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Generalizations of Aitken's Process for Accelerating the Convergence Of “main” — 2007/6/26 — 18:15 — page 171 — #1 Volume 26, N. 2, pp. 171–189, 2007 Copyright © 2007 SBMAC ISSN 0101-8205 www.scielo.br/cam Generalizations of Aitken’s process for accelerating the convergence of sequences CLAUDE BREZINSKI1 and MICHELA REDIVO ZAGLIA2 1Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de Mathématiques Pures et Appliquées, Université des Sciences et Technologies de Lille, 59655 – Villeneuve d’Ascq cedex, France 2Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata Via Trieste 63, 35121 – Padova, Italy E-mails: [email protected] / [email protected] Abstract. When a sequence or an iterative process is slowly converging, a convergence accel- eration process has to be used. It consists in transforming the slowly converging sequence into a new one which, under some assumptions, converges faster to the same limit. In this paper, new scalar sequence transformations having a kernel (the set of sequences transformed into a constant sequence) generalizing the kernel of the Aitken’s 12 process are constructed. Then, these trans- formations are extended to vector sequences. They also lead to new fixed point methods which are studied. Mathematical subject classification: Primary: 65B05, 65B99; Secondary: 65H10. Key words: convergence acceleration, Aitken process, extrapolation, fixed point methods. 1 Introduction When a sequence (Sn) of real or complex numbers is slowly converging, it can be transformed into a new sequence (Tn) by a sequence transformation. Aitken’s 12 process and Richardson extrapolation (which gives rise to Romberg’s method for accelerating the convergence of the trapezoidal rule) are the most well known sequence transformations. It has been proved that a sequence transformation able to accelerate the convergence of all sequences cannot exist [8] (see also [7]). #682/06. Received: 09/X/06. Accepted: 05/III/07. “main” — 2007/6/26 — 18:15 — page 172 — #2 172 GENERALIZATIONS OF AITKEN’S PROCESS In fact, each transformation is only able to accelerate the convergence of special classes of sequences. This is the reason why several sequence transformations have to be constructed and studied. For constructing a new sequence transformation, an important object is its kernel (we will explain why below). It is the set of sequences (Sn), characterized by a particular expression or satisfying a particular relation between its terms, both involving an unknown parameter S (the limit of the sequence if it converges or its antilimit if it does not converge), that are transformed into the constant sequence (T S). For example, the kernel of the Aitken’s 12 process (see its n = definition below) is the set of sequences of the form S S aλn, n 0, 1,..., n = + = where a 0 and λ 1 or, equivalently, satisfying the relation a (S S) 6= 6= 0 n − + a1(Sn 1 S), for n 0, 1,..., with a0a1 0 and a0 a1 0. If λ < 1, + − = 6= + 6= | | then (Sn) converges to its limit S. Otherwise, S is called the antilimit of the sequence (Sn). The construction of a sequence transformation having a specific kernel consists in giving the exact expression of the parameter S for any sequence belonging to this kernel. This expression makes use of several consecutive terms of the sequence starting from Sn, and it is valid for all n. Thus, by construction, for all n, T S. When applied to a sequence not belonging to its kernel, n = the transformation produces a sequence (Tn) which, under some assumptions, converges to S faster than (Sn), that is Tn S lim − 0. n Sn S = →∞ − In that case, it is said that the transformation accelerates the convergence of (Sn). In fact, a sequence transformation is based on interpolation followed by ex- trapolation. For example, the parameters a, λ and S appearing in the kernel of n i the Aitken’s process are computed by solving the system Sn i S aλ + + = + for i 0, 1, 2. If the sequence (Sn) to be transformed does not belong to the = kernel, then the value of S (and also those of a and λ) obtained from this system depends of n and it is denoted by Tn. In order to fully understand the procedure followed for obtaining the transformations given in this paper, let us explain in details how the Aitken’s 12 process is derived from its kernel. The sequences Comp. Appl. Math., Vol. 26, N. 2, 2007 “main” — 2007/6/26 — 18:15 — page 173 — #3 CLAUDE BREZINSKI and MICHELA REDIVO ZAGLIA 173 of its kernel satisfy (S S)/λn a. Applying the usual forward difference n − = operator 1 (it is an annihilation operator as will be explained below) to both n sides leads to 1((Sn S)/λ ) 1a 0. Thus Sn 1 S λ(Sn S) and it − = = + − = − follows S (Sn 1 λSn)/(1 λ). The problem is now to compute λ. Apply- = + − − ing the operator 1 to S S aλn gives 1S aλn(λ 1), and we obtain n = + n = − λ 1Sn 1/1Sn. Thus, replacing λ by this expression in the formula for S, = + leads to the transformation 2 Sn Sn 2 Sn 1 Tn + − + , n 0, 1,... (1) = Sn 2 2Sn 1 Sn = + − + + which, by construction, has a kernel including all sequences of the form Sn n = S aλ or, equivalently, such that Sn 1 S λ(Sn S) for all n. We see that + + 2 − = − the denominator in this formula is 1 Sn, thus the name of the transformation. An important point to notice for numerical applications is that Formula (1) is numerically unstable. It has to be put under one of the equivalent forms 2 (1Sn) Tn Sn = − Sn 2 2Sn 1 Sn + − + + 1Sn1Sn 1 Sn 1 + = + − Sn 2 2Sn 1 Sn + − + + 2 (1Sn 1) Sn 2 + = + − Sn 2 2Sn 1 Sn + − + + which are more stable. Indeed, when the terms Sn, Sn 1 and Sn 2 are close to S, a + + cancellation appears in Formula (1). Its numerator and its denominator are close to zero, thus producing a first order cancellation. A cancellation also appears in the three preceding formulae, but it is a cancellation on a correcting term, that is, in some sense, a second order cancellation (see [5, pp. 400–403] for an extensive the discussion). Similarly, in the sequel, when a transformation is written as T N /D , it is n = n n usually unstable. If the computation of Tn makes use of Sn,..., Sn k, then Tn + can also be put under one of the forms Tn Sn i (Sn i Dn Nn)/Dn, for any = + − + − i 0,..., k, which, after simplification in the numerator Sn i Dn Nn, is more = + − stable. Comp. Appl. Math., Vol. 26, N. 2, 2007 “main” — 2007/6/26 — 18:15 — page 174 — #4 174 GENERALIZATIONS OF AITKEN’S PROCESS By construction, we saw that, for all n, T S, if (S ) belongs to the kernel n = n of the transformation. To prove that this condition is also necessary is more difficult. One has to start from the condition forall n, Tn S, and then to show = that it implies that the relation defining the kernel is satisfied. This is why, inthis paper, we will only say that the kernels of the transformations studied include all sequences having the corresponding form since additional sequences can also belong to the kernel. Let us mention that, for the Aitken’s process, the condition is necessary and sufficient. Of course, one can ask why the notion of kernel is an important one. Although this result was never proved, it is hoped (and it was experimentally verified) that if a sequence is not “too far away” from the kernel of a certain transformation, then this transformation will accelerate its convergence. For example, the kernel of the Aitken’s process can also be described as the set of sequences such that for all n, (Sn 1 S)/(Sn S) λ 1. It is easy to prove that this process + − − = 6= accelerates the convergence of all sequences for which there exists λ 1 such 6= that limn (Sn 1 S)/(Sn S) λ. On sequence transformations, their →∞ + − − = kernels, and extrapolation methods see, for example, [5, 14, 16]. The Aitken’s 12 process is one of the most popular and effective convergence acceleration method. In this paper we will construct scalar sequence transforma- tions whose kernels generalize the kernel of this transformation which consists, as we saw above, of sequences such that S S aλn for n 0, 1,.... Defining n = + = and studying generalizations of Aitken’s process leads to interesting applica- tions as those described in [11] and [12]. In this paper, we will consider two new generalizations of the kernel of Aitken’s process, namely consisting of se- n n quences of the form Sn S (a bxn)λ (Section 2) and Sn S λ /(a bxn) = + + = + + (Section 3), where (xn) is a given known sequence. Compared to the sequences in the kernel of Aitken’s process, the additional term bxn can completely change the behaviour since non monotonic sequences or non strictly alternating ones are now included into the kernel. According to the value of λ and to the choice of (xn), we can have sequences whose error (in absolute value) first increases, and then tends to zero, or divergent sequences whose error (in absolute value) begins to approach zero, and then goes to infinity, thus imitating asymptotic series. This extra term can also be considered as a subdominant contribution Comp. Appl. Math., Vol. 26, N. 2, 2007 “main” — 2007/6/26 — 18:15 — page 175 — #5 CLAUDE BREZINSKI and MICHELA REDIVO ZAGLIA 175 (that is a kind of second order term) to the sequence as pointed out by Weniger [15].
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