Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed. Some of the sequence transformations of this report as for instance Wynn's $\epsilon$ algorithm or Levin's sequence transformation are well established in the literature on convergence acceleration, but the majority of them is new. Efficient algorithms for the evaluation of these transformations are derived. The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed. Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and divergent series, which are hopefully realistic models for a large part of the slowly convergent or divergent series that can occur in scientific problems and in applied mathematics.

Quadrature rules with maximal even trigonometric degree of exactness are considered. We give a brief historical survey on such quadrature rules. Special attention is given on an approach given by Turetzkii [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (3) (2005) 337-359. Translation in English from Uchenye Zapiski, Vypusk 1 (149). Seria Math. Theory of Functions, Collection of papers, Izdatel'stvo Belgosuniversiteta imeni V.I. Lenina, Minsk, 1959, pp. 31-54]. The main part of the topic is orthogonal trigonometric systems on [0,[email protected]) (or on [[email protected],@p)) with respect to some weight functions w(x). We prove that the so-called orthogonal trigonometric polynomials of semi-integer degree satisfy a five-term recurrence relation. In particular, we study some cases with symmetric weight functions. Also, we present a numerical method for constructing the corresponding quadratures of Gaussian type. Finally, we give some numerical examples. Also, we compare our method with other available methods.

http://www.mi.sanu.ac.rs/~gvm/radovi/Trig2008.pdf

Trigonometric orthogonal systems and quadrature formulae

An extension of the theory of orthogonal polynomials and Gaussian quadrature to trigonometric and hyperbolic polynomials

An alternative to Lyche's recently proposed scheme for trigonometric divided differences and a Newton-type interpolation formula, obtained through the development of an earlier suggestion of the writer, appears to be simpler, neater in format, more symmetrical and easier to compute.

Herbert E. Salzer

Papers

A new form of trigonometric orthogonality and Gaussian-type quadrature

Note on the Dočev-Grosswald asymptotic series for generalized Bessel polynomials

Short communication: Inverse laplace transforms of osculatory and hyperosculatory interpolation polynomials using “near-Chebyshev” nodes

Minimal error difference formulas