About Henrici's transformation for accelerating vector sequences
The paper is concerned with a transformation of a sequence s(n) \((n=0,1,...)\) of vectors in \({\mathbb{C}}^ p\) (p\(\geq 1)\). The transformation has the form \(h(n)=s(n)-\Omega (1,n)\Omega (2,n)^{-1}\Delta s(n)\) where \(\Delta s(n)=s(n+1)-s(n),\) \(\Omega\) (1,n) is the \(p\times p\) matrix whose \((\nu +1)th\) column is \(\Delta s(n+\nu)\) \((0\leq \nu <p)\) and \(\Omega\) (2,n) is formed from \(\Delta s(n+\nu +1)-\Delta s(n+\nu)\) in the same way. The following exemplifies the results obtained. Let s(n) converge to s and satisfy the condition \(s(n+1)-s=\{B+B(n)\}\{s(n)-s\}\) where, for the spectral radius \(\rho\) (B) of B, \(\rho (B)<1\), and \(\lim B(n)=0\quad (n\to \infty).\) Let \(\det \Omega (1,n)\geq c\Pi \| \Delta s(n+\nu)\| \quad (0\leq \nu <p)\) for all \(n>N\) for some \(N\geq 0\) and \(c>0\), where \(\| \cdot \|\) denotes the Euclidean norm. Then \(\lim \| h(n)- s\| /\| s(n)-s\| =0\quad (n\to \infty).\)
scientific article; zbMATH DE number 4124876
scientific article