Scalar Levin-Type Sequence Transformations

Abstract

Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Transformations that depend not only on the sequence elements or partial sums $s_n$ but also on an auxiliary sequence of so-called remainder estimates ${\omega }_n$ are of Levin-type if they are linear in the $s_n$, and nonlinear in the ${\omega }_n$. Known Levin-type sequence transformations are reviewed and put into 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.

59 pages, LaTeX, invited review for J. Comput. Applied Math., abstract shortened

Cited by in corpus (16)

• From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions
• Robust Computation of Dipole Electromagnetic Fields in Arbitrarily-Anisotropic, Planar-Stratified Environments
• Mathematical Properties of a New Levin-Type Sequence Transformation Introduced by Č\'ıžek, Zamastil, and Skála. I. Algebraic Theory
• Implementation of the Combined--Nonlinear Condensation Transformation
• On the Analyticity of Laguerre Series
• Irregular Input Data in Convergence Acceleration and Summation Processes: General Considerations and Some Special Gaussian Hypergeometric Series as Model Problems
• Tensor Green's Function Evaluation in Arbitrarily Anisotropic, Layered Media using Complex-Plane Gauss-Laguerre Quadrature
• Complex-Plane Generalization of Scalar Levin Transforms: A Robust, Rapidly Convergent Method to Compute Potentials and Fields in Multi-Layered Media
• Level truncation and the quartic tachyon coupling
• A Rational Approximant for the Digamma Function
• Acceleration of generalized hypergeometric functions through precise remainder asymptotics
• Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series
• Coulomb matrix elements of bilayers of confined charge carriers with arbitrary spatial separation
• Extended Comment on "One-Range Addition Theorems for Coulomb Interaction Potential and Its Derivatives" by I. I. Guseinov (Chem. Phys. Vol. 309 (2005), pp. 209 - 213)
• Application of Levin's transformations to virial series
• Construction of New Generalizations of Wynn's Epsilon and Rho Algorithm by Solving Finite Difference Equations in the Transformation Order