NONLINEAR SEQUENCE TRANSFORMATIONS FOR THE ACCELERATION OF CONVERGENCE AND THE SUMMATION OF DIVERGENT SERIES Ernst Joachim Weniger Institut f¨ur Physikalische und Theoretische Chemie Universit¨at Regensburg D-8400 Regensburg Federal Republic of Germany Abstract Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly arXiv:math/0306302v1 [math.NA] 19 Jun 2003 nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed. Some of the sequence transformations of this report as for instance Wynn’s ǫ algorithm or Levin’s sequence transformation are well established in the literature on convergence acceleration, but the majority of them is new. Efficient algorithms for the evaluation of these transformations are derived. The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed. Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and divergent series, which are hopefully realistic models for a large part of the slowly convergent or divergent series that can occur in scientific problems and in applied mathematics. Published as Computer Physics Reports Vol. 10, 189 - 371 (1989) i Contents 1. Introduction ................................ 1 1.1. Infinite series and their evaluation . ..... 1 1.2. A short history of sequence transformations . ........ 2 1.3.Organizationofthisreport . 4 2. Terminology ................................ 7 2.1. Special mathematical symbols and special functions . ........... 7 2.2.Ordersymbols .............................. 7 2.3. Asymptotic sequences and asymptotic expansions . ......... 8 2.4.Finitedifferences . 9 2.5.Specialsequences . 9 2.6.Typesofconvergence . 10 2.7.Sequencetransformations . 12 3. On the derivation of sequence transformations ............... 14 3.1. General properties of nonlinear sequence transformations .......... 14 3.2. An example: Convergence acceleration of alternating series ......... 15 3.3. The general extrapolation algorithm by Brezinski and H˚avie . .. 18 3.4. Iterated sequence transformations . 19 4. The epsilon algorithm and related topics .................. 20 4.1.TheShankstransformation . 20 4.2.Wynn’sepsilonalgorithm . 21 4.3. Programming the epsilon algorithm . 23 2 5. The iteration of Aitken’s ∆ process .................... 27 5.1. Aitken’s ∆2 transformation and its iteration . 27 5.2. Programming Aitken’s iterated ∆2 process ................ 29 6. Wynn’s rho algorithm and related topics .................. 32 6.1. Polynomial and rational extrapolation . ..... 32 6.2.Wynn’srhoalgorithm . 34 6.3. The iteration of Wynn’s rho algorithm . 36 7. The Levin transformation ......................... 39 7.1. The derivation of Levin’s sequence transformation . .......... 39 7.2. Recursive computation of the Levin transformation . ......... 41 7.3. Remainder estimates for the Levin transformation . ......... 42 7.4. Sidi’s generalization of Levin’s sequence transformation ........... 45 7.5. Programming the Levin transformation . 49 8. Sequence transformations based upon factorial series ............ 53 8.1.Factorialseries . 53 8.2. A factorial series analogue of Levin’s transformation ............ 55 8.3.Recurrenceformulas . 58 8.4. Explicit remainder estimates . 59 ii 9. Other generalizations of Levin’s sequence transformation .......... 63 9.1. Asymptotic approximations based upon Pochhammer symbols ........ 63 9.2. New sequence transformations based upon Pochhammer symbols . 64 9.3.Recurrenceformulas . 65 9.4. Explicit remainder estimates . 67 9.5. Drummond’ssequencetransformation . 70 10. Brezinski’s theta algorithm and related topics ............... 72 10.1. The derivation of Brezinski’s theta algorithm . ......... 72 10.2. Programming Brezinski’s theta algorithm . ...... 74 (n) 10.3. The iteration of ϑ2 ......................... 79 10.4. Programming the iterated theta algorithm . ...... 80 11. On the derivation of theta-type algorithms ................. 83 11.1. New sequence transformations based upon Aitken’s iterated ∆2 process . 83 11.2. New nonlinear sequence transformations obtained from linear transformations . .. .. 87 12. A theoretical analysis of sequence transformations ............. 90 12.1. Germain-Bonne’s formal theory of convergence acceleration . 90 12.2. Applications of Germain-Bonne’s theory . ...... 95 12.3. A modification of Germain-Bonne’s theory for sequence transformations involvingremainderestimates. 99 12.4. A critical assessment of Germain-Bonne’s theory . ......... 107 13. Summation and convergence acceleration of Stieltjes series ......... 109 13.1. Stieltjes series and Stieltjes functions . ........ 109 13.2. Theoretical error estimates . 112 13.3.SummationoftheEulerseries . 119 13.4. A Stieltjes series with a finite radius of convergence . ........... 125 14. The acceleration of logarithmic convergence ................ 132 14.1. Properties of logarithmically convergent sequences and series . 132 14.2. Exactness results and error estimates . 134 14.3.Somenumericaltestseries . 137 14.4.Numericalexamples . 142 15. Synopsis ................................. 152 15.1.Generalconsiderations . 152 15.2. Wynn’s epsilon algorithm and related transformations ........... 153 15.3. Wynn’s rho algorithm and related transformations . ......... 154 15.4. Levin’s sequence transformation and related transformations . 155 15.5. Brezinski’s theta algorithm and related transformations .......... 157 References .................................. 159 1 1. Introduction 1.1. Infinite series and their evaluation Infinite series are ubiquitous in the mathematical analysis of scientific problems. They naturally appear in the evaluation of integrals, in the solutions of differential and integral equations, or as Fourier series. They are also used for both the definition and the evaluation of many of the special functions of mathematical physics. The conventional approach for the evaluation of an infinite series consists in computing a finite sequence of partial sums n sn = ak (1.1-1) Xk=0 by adding up one term after the other. Then, the magnitude of the truncation error is estimated. If the sequence of partial sums s0,...,sn has not converged yet to the desired accuracy, additional terms must be added until convergence has finally been achieved. With this approach it is at least in principle possible to determine the value of an infinite series as accurately as one likes provided that one is able to compute a sufficiently large number of terms accurately enough to overcome eventual numerical instabilities. However, in many scientific problems one will only be able to compute a relatively small number of terms. In addition, particularly the series terms with higher summation indices are often affected by serious inaccuracies which may lead to a catastrophic accumulation of round-off errors. Consequently, if an infinite series is to be evaluated by adding one term after the other, an infinite series will be of practical use only if it converges after a sufficiently small number of terms. Unfortunately, many counterexamples are known in which alternative methods for the evaluation of infinite series must be used since in these cases the conventional approach of evaluating an infinite series does not suffice. For instance, when Haywood and Morgan [1] performed a discrete basis-set calculation of the Bethe logarithm of the 1s state of the hydrogen atom, they found that even 120 basis functions gave no more than 2 – 3 decimal digits and they estimated that approximately 1010 basis functions would be needed to obtain an accuracy of more than 10 decimal digits. Haywood and Morgan also showed that with the help of a suitable convergence acceleration method an accuracy of more than 13 decimal digits can be extracted from their data. A good mathematical model for the convergence problems which Haywood and Morgan [1] encountered in their calculation of the Bethe logarithm is the following series expansion for the Riemann zeta function: ∞ ζ(z) = (n + 1)−z . (1.1-2) nX=0 It is well known that this infinite series converges if Re(z) > 1 holds. However, if Re(z) is only slightly larger than one, the rate of convergence becomes extremely slow. For instance, Bender and Orszag remark in their book (see p. 379 of ref. [2]) that about 1020 terms of the above series expansion would be needed to compute ζ(1.1) accurate to one percent. Bender and Orszag also show that only 10 terms of the series in connection with a specially designed acceleration method are needed to compute ζ(1.1) to 26 decimal digits (see table 8.7 on p. 380 of ref. [2]). Even more striking examples for the inadequacy of the conventional approach towards the evaluation of infinite series are some Rayleigh-Schr¨odinger perturbation expansions of elementary quantum mechanical systems. For instance, if the following normalization for the Hamiltonian of the quartic anharmonic oscillator is used, Hˆ =p ˆ2 +x ˆ2 + βxˆ4 , (1.1-3) 2 then it follows from the results obtained by Bender and Wu (see eq. (1.8) of ref. [3]) that the coefficients cn of the power series in the coupling constant β for the ground state energy eigenvalue E0(β) of the quartic anharmonic oscillator, ∞ n E0(β) = cn β , (1.1-4) Xn=0 possess the following asymptotic behaviour: n+1 n cn ∼ (−1) (3/2) Γ(n + 1/2) , n →∞ . (1.1-5) The radius of convergence of the above Rayleigh-Schr¨odinger