Convergence Acceleration Methods Research Articles

Evaluation of Higher Order Matrix Elements in Atomic Physics: Using Convergence Acceleration Methods

Evaluation of Higher Order Matrix Elements in Atomic Physics: Using Convergence Acceleration Methods

A Convergence Acceleration Method for Some Logarithmically Convergent Sequences

A convergence acceleration method with one parameter for a sequence $\{ {S_n } \}$ satisfying \[ S_n \sim S + n^\theta \left( {c_0 + \frac{{c_1 }}{n} + \frac{{c_2 }}{{n^2 }} + \cdots } \right)\quad {\text{as }}n \to \infty ,\] is proposed. When the parameter is equal to $ - 1$, the method agrees with Wynn’s $\rho $-algorithm. Hence the method is designated as a generalized $\rho $-algorithm. When $S_n $ is the nth partial sum of a series whose nth term is asymptotic to $An^{\theta - 1} $, where $\theta $ is negative, the generalized $\rho $-algorithm with parameter $\theta $ agrees essentially with the modified Wynn’s $\rho $-algorithm taken up by Drummond. Error estimates and numerical examples of the generalized $\rho $-algorithm with parameter $\theta $ are given.The exponent $\theta $ in the expansion above can be estimated using the generalized $\rho $-algorithm with parameter $ - 2$. In practice, by estimating the exponent $\theta $, the generalized $\rho $-algorithm can be automatically applied to ...

A Convergence Acceleration Method based on a good Estimation of the Absolute value of the Error

Suppose a sequence (Sn)n, converges to a limit S. In this paper we construct, from a good estimate (Dn)n of (∣Sn−S∣), an acceleration method for (Sn)n based on: (a) an automatic selection between two transformations if (Dn)n converges logarithmically; (b) composite sequence transformations if (Dn)n converges linearly.

Improving a method for computing non-dominant solutions of certain second-order recurrence relations of Poincar�-type

In this paper we present a method of convergence acceleration for the calculation of non-dominant solutions of second-order linear recurrence relations for which the coefficients satisfy certain asymptotic conditions. It represents an improvement of the method recently proposed by Jacobsen and Waadeland [3, 4] for limit periodic continued fractions. For continued fractions the method corresponds to a repeated application of the Bauer-Muir transformation. Some examples and a generalization to non-homogeneous recurrence relations are given.

A note on two convergence acceleration methods for ordinary continued fractions

In this note we compare two recent methods of convergence acceleration for ordinary continued fractions, the first one introduced by Thron and Waadeland [13], and further developed by Brezinski [1], the second one by Jacobsen and Waadeland [4,5].

Wave propagation in turbulent media: use of convergence acceleration method

We propose the use of convergence acceleration methods for the evaluation of integral expressions of an oscillatory nature, often encountered in the study of optical wave propagation in the turbulent atmosphere. These techniques offer substantial savings in computation time with appreciable gain in accuracy. As an example, we apply the Levin u acceleration scheme to the problem of remote sensing of transversal wind profiles.

Convergence acceleration of limit periodic continued fractions under asymptotic side conditions

We present a method of convergence acceleration for limitk-periodic continued fractionsK(an/1) orK(1/bn) satisfying certain asymptotic side conditions. The method represents an improvement of the “fixed point modification” considered by Thron and Waadeland [8], under these conditions. The regularC-fraction expansions of hypergeometric functions2F1(a, 1;c; z) and2F1(a, b; c; z)/2F1(a, b+1;c+1;z) are examples of continued fractions satisfying these conditions.

The summation of alternating and monotonic series

In previous papers the author has treated the question of convergence acceleration of alternating series, and of power series in x having positive coefficients. In the latter case it was assumed that 0 < x < 1, but even for x extremely close (but not equal) to 1, a powerful convergence acceleration was obtained. The present paper propounds an improved convergence acceleration method for alternating series, and also an efficient convergence acceleration technique which is valid for many monotonic series of importance in applied mathematics, i.e. the case x = 1 is now treated. In essence this technique obtains from the positive terms of the monotonic series an alternating sequence of convergents, which are then handled by the method for alternating series or sequences. It is assumed that the coefficients (or terms) of the series treated in this paper are moments of a function itf( u) over the interval 0 ⩽ u ⩽ 1, and the methods are based on properties of the shifted Legendre polynomials P ∗ n ( u), and on the author's earlier summation formula for power series having moment coefficients. The function itf( u) does not need to be known.

A Comparison of Some Taylor and Chebyshev Series

A function is approximated in the interval $- 1 \leq x \leq 1$ by (i) a Taylor series in x; (ii) a Taylor series in $y = (x + \lambda )/(1 + \lambda x)$; (iii) a Chebyshev series in x; and (iv) a Chebyshev series in $z = (x + \mu )/(1 + \mu x)$. The convergence of all four series is discussed, and a method is given for finding the values of $\lambda$ and $\mu$ which optimize convergence. Methods are also given for transforming one of the above series into another, some of which provide effective methods for acceleration of convergence. The application of the theory to even and odd functions is also discussed.

A comparison of some Taylor and Chebyshev series

A function is approximated in the interval − 1 ≤ x ≤ 1 - 1 \leq x \leq 1 by (i) a Taylor series in x; (ii) a Taylor series in y = ( x + λ ) / ( 1 + λ x ) y = (x + \lambda )/(1 + \lambda x) ; (iii) a Chebyshev series in x; and (iv) a Chebyshev series in z = ( x + μ ) / ( 1 + μ x ) z = (x + \mu )/(1 + \mu x) . The convergence of all four series is discussed, and a method is given for finding the values of λ \lambda and μ \mu which optimize convergence. Methods are also given for transforming one of the above series into another, some of which provide effective methods for acceleration of convergence. The application of the theory to even and odd functions is also discussed.

Acceleration of Convergence of Vector Sequences

A general approach to the construction of convergence acceleration methods for vector sequences is proposed. Using this approach, one can generate some known methods, such as the minimal polynomial extrapolation, the reduced rank extrapolation, and the topological epsilon algorithm, and also some new ones. Some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties. The convergence analysis of these new methods is carried out, and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iteratively. A stability analysis is also given, and numerical examples are provided. The convergence and stability properties of the topological epsilon algorithm are likewise given.

Prediction properties of some extrapolation methods

Convergence acceleration methods consist in the construction of a sequence converging faster than the initial sequence. Each member of the new sequence is a guess for the limit and it is computed from a restricted number of terms of the initial sequence. It is shown herein how convergence acceleration methods can be used to predict the next (unknown) term of the initial sequence instead of its limit. Particular emphasis on Aitken's △ 2 process and the E-algorithm is placed.

Estimates of the electrongfactor: Application of convergence acceleration methods to the QED series

We have applied a variety of different convergence acceleration methods to the accurately known (first four) terms of the QED perturbation series for the electron g factor g/2=${\mathcal{J}}_{n=0}^{\ensuremath{\infty}}$${c}_{n}$(\ensuremath{\alpha}/\ensuremath{\pi}${)}^{n}$ so as to obtain eight estimates of its value. All of the estimates agree with the most recent experimental value of g/2 to better than one standard deviation when we use the new NBS condensed-matter value of \ensuremath{\alpha}. These convergence acceleration methods also provide as estimates of the higher-order expansion coefficients ${c}_{4}$=-5.65\ifmmode\pm\else\textpm\fi{}0.64, ${c}_{5}$=31\ifmmode\pm\else\textpm\fi{}6. .AE

Convergence acceleration methods: The past decade

The aim of this paper is to present a review of the most significant results obtained the past ten years on convergence acceleration methods. It is divided into four sections dealing respectively with the theory of sequence transformations, the algorithms for such transformations, the practical problems related to their implementation and their applications to various subjects of numerical analysis.

On the stability of a class of convergence acceleration methods for power series

In this paper we study the problem of evaluating the sum of a power series whose terms are given numerically with a moderate accuracy. For a large class of divergent series a sum may be defined using analytic continuation. This sum may be estimated using the values of a finite number of terms. However, it is established here that the accuracy of this estimate will generally deteriorate if we use an ever-growing number of terms. A result on the stability of product quadrature is also obtained as a corollary of our main stability theorem.

Broyden's method for self-consistent field convergence acceleration

Broyden's method for self-consistent field convergence acceleration

Broyden's method for self-consistent field convergence acceleration

Broyden's method (1965) for self-consistent field convergence acceleration, recently used by Bendt and Zunger (1982) in the band-structure calculation of solids, is reviewed and a simpler computational scheme is presented.

Calculation of voltage and space charge distributions in a wire-plate electrostatic precipitator

Several convergence acceleration methods have been introduced into procedures developed by the Southern Research Institute for calculating voltage and space charge distributions in parallel plate electrostatic precipitators. A dominant eigenvalue method is used in conjunction with several other techniques to promote convergence of an iterative calculation of these quantities. In each of two test problems, the effect of the acceleration step was to reduce the required computation time to roughly 5% of its original value.

On a relationship between Quasi-Newton and dominant eigenvalue methods for the numerical solution of non-linear equations

In order to compare Quasi-Newton methods with the dominant eigenvalue (DE) method of convergence acceleration, the latter is modified so that an acceleration step is taken at every iteration and the method of calculation of the required coefficients can be suitably chosen. A relationship is derived between the class of rank-one Quasi-Newton methods (QN1) and modified DE methods such that each QN1 method corresponds to a particular modified DE method. However, it was not possible to find a QN1 formula which would correspond to the DE method applied at every iteration. Such a formula would appear to require information in advance, from iteration yet to be done. The advantages of using the equivalent modified DE method instead of Broyden's QN1 method are examined. A new algorithm of QN1 methods is proposed, based on the equivalent modified DE method, which was at least one order of magnitude faster, for Broyden's mehtod[1] on an example problem, than the conventional implementation.

A method for accelerating SCF convergence

A new efficient semi-iterative convergence acceleration method is presented and illustrated by examples of SCF iteration.