Nonlinear Sequence Transformations Research Articles

Self-similar sequence transformation for critical exponents

Self-similar sequence transformation is an original type of nonlinear sequence transformations allowing for defining effective limits of asymptotic sequences. The method of self-similar factor transformations is shown to be regular. This method is applied for calculating the critical exponents of the O(N)-symmetric φ4 theory in three dimensions by summing asymptotic ε expansions. It is shown that this method is straightforward and essentially simpler than other summation techniques involving complicated numerical calculations, while enjoying comparable accuracy.

A family of integral representations for the voltammetric responses of reversible electrochemical reactions, arising from Abel, Lindelöf and Euler-Ramanujan summations of the alternating series solution

Given the closed-form expression for the mass transfer function M(p) pertaining to one-dimensional mass transport conditions in the electrolyte, the linear-sweep-voltammetry response of reversible electrochemical reactions involving soluble species is derived as an infinite series that is conditionally convergent, depending on the electrode potential imposed. The alternating feature of this series greatly simplifies its summation process, even in its divergence domain, using appropriate nonlinear sequence transformations (NLST) together with the high-precision computing mode of computer algebra systems (CAS). Alternatively, the use of Abel, Lindelöf and Euler-Ramanujan summation formulae enables the Faradaic current to be modelled as three integral representations, which can be evaluated numerically using self-adaptive numerical integration procedures implemented in CAS software. The highly accurate data, either obtained from NLST summation of the infinite series solution, or evaluated numerically from the associated integral representations, could be used as benchmark data for checking the computation accuracy of other numerical methods.

Nonlinear Sequence Transformation-Based Continuous-Time Wavelet Filter Approximation

Time-domain synthesis is used to design continuous-time filters when their time-domain response for a given excitation is known. A case in point is the design of a continuous-time wavelet filter which requires a chosen wavelet as the filter’s impulse response. A key step in the design of a continuous-time wavelet filter is the approximation of its transfer function to a proper rational function. This approximation problem is addressed using methods that can be broadly categorized as closed-form methods and numerical optimization methods. While optimization methods are very effective, closed-form solutions are attractive because they are easy to design. In this work, we propose variants of nonlinear sequence transformation (a closed-form method) that obtain proper rational approximations of wavelet functions which are stable and also perform better in terms of mean square error when compared to those obtained by other closed-form methods. It is shown that the model-order reduction of the proposed variants leads to either similar or better performance compared to \(L_2\) optimization method. These variants are also shown to act as good starting points for optimization methods that use local search routines.

Convergence acceleration of polynomial chaos solutions via sequence transformation

We investigate convergence acceleration of the solution of stochastic differential equations characterized by their polynomial chaos expansions. Specifically, nonlinear sequence transformations are adapted to these expansions, viewed as a one-parameter family of functions with the parameter being the polynomial degree of the expansion. These transformations can be generally viewed as nonlinear generalizations of Richardson Extrapolation and permit the estimation of coefficients in higher order expansions having knowledge of the coefficients in lower order ones. Stochastic Galerkin closure that typically characterizes the solution of such equations yields polynomial chaos representations that have the requisite analytical properties to ensure suitable convergence of these nonlinear sequence transformations. We investigate specifically Shanks and Levin transformations, and explore their properties in the context of a stochastic initial value problem and a stochastic elliptic problem.

A closed-form solution to American options under general diffusion processes

This paper investigates American option pricing under general diffusion processes. Specifically, the underlying asset price is assumed to follow a diffusion process in which both the dividend yield and volatility are functions of time and the underlying asset price. Using the generalized homotopy analysis method, the determination of the early exercise boundary is separated from the valuation procedure of American options. Then, an exact and explicit solution for American options on a dividend-paying stock is derived as a Maclaurin series. In addition, the corresponding optimal early exercise boundary and the Greeks are obtained in closed-form solutions. A nonlinear sequence transformation, the Padé technique, is used to effectively accelerate the convergence of the partial sums of the infinite series. As the homotopy constructed in this paper is based on a generalized deformation with a shape parameter and kernel function, the error of the homotopic approximation could be reduced further for a fixed order. Numerical examples demonstrate the validity, effectiveness, and flexibility of the proposed approach.

Is the fast Hankel transform faster than quadrature?

The fast Hankel transform (FHT) implemented with digital filters has been the algorithm of choice in EM geophysics for a few decades. However, other disciplines have predominantly relied on methods that break up the Hankel transform integral into a sum of partial integrals that are each evaluated with quadrature. The convergence of the partial sums is then accelerated through a nonlinear sequence transformation. While such a method was proposed for geophysics nearly three decades ago, it was demonstrated to be much slower than the FHT. This work revisits this problem by presenting a new algorithm named quadrature-with-extrapolation (QWE). The QWE method recasts the quadrature sum into a form conceptually similar to the FHT approach by using a fixed-point quadrature rule. The sum of partial integrals is efficiently accelerated using the Shanks transformation computed with Wynn’s [Formula: see text] algorithm. A Matlab implementation of the QWE algorithm is compared with the FHT method for accuracy and speed on a suite of relevant modeling problems including frequency-domain controlled-source EM, time-domain EM, and a large-loop magnetic source problem. Surprisingly, the QWE method is faster than the FHT for all three problems. However, when the integral needs to be evaluated at many offsets and the lagged convolution variant of the FHT is applicable, the FHT is significantly faster than the QWE method. For divergent integrals such as those encountered in the large loop problem, the QWE method can provide an accurate answer when the FHT method fails.

Numerical calculation of Bessel, Hankel and Airy functions

The numerical evaluation of an individual Bessel or Hankel function of large order and large argument is a notoriously problematic issue in physics. Recurrence relations are inefficient when an individual function of high order and argument is to be evaluated. The coefficients in the well-known uniform asymptotic expansions have a complex mathematical structure which involves Airy functions. For Bessel and Hankel functions, we present an adapted algorithm which relies on a combination of three methods: (i) numerical evaluation of Debye polynomials, (ii) calculation of Airy functions with special emphasis on their Stokes lines, and (iii) resummation of the entire uniform asymptotic expansion of the Bessel and Hankel functions by nonlinear sequence transformations. In general, for an evaluation of a special function, we advocate the use of nonlinear sequence transformations in order to bridge the gap between the asymptotic expansion for large argument and the Taylor expansion for small argument (“principle of asymptotic overlap”). This general principle needs to be strongly adapted to the current case, taking into account the complex phase of the argument. Combining the indicated techniques, we observe that it possible to extend the range of applicability of existing algorithms. Numerical examples and reference values are given.

An introduction to the topics presented at the conference “Approximation and extrapolation of convergent and divergent sequences and series” CIRM Luminy: September 28, 2009–October 2, 2009

The conference “Approximation and extrapolation of convergent and divergent sequences and series” discusses some very old problems of mathematical analysis and modern approaches for their solution based on so-called nonlinear sequence transformations. In spite of all the advances in computer hard and software, slowly convergent or divergent sequences and series are still annoying obstacles not only in mathematics, but in particular also in other mathematically oriented disciplines. This articles tries to give a highly condensed survey of the topics treated at this conference.

Method of summation of some slowly convergent series

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 introduced by Čížek et al. [J. Čížek, J. Zamastil, L. Skála, New summation technique for rapidly 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.

On the analyticity of Laguerre series

The transformation of a Laguerre series f(z) = ∑∞n=0λ(α)nL(α)n(z) to a power series f(z) = ∑∞n=0γnzn is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power series coefficients that are infinite in magnitude. Simple sufficient conditions based on the decay rates and sign patterns of the Laguerre series coefficients λ(α)n as n → ∞ can be formulated which guarantee that the resulting power series represents an analytic function. The transformation produces a mathematically meaningful result if the coefficients λ(α)n either decay exponentially or factorially as n → ∞. The situation is much more complicated—but also much more interesting—if the λ(α)n decay only algebraically as n → ∞. If the λ(α)n ultimately have the same sign, the series expansions for the power series coefficients diverge, and the corresponding function is not analytic at the origin. If the λ(α)n ultimately have strictly alternating signs, the series expansions for the power series coefficients still diverge, but are summable to something finite, and the resulting power series represents an analytic function. If algebraically decaying and ultimately alternating Laguerre series coefficients λ(α)n possess sufficiently simple explicit analytical expressions, the summation of the divergent series for the power series coefficients can often be accomplished with the help of analytic continuation formulae for hypergeometric series p+1Fp, but if the λ(α)n have a complicated structure or if only their numerical values are available, numerical summation techniques have to be employed. It is shown that certain nonlinear sequence transformations—in particular the so-called delta transformation (Weniger 1989 Comput. Phys. Rep. 10 189–371 (equation (8.4-4)))—are able to sum the divergent series occurring in this context effectively. As a physical application of the results of this paper, the legitimacy of the rearrangement of certain one-range addition theorems for Slater-type functions (Guseinov 1980 Phy. Rev. A 22 369–71, Guseinov 2001 Int. J. Quantum Chem. 81 126–29, Guseinov 2002 Int. J. Quantum Chem. 90 114–8) is investigated.

Evaluation of diffraction catastrophes by using Weniger transformation

Weniger transformation is a powerful nonlinear sequence transformation that, when applied to the sequence of the partial sums of a divergent or a slowly convergent series, can convert it to a fast-converging sequence. Weniger transformation is not yet well known in optics. Diffraction catastrophes are fundamental tools for evaluating an optical field in proximity to caustics and singularities. The action of the Weniger transformation on the power series representation of diffraction catastrophes is numerically studied for two particular cases, corresponding to the Airy and the Pearcey functions. The obtained results clearly show that Weniger transformation could become a computational tool of great importance for summing several types of series expansions in optics.

Scalar Levin-type sequence transformations

Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. The basic idea is to construct from a given sequence {{ s n }} a new sequence {{s n′}}= T({{s n}}) where each s n ′ depends on a finite number of elements s n 1 ,…, s n m . Often, the s n are the partial sums of an infinite series. The aim is to find transformations such that {{ s n ′}} converges faster than (or sums) {{ s n }}. Transformations T({{s n}},{{ω n}}) that depend not only on the sequence elements or partial sums s n but also on an auxiliary sequence of the so-called remainder estimates ω n are of Levin-type if they are linear in the s n , and nonlinear in the ω n . Such remainder estimates provide an easy-to-use possibility to use asymptotic information on the problem sequence for the construction of highly efficient sequence transformations. As shown first by Levin, it is possible to obtain such asymptotic information easily for large classes of sequences in such a way that the ω n are simple functions of a few sequence elements s n . Then, nonlinear sequence transformations are obtained. Special cases of such Levin-type transformations belong to the most powerful currently known extrapolation methods for scalar sequences and series. Here, we review known Levin-type sequence transformations and put them in a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.

Resummation of QED perturbation series by sequence transformations and the prediction of perturbative coefficients

We propose a method for the resummation of divergent perturbative expansions in quantum electrodynamics and related field theories. The method is based on a nonlinear sequence transformation and uses as input data only the numerical values of a finite number of perturbative coefficients. The results obtained in this way are for alternating series superior to those obtained using Pade approximants. The nonlinear sequence transformation fulfills an accuracy-through-order relation and can be used to predict perturbative coefficients. In many cases, these predictions are closer to available analytic results than predictions obtained using the Pade method.

Convergence acceleration via combined nonlinear-condensation transformations

A method of numerically evaluating slowly convergent monotone series is described. First, we apply a condensation transformation due to Van Wijngaarden to the original series. This transforms the original monotone series into an alternating series. In the second step, the convergence of the transformed series is accelerated with the help of suitable nonlinear sequence transformations that are known to be particularly powerful for alternating series. Some theoretical aspects of our approach are discussed. The efficiency, numerical stability, and wide applicability of the combined nonlinear-condensation transformation is illustrated by a number of examples. We discuss the evaluation of special functions close to or on the boundary of the circle of convergence, even in the vicinity of singularities. We also consider a series of products of spherical Bessel functions, which serves as a model for partial wave expansions occurring in quantum electrodynamic bound state calculations.

Static and dynamic polarizability calculations for the polyyne series (C2nH2) with extrapolation to the infinite chain

We have calculated static and dynamic longitudinal polarizabilities, αzz(−ω;ω), for the polyyne series, C2nH2, using both the uncorrelated random phase approximation and the correlated second-order polarization propagator approximation. The calculated polarizabilities are extrapolated to the value for the infinitely long chain using seven different extrapolation techniques. We employ both conventional schemes, such as the fitting of simple polynomials, as well as new schemes, such as the fitting of a Padé approximant, or purely mathematically motivated nonlinear sequence transformations which have not previously been used in connection with this property. For the direct fits, where the number of parameters is the same as the number of points, we find that the most stable and reliable extrapolation schemes are to be found among the latter.

On the stability of the J transformation

The stability of a large class of nonlinear sequence transformations is analyzed. Considered are variants of the J transformation [17]. Suitable variants of this transformation belong to the most successful extrapolation algorithms that are known [20]. Similar to recent results of Sidi, it is proved that the p{J} transformations, the Weniger S transformation, the Levin transformation and a special case of the generalized Richardson extrapolation process of Sidi are S-stable. An efficient algorithm for the calculation of stability indices is presented. A numerical example demonstrates the validity of the approach.

Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations

The computation of the Whittaker function Wκ,μ(z) via its divergent asymptotic expansion is discussed. This divergent series can be summed with the help of Padé approximants, which convert its partial sums into rational functions [C. de Izarra, O. Vallée, J. Picart, and N. Tran Minh, Comput. Phys. 9, 318 (1995)]. However, this and related divergent series, as they for instance occur in special function theory or in quantum mechanical perturbation theory, can be summed much more efficiently by some alternative rational approximants that use explicit estimates for the truncation error [D. Levin, Int. J. Comput. Math. B 3, 371 (1973); E.J. Weniger, Comput. Phys. Rep. 10, 189 (1989)]. © 1996 American Institute of Physics.

Analytical and numerical studies of the convergence behavior of the [formula omitted] transformation

A new nonlinear sequence transformation, the iterative j transformation, was proposed recently by the author (1993). For this transformation, a derivation based on the method of hierarchical consistency, alternative recursive representations, general properties, an explicit expression for the kernel, model sequences, and its relation to other sequence transformations have been given (the author, 1994). The j transformation is of similar generality as the well-known E algorithm (Brezinski, 1980; Håvie, 1979). In the present contribution, some results on convergence acceleration properties of the j transformation are proved. Numerical test results are presented which show that the j transformation is a very powerful computational tool for convergence acceleration, extrapolation, and summation of divergent series.

Nonlinear sequence transformations: A computational tool for quantum mechanical and quantum chemical calculations

Nonlinear sequence transformations: A computational tool for quantum mechanical and quantum chemical calculations

Nonlinear sequence transformations: A computational tool for quantum mechanical and quantum chemical calculations

Sequence transformations, which transform a slowly convergent or divergent sequence {sn}∞n&=0 into a new sequence {sln}∞n=0 with hopefully better numerical properties, are useful computational tools to overcome convergence and divergence problems. In this article, sequence transformations are discussed that use explicit remainder estimates [E.J. Weniger, Comput. Phys. Rep. 10, 189 (1989)]. Because of the explicit incorporation of the information contained in the remainder estimates, these transformations are potentially very powerful and well suited for the summation of strongly divergent series. The Rayleigh-Schrödinger perturbation series for the ground-state energy of the quartic, sextic, and octic anharmonic oscillator is a typical example of a perturbation series that diverges quite strongly for every nonzero coupling constant. It can be summed efficiently even in the very challenging strong coupling regime, if sequence transformations are combined with a suitable renormalization technique described by F. Vinette and J. C˘íz˘ek [J. Math. Phys. 32, 3392 (1991)]. Moreover, a renormalized strong coupling expansion for the ground-state energy of an anharmonic oscillator can be constructed, which apparently converges for all coupling constants β ϵ [0,∞) and which makes the computation the ground-state energy almost trivial. Other applications of sequence transformations in quantum mechanical and quantum chemical calculations are also reviewed. © 1996 John Wiley & Sons, Inc.