Strong Μ‐faster Convergence and Strong Μ‐acceleration of Convergence by Regular Matrices

Abstract

Abstract. 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.

Some alternative methods of convergence acceleration

“…It is easy to see that for all μ we have l k = o (1) for each x ∈ ϕ. First we notice that there exists no convergent sequence, converging faster or μ-faster than any sequence of ϕ.…”

“…We note that in [1] the concepts of strong μ-faster convergence and strong μ-acceleration by a regular matrix are introduced. There are also the relationship of these concepts with usual faster convergence and usual convergence acceleration, and with μ-faster convergence and μ-acceleration by a regular matrix are established.…”