ðrðnÞEÞ ¼ ðE P b ð1Þ  DÞðrðnÞIÞ: P

ðbÞ

One can also check the following property:

b ðkþ1Þ D ¼ D P b ðkÞ ; P

k 2 N:

ðcÞ

The proof of (5.17) is by induction on m 2 N. By (5.16), it holds for m ¼ 1. Suppose that (5.17) holds for some m 2 N. Observing

b ð1Þ ½rðnÞðan þ anþ1 þ    þ anþm Þ ¼ P b ðmþ1Þ P b ð2Þ P b ðmþ1Þ P b ðmÞ    P b ðmÞ    P b ð1Þ ðrðnÞ anþm Þ; P we will show that right side of above formula vanishes. Namely, using the properties (a), (b), (c), we obtain

b ðmþ1Þ P b ðmÞ    P b ð1Þ ðrðnÞ anþm Þ ¼ ½ P b ð1Þ ðrðnÞEÞðanþm1 Þ b ð2Þ P b ðmþ1Þ    P b ð2Þ P P ðbÞ

b ðmþ1Þ    P b ð2Þ ðE P b ð1Þ  DÞðrðnÞanþm1 Þ ¼ ½P b ðmþ1Þ    P b ð3Þ E0 P b ð2Þ E P b ð2Þ DÞðrðnÞanþm1 Þ b ð4Þ P b ð1Þ Þ  ð P b ðmþ1Þ    P b ð3Þ P ¼ ½ð P

ða;cÞ

b ð3Þ E2 P b ð2Þ P b ð1Þ Þ  ð P b ðmþ1Þ    P b ð4Þ E1 P b ðmþ1Þ    P b ð3Þ D P b ð1Þ ÞðrðnÞanþm1 Þ ¼ ½ð P

ða;cÞ

ða;cÞ b ð2Þ P b ðmÞ    P b ð1Þ ÞðrðnÞanþm1 Þ b ðmþ1Þ Em P b ðmÞ    P b ð1Þ Þ  ðD P ¼ . . . ¼ ½ðE1m P b ðmþ1Þ Em  DÞ P b ð1Þ ðanþm1 Þ: b ðmÞ    P b ð2Þ P ¼ ðE1m P

Now, observe that

b ð2Þ P b ðmÞ    P b ð2Þ P b ðmÞ    P b ð1Þ ðanþm1 Þ ¼ P b ð1Þ ða1 þ a2 þ    þ anþm1 Þ ¼ 0: P This completes the proof. h ðmÞ Theorem 12. Let the sequences bn ; W n ðxÞ, and the functions AnðmÞ ; BðmÞ n , and C n ; m 2 N, be defined as follows:

Case

I

II

III

bn

1 zn ðnþtÞv

1 zn

1 zn

W n ðxÞ

T n ðxÞ; U n ðxÞ

Pn ðxÞ

AðmÞ n BðmÞ n C ðmÞ n

nþmþtþv 2

nþm

LnðaÞ ðxÞ nþmþa

Here x; z; t 2 C and

2zxðn þ 2m þ t þ v  2Þ

2zxðn þ 2m  1=2Þ

2zðn þ 2m þ ða  x  1Þ=2Þ

z2 ðn þ 3m þ t þ v  2Þ

z2 ðn þ 3m  1Þ

z2 ðn þ 3m  1Þ

v 2 N0 . Further, define the second-order operators

b ðmÞ :¼ AðmÞ I þ BðmÞ E þ C ðmÞ E2 ; P n n n

m 2 N:

ð5:18Þ

b ð1Þ ððn þ tÞ IÞ; in the case I; b ðm1Þ    P b ðmÞ P P v 1 ðmÞ b ðm1Þ b ð1Þ ; b  P in the cases II; III; P P

ð5:19Þ

Then the operators

( b L ðmÞ :¼ satisfy

b L ðmÞ ðan þ anþ1 þ    þ anþm1 Þ ¼ 0;

m 2 N;

ð5:20Þ

with an ¼ bn W n ðxÞ. Proof. In all the cases, the proof follows by Lemma 11. Let us put in (5.14)

ðd; 1Þ :¼



ðv  1; ðtÞv 1 Þ; in the case I; ð0; 1Þ;

in the cases II; III;

where the number A, and consequently the numbers u; w; v; B; C, are such that definition (5.15) agrees with (5.18). Observe that

rðnÞ :¼



ðn þ tÞv 1 ; in the case I; 1;

in the cases II; III:

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1638

Since

b ð1Þ ¼ ðjn I þ kn E þ l E2 Þðzn nðnÞ  IÞ with nðnÞ :¼ zn P n



n þ t þ v  1; in the case I; 1;

in the cases II; III

(cf. (5.3)–(5.5)), one can see that assumption (5.16) is satisfied, too. h Example 5.1. Consider the series

2 

p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 þ 2x ¼

2

1 X

bn T n ðxÞ;

bn :¼

n¼0

ð1Þn n2  1=4

ð5:21Þ

(cf. [21, (2), p. 143]). Depending on x 2 ½1; 1, its partial sums may behave very irregularly, so the classic methods of convergence acceleration usually fail. By Theorem 12 (cf. case I with z ¼ 1; t ¼ 1=2; v ¼ 2), we obtain the operators

b ðmÞ :¼ ðn þ m  1=2ÞI þ 2xðn þ 2m  1=2ÞE þ ðn þ 3m  1=2ÞE2 ; P satisfying

b ðmÞ    P b ð1Þ ½ðn  1=2Þðbn T n ðxÞ þ bnþ1 T nþ1 ðxÞ þ    þ bnþm1 T nþm1 ðxÞÞ ¼ 0: b ð2Þ P P

ð5:22Þ

Let us define the operators

b ð1Þ ½ðn  1=2ÞI; Pð1Þ :¼ P

b ðmÞ ; PðmÞ :¼ P

m P 2:

ð5:23Þ

For the summation of the series (5.21) one can use Wynn’s e-algorithm or, among others, the K transformation (5.11) with the parameters dðkÞ n :¼ 1=ðn þ 1Þ and the remainder estimates xn :¼ bn . However, the Algorithm 1 of performing the Q transformation, determined by operators (5.23), looks similar to the H transformation with parameters b ¼ 1=2; cos a ¼ x, and the remainder estimates xn :¼ ðn  1=2Þ2 ; see (5.9). In the sequel we test these methods, as well as some techniques given by Paszkowski [8]. A. Let x ¼ 1. Since T n ð1Þ ¼ ð1Þn , the partial sums of the series (5.21) are as follows

sn ¼

n1 X

bk T k ð1Þ ¼

n1 X

1

k¼0

k  1=4

2

k¼0

¼ 2 

2 ; 2n  1

and hence s ¼ limn!1 sn ¼ 2. One can check that the Qð1Þ transformation, applied to the series (5.21), is exact, i.e.,

Qnð1Þ ¼ 2 for all n 2 N: The H and K transformations, as well as the variants u; v of Levin transformation, are exact, too. However, the series is convergent logarithmically:

accðs10 Þ ¼ 1:28;

accðs100 Þ ¼ 2:30;

accðs600 Þ ¼ 3:08;

accðs1000 Þ ¼ 3:30;

ð1Þ 30 Þ

and e-algorithm yields only accðe ¼ 2:70. B. Let x ¼ 0. The convergence of the series (5.21) is slow again, since we have

accðs10 Þ ¼ 2:85;

accðs100 Þ ¼ 4:92;

accðs400 Þ ¼ 6:13;

accðs1000 Þ ¼ 6:93:

Using Q transformation, one can compute some approximation of the sum of the series. Namely, using the operators (5.23) in ð15Þ ð15Þ Algorithm 1 yields accðQ1 Þ ¼ 23:18. Using Homeier’s transformations we obtain accðK1 Þ ¼ 17:07 and ð15Þ accðH1 Þ ¼ 20:37. Since T 2nþ1 ð0Þ ¼ 0 for n 2 N0 , the e-algorithm is unable to accelerate the convergence of the series; ð1Þ namely, we obtain only accðe30 Þ ¼ 3:28. But application of e-algorithm to the simplified form of the series (5.21), i.e.,

pffiffiffi

2 

p 2 2

¼s¼

1 X

ð1Þn b2n ¼

n¼0

1 X n¼0

ð1Þn ; 4n2  1=4 ð1Þ

results in better acceleration of the convergence; namely we obtain accðe14 Þ ¼ 12:06. C. Now, let x ¼ 0:5. Then we obtain

accðs10 Þ ¼ 2:95;

accðs100 Þ ¼ 4:97;

accðs600 Þ ¼ 6:53;

accðs800 Þ ¼ 9:86;

accðs200 Þ ¼ 8:05;

accðs400 Þ ¼ 6:18;

accðs1000 Þ ¼ 6:97: ðkÞ

Using Algorithm 1 with operators (5.23), we obtain the following values of accðQ1 Þ for k ¼ 1; 3; . . . ; 15:

3:10;

6:38;

9:63;

13:18;

17:10;

20:64;

23:76;

27:26:

ð15Þ Q1

ð5:24Þ

Notice that computing requires the knowledge of partial sums s1 ; s2 ; . . . ; s31 . By Wynn’s e-algorithm the following values ð1Þ of quantities accðek Þ; k ¼ 2; 6; . . . ; 30, were obtained:

1:62;

4:10;

6:87;

8:31;

10:07;

12:17;

14:26;

16:01:

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1639

Much worse results are obtained by using the variants t; u; v of Levin transformation, e.g., ð30Þ

accðt L1 Þ ¼ 4:05;

ð30Þ

accðu L1 Þ ¼ 3:94;

ð30Þ

accðv L1 Þ ¼ 2:14:

We also tried the method Wðr; xj Þ recently proposed by Paszkowski in [8]. We set the parameters r ¼ 0; xj ¼ bjþ1 , and obðkÞ tained the following accuracy of quantities s1 ; k ¼ 2; 6; . . . ; 30:

4:37;

9:00;

14:15;

17:84;

21:44;

24:90;

28:27;

31:58:

ð30Þ accðs1 Þ

ð5:25Þ ð15Þ accðQ1 Þ

D. Let x ¼ 1. Then using first 31 partial sums, we obtained ¼ 32:77 and ¼ 33:89. Thus our method is better than Paszkowski’s method. If we use first 50 (resp. 100) partial sums, then our method becomes better for x P 0:75 (resp. x P 0:45). We made numerical computations for x ¼ 0:95; 0:9; . . . ; 1:0 and compared the following methods of summation of the series (5.21), using first 50 partial sums (after colon, we give the notation of quantities tested):  Usual summation: s50 . ð1Þ  Wynn’s e-algorithm: e48 . ð24Þ  Homeier’s Hða; b; fsn g; fxn gÞ transformation with cos a ¼ x; b ¼ 1=2 and xn ¼ ðn  1=2Þ2 : H1 . ð24Þ ðkÞ ðkÞ  Homeier’s Kðfsn g; fxn g; fdn g; f1; 2x; 1gÞ transformation with dn ¼ 1=ðn þ 1Þ and xn ¼ bn : K1 . ð24Þ  Q transformation, determined by operators (5.23): Q1 . ð50Þ  Paszkowski’s method Wðr; xj Þ with r ¼ 0; xj ¼ bjþ1 : s1 . On Fig. 1, we present the accuracies obtained. We observed that e-algorithm fails for x ¼ 0, exceptionally. Computing of ð24Þ ð1Þ Q1 was performed by Algorithm 1 and lasted about 3:5 times longer than in the case of computing e24 . Let us notice that computations using Paszkowski’s method lasted about 30 times longer than e-algorithm. Remark 13. In Example 5.1 we considered the Chebyshev series with coefficients bn of the form

bn ¼

1 ; zn ðn þ tÞv

ð5:26Þ

where z ¼ 1; t ¼ 1=2; v ¼ 2. If we choose bigger value of parameter t, then Q transformation (determined by the operators corresponding to the case I of Theorem 12) can yield much better results than the method Wð0; bjþ1 Þ of Paszkowski [8]. For instance, using first 50 partial sums of the series

Fig. 1. Accuracy of the summation of the series (5.21) for x ¼ 0:95; 0:9; . . . ; 1.

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1640 1 X n¼0

ð1Þn T n ðxÞ ðthe choice z ¼ 1; t ¼ 80; v ¼ 2Þ; ðn þ 80Þðn þ 81Þ

ð5:27Þ

our transformation is more efficient for x J  0:55; see Fig. 2. The choice t ¼ 130 makes our method better for all x ¼ 0:95; 0:9; . . . ; 1. Results presented on Fig. 2 correspond to the following methods:      

Usual summation: s50 . ð1Þ Wynn’s e-algorithm: e48 . ð24Þ Homeier’s Hða; b; fsn g; fxn gÞ transformation with cos a ¼ x; b ¼ t and xn ¼ ðn þ tÞ2 : H1 . ð24Þ ðkÞ ðkÞ Homeier’s Kðfsn g; fxn g; fdn g; f1; 2x; 1gÞ transformation with dn ¼ 1=ðn þ 1Þ, and xn ¼ bn : K1 . ð24Þ Q transformation, determined by operators given by Theorem 12: Q1 . ð50Þ Paszkowski’s method Wðr; xj Þ with r ¼ 0, xj ¼ bjþ1 : s1 . Analysing Fig. 2, one can also observe that Hð24Þ and Qð24Þ transformations behave very similarly.

Example 5.2. Consider the following trigonometric series: 1 X n¼0

1 f zn ðn þ tÞv n

with f n ¼ sin nx; cos nx

ð5:28Þ

(cf. case I of Theorem 12). Combining (5.3) with (5.8), in the case of series (5.28), one can apply the Q transformation deterhas cos x instead of x. For mined by the operators corresponding to the case I of Theorem 12, where the definition of BðmÞ n instance, let us consider the series 1 X n¼0

cos nx : nþbþ1

ð5:29Þ

Observing that (5.18) are of the form

b ðmÞ ¼ ðn þ m þ bÞI  2 cos xðn þ 2m þ bÞE þ ðn þ 3m þ bÞE2 ; P

Fig. 2. Accuracy of the summation of the series (5.27) for x ¼ 0:95; 0:9; . . . ; 1.

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1641

Table 5 Accuracy of the summation of the series (5.29) with b ¼ 10. 0:1p

x ð10Þ accðQ1 Þ ð10Þ accðH1 Þ ð21Þ accðs1 Þ ð21Þ accð~s1 Þ

0:2p

0:3p

0:4p

0:5p

0:6p

0:7p

0:8p

0:9p

5.81

10.28

13.37

15.41

18.15

19.44

22.63

23.71

26.39

p 27.43

4.91

9.44

11.88

14.29

16.50

18.65

20.93

24.44

26.65

27.48

12.00

18.18

21.74

24.88

26.96

29.25

30.02

32.43

31.36

31.43

12.13

19.16

22.18

24.81

27.76

28.64

30.65

32.35

30.92

30.84

we obtain the following form of Algorithm 1:

Nð0Þ n ¼ sn ;

Dð0Þ n ¼ 1; ðk1Þ

ðk1Þ

ðk1Þ N ðkÞ  2 cos x ðn þ 2k þ bÞNnþ1 þ ðn þ 3k þ bÞNnþ2 ; n ¼ ðn þ k þ bÞN n ðk1Þ DðkÞ  2 cos x ðn þ 2k þ n ¼ ðn þ k þ bÞDn

ðk1Þ bÞDnþ1

þ ðn þ 3k þ

ðk1Þ bÞDnþ2 ;

ð5:30Þ

ðkÞ ðkÞ QðkÞ n :¼ N n =Dn ;

which is similar to the algorithm (5.9) of performing HðkÞ n ðx; b; fsn g; fxn gÞ. We applied the following methods to the case of summation of the series (5.29): (a) Q transformation given by (5.30), (b) transformation HðkÞ n ðx; b; fsn g; fxn gÞ of Homeier [10], (c) method Wðr; xj Þ of Paszkowski [8]. In the case of Homeier’s transformation, considering classic variants of remainder estimates xn , does not accelerate the convergence, at all. We choose remainder estimates:

xn :¼

1 : nþbþ1

We made a numerical computations for b ¼ 10 and b ¼ 100. In the case of Paszkowski’s method, we put r ¼ 0, xj ¼ 1=ðj þ b þ 1Þ (cf. [8, Case (1), p. 41]) and r ¼ 1; xj  1 (cf. [8, Case (2), p. 41]). The arrays obtained we denote by sðkÞ n ðkÞ and ~sn , respectively. Putting x ¼ 0:1 p; 0:2 p; . . . ; p, we obtain the accuracies presented in the Table 5 ðb ¼ 10Þ and Table 6 ðb ¼ 100Þ. The input for all algorithms was the set of the partial sums s1 ; s2 ; . . . ; s21 of the series (5.29). One can observe that Paszkowski’s method results in the best convergence acceleration, in case b ¼ 10. In each case, the transformation (5.30) gives accuracies growing with x and permits us to obtain similar accuracy as Homeier’s does. In the case b ¼ 100, both methods yield much higher accuracy than Paszkowski’s Wðr; xj Þ method. 6. Iterated methods Knowing the explicit form of operators PðmÞ satisfying (2.2), allows us to apply the transformation

QðmÞ n ðsn Þ ¼

LðmÞ ðsn Þ ; LðmÞ ð1Þ

LðmÞ ¼ PðmÞ Pðm1Þ    Pð1Þ ;

ð6:1Þ

in a very efficient way, using the Algorithm 1. Usually, it is difficult to obtain the operators PðmÞ ; m 2 N, explicitly. In most cases, it is enough to obtain the operators Pð1Þ ; Pð2Þ ; . . . ; Pðm0 Þ , m0 being a small number. But if it is still hard task, then we propose to use one of the following two iterated methods. I. Let us consider the classic idea to iterate the Qð1Þ transformation. We define the U transformation by

Uð0Þ n :¼ sn ;

UðmÞ :¼ n

Pð1Þ ðUðm1Þ Þ n ; ð1Þ P ð1Þ

m 2 N:

ð6:2Þ

Notice that such classic methods as iterated D2 Aitken’s process, or Brezinski’s #-algorithm [1, (10.1-9)], were obtained using such idea. II. We propose the following simplification: establish Pð1Þ and put

PðmÞ :¼ Pð1Þ ;

m ¼ 2; 3 . . . :

Then the transformation (6.1) simplifies to the V transformation, defined by

VðmÞ :¼ n

½Pð1Þ m ðsn Þ : ½Pð1Þ m ð1Þ

ð6:3Þ

Notice that both iterated methods require only the operator Pð1Þ . Since Pð1Þ ¼ Lð1Þ , it may be obtained as in the Section 2.1.

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1642

Table 6 Accuracy of the summation of the series (5.29) with b ¼ 100. x

0:1p

0:2p

0:3p

0:4p

0:5p

0:6p

0:7p

0:8p

0:9p

ð10Þ accðQ1 Þ ð10Þ accðH1 Þ ð21Þ accðs1 Þ ð21Þ accð~s1 Þ

15.95

21.18

24.66

28.54

29.67

32.07

33.73

36.31

39.23

p 46.15

14.65

21.10

25.26

27.63

29.91

32.09

34.33

36.92

40.77

47.69

10.02

13.78

16.26

18.10

19.38

20.35

21.21

22.52

22.11

22.02

7.40

10.62

12.63

14.22

15.36

16.24

17.04

18.51

17.80

17.71

The following example shows that V transformation can yield not much worse results than original Q transformation (with proper operators PðmÞ ). Of course, such behavior is quite unpredictable and, in general, it is hard to say if iterated transformation can accelerate the convergence of considered series. The following example shows that the V transformation performs better in some cases, while U – in other cases. Next we give a remark considering the case of summation of some orthogonal polynomial series given by Paszkowski [8]. They do not fit to any case of Theorem 12. In such a case, numerical tests show that the Paszkowski’s method is much more efficient than the iterated methods U and V. In the Example 6.2, we present the series which is neither orthogonal nor hypergeometric. We make a special choice of operator Pð1Þ , and obtain much better results by using the second iterated method V. Example 6.1. Consider the orthogonal polynomial series (5.21) and suppose we do not know the operators (5.23). Since

ðI  2xE þ E2 ÞðT n ðxÞÞ ¼ 0 (cf. (5.3)), one can put

b ð1Þ :¼ ðI  2xE þ E2 Þ  P



 1 I ; bn

ð6:4Þ

b ð1Þ ðbn T n ðxÞÞ ¼ 0. Now, one can use the both iterated methods with Pð1Þ ¼ P b ð1Þ . Observe that one can also consider and obtain P ð1Þ the operator P of the form

b ð1Þ Pð1Þ ¼ Q  P

ð6:5Þ

with arbitrary operator Q. Let us put x ¼ 0:5 (cf. part C of Example 5.1). We put

Q¼

ð1Þn I ð2n  1Þð2n þ 1Þ

in (6.5), and obtained

Pð1Þ ¼ I þ

2n þ 3 ð2n þ 3Þð2n þ 5Þ 2 Eþ E : 2n  1 ð2n  1Þð2n þ 1Þ

ð6:6aÞ ð15Þ

ð15Þ

We computed the transformations (6.2) and (6.3), obtaining accðU1 Þ ¼ 15:03; accðV1 Þ ¼ 12:55. One can also consider

Q¼

ð1Þn1 E1 ; ð2n  3Þð2n  1Þ

Pð1Þ ¼ E1 þ

2n þ 1 ð2n þ 1Þð2n þ 3Þ Iþ E; 2n  3 ð2n  3Þð2n  1Þ

ð6:6bÞ

Q¼

ð1Þn E2 ; ð2n  5Þð2n  3Þ

Pð1Þ ¼ E2 þ

2n  1 1 ð2n  1Þð2n þ 1Þ E þ I: 2n  5 ð2n  5Þð2n  3Þ

ð6:6cÞ

or

However, the best results are obtained for the choice (6.6a); see Table 7.

Table 7 Accuracy of the summation of the series (5.21) for x ¼ 0:5, obtained by U; V transformations, determined by operators Pð1Þ of the form (6.6c) and (6.7c). Pð1Þ

n

accðUn

(6.6a) (6.6b) (6.6c)

1 15 31

15.03 13.87 11.48

12.55 9.89 9.62

(6.7a) (6.7b) (6.7c)

1 15 31

15.03 13.87 11.48

21.86 25.26 26.87

ð15Þ

Þ

ð15Þ

accðVn

Þ

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1643

We considered also the operators Q of the form

Q 2 fð1Þn I; ð1Þn1 E1 ; ð1Þn E2 g; and get the following variants of the operator Pð1Þ with polynomial coefficients:

Pð1Þ ¼ ð2n  1Þð2n þ 1ÞI þ ð2n þ 1Þð2n þ 3ÞE þ ð2n þ 3Þð2n þ 5ÞE2 ;

ð6:7aÞ

Pð1Þ ¼ ð2n  3Þð2n  1ÞE1 þ ð2n  1Þð2n þ 1ÞI þ ð2n þ 1Þð2n þ 3ÞE;

ð6:7bÞ

Pð1Þ ¼ ð2n  5Þð2n  3ÞE2 þ ð2n  3Þð2n  1ÞE1 þ ð2n  1Þð2n þ 1ÞI;

ð6:7cÞ

respectively. The obtained accuracies are shown in Table 7. One can observe that, in the case of using operators (6.6), U transformation yields better results than V. But when operators (6.7) are used, the V transformation gives essentially better results. The normalization of operators (6.6) into (6.7) does matter in the case of V transformation, and does not in the case of U transformation. Remark 14. In the case of orthogonal series, the methods given by Paszkowski [8] are more general than our transformation, and can be successfully applied in the case of many important orthogonal series of the form (5.1). We are not able to obtain explicit forms of operators PðmÞ in the case of Legendre series: 1 X

1 P n ðxÞ; nþ1

1 X

1 P n ðxÞ; ð2n  1Þð2n þ 3Þ n¼1 n¼1 1 1 X X logð1 þ 2n1 ÞPn ðxÞ; ðDÞ ¼ ð2n þ 1Þzn Pn ðxÞ ðCÞ ¼

ðAÞ ¼

ðBÞ ¼

n¼1

n¼1

(see [8, (10–13)]). However, we can apply the iterated methods, since for any orthogonal series (5.1), one can choose the following operator Pð1Þ :

  1 Pð1Þ :¼ ðjn I þ kn E þ ln E2 Þ I : bn

ð6:8Þ

We observed that better results can be obtained by using the transformation

RðmÞ :¼ n

LðmÞ ðsn Þ ; LðmÞ ð1Þ

LðmÞ :¼ PðmÞ Pðm1Þ    Pð1Þ ;

ð6:9Þ

determined by operators

  b ð1Þ 1 I ; Pð1Þ :¼ P bn

b ðmÞ ; m P 2; PðmÞ :¼ P

ð6:10Þ

b ðmÞ ; m P 1, are given by (5.15) and u; w; v; A; B; C are such that Pð1Þ fits to (6.8). Notice that this is possible where operators P in each case (5.3)–(5.6) since jn ; kn ; ln are linear polynomials in n. We introduced the notation RðmÞ since the operators LðmÞ , given in (6.9), do not satisfy (2.2), as it is required in Q transformation. One can check that in the case of Legendre polynomial series, the R transformation is determined by operators (6.10), where

b ðmÞ ¼ ðn þ mÞI  2xðn þ 2m  1=2ÞE þ ðn þ 3m  1ÞE2 P (cf. (5.4)). For instance, let x ¼ 0:4 and z ¼ 0:8. We tested the summation of the series (A)–(D) by both iterated methods, determined by operator (6.8), and by the R transformation. We also tested Homeier’s K transformation. Results are presented in the Table 8. The last row corresponds to the case Wð0; bjþ1 Þ of Paszkowski’s method [8], where bn are the coefficients of P n ðxÞ in (A)–(D). In the case of K transformation, we put the parameters dðkÞ n ¼ 1=ðn þ 1Þ; xn ¼ bn (cf. (5.11)). Both iterated methods and R transformation perform much worse than Paszkowski’s method. It may be caused by wrong choice of operators PðmÞ , in the sense that the operators (6.10) do not satisfy the assumption in Algorithm 1, in the case of the series (A)–(D). More precisely, (2.2) holds only for m ¼ 1. In the case (D), we observed that Homeier’s transformation performs better than R transformation if z is closer and closer 1. In contrast with our transformation, Paszkowski’s method can be successfully applied also for x close to 1.

Table 8 ð20Þ ð20Þ ð20Þ ð41Þ Accuracies of U1 ; V1 ; R1 and s1 , in the case of summation of the series (A)–(D) with x ¼ 0:4 and z ¼ 0:8. (A) ð20Þ accðU1 Þ ð20Þ accðV1 Þ ð20Þ accðR1 Þ ð20Þ accðK1 Þ ð41Þ accðs1 Þ

(B)

(C)

(D)

8.68

9.96

7.39

3.30

20.12

11.51

18.86

6.78

22.18

20.95

18.79

19.52

21.13

21.74

19.89

24.42

47.82

34.38

32.54

33.06

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645

1644

Example 6.2. Consider the series 1 X

an ;

an :¼

n¼0

nn pffiffiffiffiffiffiffiffiffiffiffiffi ðn þ 2Þ n þ 1

ðn ¼ 1Þ:

ð6:11Þ

A. Let n ¼ 1. Then (6.11) is the alternating series, and its convergence can be accelerated by classic method as Wynn’s ealgorithm, or Aitken’s iterated D2 process, very well. For instance, using first 21 partial sums, we obtain ð1Þ

ð1Þ

accðe20 Þ ¼ 16:0;

accðA10 Þ ¼ 20:1:

Consider the operator

  1 Pð1Þ ¼ Lð1Þ :¼ D I an

ð6:12Þ

(cf. (2.4)). We are not able to find explicit forms of operators PðmÞ given by the method described in Section 2.1. But even if we do, results obtained by Q transformation will be the same as the ones given by e-algorithm: ð20Þ

ð1Þ

accðQ1 Þ ¼ accðe40 Þ ¼ 31:30; see Section 2.3. Using the iterated methods determined by operator (6.12) yields ð20Þ

accðU1 Þ ¼ 13:1;

ð20Þ

accðV1 Þ ¼ 5:3:

The efficiency of the U transformation is higher probably because it is based on an idea similar to the one used in the Aitken’s process. If we use D2 instead of D in (6.12), then we obtain the second-order operator Pð1Þ which results in a little bit better ð10Þ ð10Þ efficiency of both iterated methods: accðU1 Þ ¼ 13:9, accðV1 Þ ¼ 8:3. We observed that much better results one can obtain by using V transformation, determined by the following second-order operator Pð1Þ :

pffiffiffiffiffiffiffiffiffiffiffiffi  Pð1Þ :¼ Dððn þ 2Þðn þ 3ÞDÞ n þ 1I ;

ð6:13Þ

where D :¼ E  nI. One can check that results obtained by the first iterated method remain the same. Namely, we obtain ð10Þ

accðU1 Þ ¼ 13:9;

ð10Þ

accðV1 Þ ¼ 22:6:

B. Now, let n ¼ 1. Then the series is convergent logarithmically, and thus very slowly. Wynn’s and Aitken’s methods do not accelerate its convergence, at all: ð1Þ

accðe20 Þ ¼ 1:1;

ð1Þ

accðA10 Þ ¼ 1:7:

Using the iterated methods determined by operator Pð1Þ of the form (6.12), or with D2 instead of D, does not result in better ð10Þ accuracies. However, the second iterated method determined by operator (6.13) yields accðV1 Þ ¼ 8:3. Let us notice that usual summation of the first million terms of the series (6.11) yields less than three exact significant decimal digits of its limit. Acknowledgements The authors wish to express their thanks to Prof. E.J. Weniger for his interest in this research, and valuable comments and suggestions. They also wish to thank their supervisor Prof. S. Lewanowicz for his pertinent comments and helpful discussion. Finally, they are grateful to the anonymous referees for their suggestions and remarks which helped to improve the paper. References [1] E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Rep. 10 (1989) 189–371. [2] J. Cˇízˇek, J. Zamastil, L. Skála, New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field, J. Math. Phys. 44 (3) (2003) 962–968. [3] C. Brezinski, M.R. Zaglia (Eds.), Extrapolation methods: theory and practice, Studies in Computational Mathematics, vol. 2, North-Holland, 1991. [4] H.H. Homeier, Scalar Levin-type sequence transformations, J. Comput. Appl. Math. 122 (1–2) (2000) 81–147. [5] J. Wimp, Sequence transformations and their applications, Mathematics in Science and Engineering, vol. 154, Academic Press, New York – London, 1981. [6] C. Brezinski, A.C. Matos, A derivation of extrapolation algorithms based on error estimates, J. Comput. Appl. Math. 66 (1–2) (1996) 5–26. [7] A.C. Matos, Linear difference operators and acceleration methods, IMA J. Numer. Anal. 20 (3) (2000) 359–388. [8] S. Paszkowski, Convergence acceleration of orthogonal series, Numer. Algorithms 47 (1) (2008) 35–62. ˇ ízˇek, Zamastil, Skála. I. Algebraic theory, J. Math. [9] E.J. Weniger, Mathematical properties of a new Levin-type sequence transformation introduced by C Phys. 45 (3) (2004) 1209–1246. [10] H.H.H. Homeier, A Levin-type algorithm for accelerating the convergence of Fourier series, Numer. Algorithms 3 (1992) 245–254. [11] H.H.H. Homeier, Some applications of nonlinear convergence accelerators, Int. J. Quantum Chem. 45 (6) (1993) 545–562. [12] H.H.H. Homeier, Nonlinear convergence acceleration for orthogonal series, in: R. Gruber, M. Tomassini (Eds.), Proceedings of the 6th Joint EPS–APS International Conference on Physics Computing, Physics Computing’94, European Physical Society, Boite Postale 69, CH-1213 Petit-Lancy, Geneva, Switzerland, 1994, pp. 47–50. [13] C. Brezinski, M. Redivo-Zaglia, On the kernel of sequence transformations, Appl. Numer. Math. 16 (1994) 239–244.

P. Woz´ny, R. Nowak / Applied Mathematics and Computation 215 (2009) 1622–1645 [14] [15] [16] [17] [18] [19] [20] [21]

1645

D. Shanks, Nonlinear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955) 1–42. G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999. G.M. Phillips, Interpolation and approximation by polynomials, CMS Books in Mathematics, vol. 14, Springer, New York, 2003. S. Lewanowicz, S. Paszkowski, An analytic method for convergence acceleration of certain hypergeometric series, Math. Comput. 64 (210) (1995) 691– 713. G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, second ed., vol. 96, Cambridge University Press, 2004. R. Koekoek, R.F. Swarttouw, The Askey-scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue, Technical Report, Delft University of Technology, 1998. C. Oleksy, A convergence acceleration method of fourier series, Comput. Phys. Commun. 96 (1) (1996) 17–26. S. Paszkowski, Numerical applications of Chebyshev polynomials and series (Zastosowania numeryczne wielomianow i szeregow Czebyszewa.), PWN, Warszawa, 1975.