\(\text{GL}(E)\)-quasilinear transformations and acceleration
type: Item
instance of: scholarly article
title: \(\text{GL}(E)\)-quasilinear transformations and acceleration
zbMATH Open document ID: 0937.65001
DOI: 10.1016/S0168-9274(98)00003-8
author: A. Benazzouz
published in: Applied Numerical Mathematics
publication date: 2000-05-17T00:00:00Z
review text: The paper is concerned with sequences of \( p \) real-dimensional vectors. \(\lambda\) being such a vector, the sequence \( {\mathbf s}(n)\) \((n \geq 0) \) for which a) \( {\mathbf s}(n) \neq\lambda\) \((n \geq N) \) for some integer \( N \) and b) a \( p \times p \) nonsingular real matrix \( {\mathbb{B}} \) for which \( {\mathbf I -\mathbb{B} }\) is also nonsingular and a sequence \( {\mathbb{B}}(n)\) \((n \geq 0) \) of such matrices converging to the null matrix exists for which \( {\mathbf s}(n + 1) - {\mathbf s}(n) = [{\mathbb{B} -\mathbb{B}}(n)] [{\mathbf s}(n) -\lambda] (n \geq 0) \) is said to converge linearly to \(\lambda\). A transformation \( {\mathbf f} \) of \( k \) vectors \( {\mathbf v}(1),\ldots,{\mathbf v}(k) \) producing a single vector and obeying rules of the form \( {\mathbf f}(\mathbb{A}{\mathbf v}(1),\ldots,\mathbb{A}{\mathbf v}(k)) = \mathbb{A} f({\mathbf v})(1),\ldots,{\mathbf v}(k)) \) for all nonsingular real \( p \times p \) matrices \(\mathbb{A} \), and \( {\mathbf f}({\mathbf v}(1) + {\mathbf b},\ldots,{\mathbf v}(k) + {\mathbf b}))= {\mathbf f}({\mathbf v}(1),\ldots,{\mathbf v}(k)) + {\mathbf b} \) is said to be strictly quasilinear. Such transformations may be applied in the form \( {\mathbf f}({\mathbf s}(m),\ldots,{\mathbf s}(m + k - 1)) \) to subsequences of vectors to produce a further sequence of vectors defined for \( m \geq 0 \) which in some cases converges to the limit of \( {\mathbf s}(n) (n \geq 0) \) faster than does this sequence itself. It is shown, however, that the convergence of linearly convergent sequences is not accelerated in this way by strictly quasilinear transformations. P. Henrici proposed the following transformation. Denote by \( \nabla(i|n) \) the matrix whose successive columns are \( \Delta^{i} {\mathbf s}(n + r) (0 \leq r \leq p), \Delta \) being the forward difference operator. The transformation is \( {\mathbf f}({\mathbf s}(m),\ldots,{\mathbf s}(m + s(m + p + 1)) = {\mathbf s}(m) - \nabla(1|m) \nab…
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 1311233
zbMATH Keywords: vector sequence transformation
zbMATH Keywords: convergence acceleration
zbMATH Keywords: linear convergence
zbMATH Keywords: quasilinear transformation
reviewed by: Peter Wynn
MaRDI profile type: Publication
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cites work: About Henrici's transformation for accelerating vector sequences
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description: scientific article; zbMATH DE number 1311233