Wynn's Epsilon Algorithm Research Articles

Derivative expansion for computing critical exponents of O(N) symmetric models at next-to-next-to-leading order.

We apply the derivative expansion of the effective action in the exact renormalization group equation up to fourth order to the Z_{2} and O(N) symmetric scalar models in d=3 Euclidean dimensions. We compute the critical exponents ν, η, and ω using polynomial expansion in the field. We obtain our predictions for the exponents employing two regulators widely used in exact renormalization group computations. We apply Wynn's epsilon algorithm to improve the predictions for the critical exponents, extrapolating beyond the next-to-next-to-leading order prediction of the derivative expansion.

An Analytical Evaluation Method of the Operational Risk Using Fast Wavelet Expansion Techniques

A financial institution that adopts an advanced measurement approach (AMA) as a method of computing operational risk capital has to measure 99.9\%value-at-risk (VaR) as the amount of an operational risk. The most popular method to satisfy the AMA standards requires the evaluation of aggregate (compound) loss distribution, which is called the loss distribution approach (LDA). The Monte Carlo (MC) method is a well known method for calculating VaR under the LDA. However, when using the MC method to calculate VaR, the statistical error of VaR for the fat-tailed distribution increases and the computation time increases in proportion to the expected value of frequency distribution. Since the MC method has these problems, this paper presents a new methodology to compute VaR under the LDA using fast wavelet expansion techniques. The key features of our algorithm are follows: (i) Scale transformation technique for loss distributions (ii) Double exponential (DE) transformation for oscillatory integrals (iii) Finite series expansion of the wavelet scaling coefficients (iv) Wynn's epsilon algorithm to accelerate the convergence of those series (v) Efficient cubic spline interpolation method to calculate the moment generating function. We illustrate the effectiveness of our algorithms through numerical examples.

Efficient algorithm for summation of some slowly convergent series

The Q transformation, introduced recently by Woźny and Nowak, may serve as a good tool for summation of slowly convergence series. This approach can be easily applied to the case of generalized or basic hypergeometric series, as well as some orthogonal polynomial expansions. It is closely related to the famous Wynn's epsilon algorithm, Weniger's or Homeier's transformations, and the method introduced by Čížek, Zamastil and Skála. However, it is difficult to use the algorithm proposed by Woźny and Nowak in the general case, because of its high complexity, and some other restrictions. We propose another realization of the Q transformation, which results in obtaining a simpler and faster algorithm. Four illustrative numerical examples are given.

A review of two algorithms for acceleration of convergence of a sequence of vectors

AbstractThe purposes of this article are threefold: (i) to review the role of Wynn's epsilon algorithm as a method for acceleration of convergence of an ordinary (scalar) series (possibly representing the value of a physical quantity), (ii) to review the role of Wynn's vector‐epsilon algorithm and another vector Padé‐type method in cases where a number of such series should be treated simultaneously, and (iii) to point out that these general methods for the acceleration of convergence of a vector‐valued series are readily applied to the solution of linear and nonlinear systems of equations. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Prediction properties of Aitken's iterated Δ2 process, of Wynn's epsilon algorithm, and of Brezinski's iterated theta algorithm

The prediction properties of Aitken's iterated Δ 2 process, Wynn's epsilon algorithm, and Brezinski's iterated theta algorithm for (formal) power series are analyzed. As a first step, the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of accuracy-through-order relationships. On the basis of these relationships, the rational approximants can be rewritten as a partial sum plus an appropriate transformation term. A Taylor expansion of such a transformation term, which is a rational function and which can be computed recursively, produces the predictions for those coefficients of the (formal) power series which were not used for the computation of the corresponding rational approximant.

Schwinger–Lanczos algorithm for calculation of off-shell [formula omitted]-matrix elements and Wynn's epsilon algorithm

A generalization of the Schwinger–Lanczos algorithm for the calculation of the off-shell T-matrix elements is presented and shown to be equivalent to the Wynn's epsilon algorithm for acceleration of the Born series. Both methods are applied to the calculation of cross section for the dissociative attachment of electrons to hydrogen molecule, e −+ H 2→ H+ H − . Although formally fully equivalent both methods behave very differently numerically. It is shown that the Schwinger–Lanczos algorithm is more stable and generally superior to all other methods discussed in this paper.

Convergence accelerators in the computation of molecular integrals over Slater-type basis functions in the two-range one-center expansion method.

The so-called nonlinear transformations may be of substantial help in many practical problems including the so-called multicenter integrals. In this work Wynn's epsilon algorithm and the Levin's u transformation have been used to improve the convergence of the series representation of three-center two-electron integrals obtained via the so-called two-range one-center expansion method.

A Levin-type algorithm for accelerating the convergence of Fourier series

A nonlinear sequence transformation is presented which is able to accelerate the convergence of Fourier series. It is tailored to be exact for a certain model sequence. As in the case of the Levin transformation and other transformations of Levin-type, in this model sequence the partial sum of the series is written as the sum of the limit (or antilimit) and a certain remainder, i.e., it is of Levin-type. The remainder is assumed to be the product of a remainder estimate and the sum of the first terms oftwo Poincare-type expansions which are premultiplied by two different phase factors. This occurrence of two phase factors is the essential difference to the Levin transformation. The model sequence for the new transformation may also be regarded as a special case of a model sequence based on several remainder estimates leading to the generalized Richardson extrapolation process introduced by Sidi. An algorithm for the recursive computation of the new transformation is presented. This algorithm can be implemented using only two one-dimensional arrays. It is proved that the sequence transformation is exact for Fourier series of geometric type which have coefficients proportional to the powers of a numberq∈ℂ, |q|<1. It is shown that under certain conditions the algorithm indeed accelerates convergence, and the order of the convergence is estimated. Finally, numerical test data are presented which show that in many cases the new sequence transformation is more powerful than Wynn's epsilon algorithm if the remainder estimates are properly chosen. However, it should be noted that in the vicinity of singularities of the Fourier series the new sequence transformation shows a larger tendency to numerical instability than the epsilon algorithm.

Convergence acceleration of viscous and inviscid hypersonic flow calculations

The convergence of inviscid and viscous hypersonic flow calculations using a two-dimensional flux-splitting code is accelerated by applying a Richardson-type overrelaxation method. Successful results are presented for various cases; and a 50 percent savings in computer time is usually achieved. An analytical formula for the overrelaxation factor is derived, and the performance of this scheme is confirmed numerically. Moreover, application of this overrelaxation scheme produces a favorable preconditioning for Wynn's epsilon-algorithm. Both techniques have been extended to viscous three-dimensional flows and applied to accelerate the convergence of the compressible Navier-Stokes code. A savings of 40 percent in computer time is achieved in this case.

Efficient computation of the free-space periodic Green's function

The application of Shanks's transform is shown to improve the convergence of the series representing the doubly infinite free-space periodic Green's function. Higher order Shanks transforms are computed via Wynn's epsilon algorithm. Numerical results confirm that a dramatic improvement in the convergence rate is obtained for the on-plane case, in which the series converges extremely slowly. In certain instances, the computation time can be reduced by as much as a factor of a few thousands. A relative error measure versus the number of terms taken in the series is plotted for various values of a convergence factor as the observation point is varied within a unit cell. Computation times are also provided. >

Application of Wynn's epsilon algorithm to periodic continued fractions

The infinite continued fractionin whichis periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.

Pade approximants and multiphonon ionisation of atomic hydrogen

Two-photon ionisation rates in hydrogen above the photoelectric threshold are computed using Pade approximants obtained directly for divergent sturmian expansions with the help of Wynn's epsilon algorithm. Numerical examples are given for initial states with 2<or=n<or=6.