Comparison of four nonlinear transforms on some classes of logarithmic fixed point sequences
Let \(D(n) = S(n) - L\) \((n \geq 0)\) be the discrepancies between the members \(S(n)\) of a sequence and its limit \(L\), and let \(F(x) = x + \{\sum \alpha (k) x^{1 + kr}\mid (k \geq 1)\}\) for sufficiently small \(x\), where \(r \geq 1\) is an integer. If the coefficients \(\alpha(k)\) satisfy certain conditions, discrepancies for which \(D(n + 1) = F\{D(n)\}\) have, as \(n\) increases, an asymptotic representation of the form \(D(n) \sim n^{-1/r}\{\sum \beta (k|n) n^{-k+1}\mid (k \geq 1)\}\) where, in particular, \(\beta(k|n)\) is for \(k \geq 1\) a polynomial of degree at most \(k - 1\) in \(\log (n)\). Applications of various nonlinear recursive convergence acceleration algorithms (a modified \(\rho\)-algorithm, the \(\theta\)-algorithm, Lubkin's and Steffenson's methods) to sequences of the above type are studied. Asymptotic error estimates in transformed sequences are given and numerical examples are provided. The methods considered have one affliction in common: gallopping instability -- and no examination of error propagation is conducted. For this reason it might have been of interest to have included in the survey a condensation method due to \textit{A. van Wijngaarden} [Course scientific computing B; process analysis, Math. Centre CR 18, Amsterdam (1965); see also: \textit{J. W. Daniel}, Math. Comp. 23, 91-96 (1969; Zbl 0183.44101)] which is relatively unaffected by instability. (Implementation of the method as a nice exercise in the use of a recursive procedure is given by \textit{P. W. Hemker} (ed.), NUMAL: numerical procedures in Algol 60, MC Syllabus 47. 1-47.7, Math. Centre, Amsterdam (1980); for a C-implementation, see: \textit{H. T. Lau} [A numerical library in C for scientists and engineers, CRC Press (1995; Zbl 0815.65001)]; a necessarily clumsy FORTRAN implementation is given by: \textit{P. Wynn} [Numal in FORTRAN, IIMAS, Universidad Nacional Autonoma de México, Comunicaciones técnicas Nos. 48.0-48.11 (1981)]).
nonlinear recursive convergence acceleration algorithms
scientific article; zbMATH DE number 825305
scientific article