Convergence acceleration of number-machine sequences

A new, nonclassical convergence acceleration concept, called \(\mu \)-acceleration of convergence (where \(\mu \) is a positive monotonically increasing sequence), is introduced and compared with the classical convergence acceleration concept. It is shown that this concept allows to compare the speeds of convergence for a larger set of sequences than the classical convergence acceleration concept. Also, regular matrix methods that improve and accelerate the convergence of sequences and series are studied. Some problems related to the speed of convergence of sequences and series with respect to matrix methods are discussed. Several theorems on the improvement and acceleration of the convergence are proved. As an application, the results obtained are used to increase the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.

Vector sequence transformations for the acceleration of logarithmic convergence

Aitken’s Δ2 method is used to accelerate convergence of sequences, e.g. sequences obtained from iterative methods. An explicit assumption in deriving Aitken’s Δ2 method and establishing acceleration (for linearly convergent sequences) is that consecutive error iterates (or their approximations) have the same sign or have an alternating sign pattern. We extend the standard Aitken’s Δ2 method to the cases in which consecutive pairs of error iterates in the sequence have alternating signs. Under suitable restrictions, acceleration of convergence is proved. Implementation of our extended method is described. Numerical examples demonstrate the process. An example is included relating our results to results obtained from Richard extrapolation.

Eleven different methods for accelerating convergence of sequences and series have been tested and compared on a wide range of test problems, including both linearly and logarithmically convergent series, monotone and alternating series. All but one of these methods are already in the literature, and they include both linear and nonlinear methods. The only methods found to accelerate convergence across the board were the u and v transforms of Levin and the $\theta $ algorithm of Brezinski. The paper gives detailed comparisons of all the tested methods on the basis of number of correct digits in the answer as a function of number of terms of the series used. A theorem of Germain-Bonne states that methods of a certain form which are exact on geometric series will accelerate linear convergence. The theorem applies to $\theta _2 $, and we have extended it to apply to Levin's transforms. No corresponding theorem is known for logarithmic convergence, but u, v, and $\theta $ are exact on certain large classes of logarithmic series, and all tested methods lacking this property failed to accelerate some logarithmically convergent series.

The present paper continues the study of acceleration of convergence started in the paper [A. Aasma, Proc. Estonian Acad. Sci. Phys. Math., 2006, 55, 4, 195–209]. The new, non‐classical convergence acceleration concept, called strong μ‐acceleration of convergence (μ is a positive monotonically increasing sequence), is introduced. It is shown that this concept allows to compare the speeds of convergence for a larger set of sequences than the classical convergence acceleration concept. Regular matrix methods are used to accelerate the convergence of sequences.

INTERVAL ACCELERATION OF CONVERGENCE