Hartley-type algebras in displacement and optimization strategies.
type: Item
instance of: scholarly article
title: Hartley-type algebras in displacement and optimization strategies.
zbMATH Open document ID: 1044.65050
DOI: 10.1016/S0024-3795(02)00499-8
author: Carmine Di Fiore
author: Filomena Lepore
author: Paolo Zellini
published in: Linear Algebra and its Applications
publication date: 2003-05-25T00:00:00Z
review text: The purpose of this paper is to show that a class of structural algebras, the Hartley-type (Ht) algebras linked in different ways to the Hartley transform, are extremly useful in solving both linear and nonlinear (minimizaton) problems. The matrix representations obtained in this paper are classical displacement formulas and lead to efficient Toeplitz inversion formulas (the inverse is rewritten in terms of a number of simple matrices which can be fastly multiplied by vectors) competitive with the well known Ammar-Gader formula based on circulant algebras. The best least squares approximation is exploited as a preconditioner in quasi-Newton methods (QN) for unconstrained minimization. For novel \(L\)QN minimization methods an algebra \(L\) (\(L=\text{Ht})\) of matrices simultnenously diagonalized by a fast unitary transform is used in the iterative scheme of generalized BFGS-type algorithm. The complexity per step of \(L\)QN is \(O(n\log n)\) and only \(O(n)\) memory allocationsare required. It is shown that \(L\)QN methods are linearly convergent.
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C05
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C53
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 1915806
zbMATH Keywords: Hartley-type algebras
zbMATH Keywords: Displacement formulas
zbMATH Keywords: Quasi-Newton methods
zbMATH Keywords: convergence
zbMATH Keywords: Hartley transform
zbMATH Keywords: Toeplitz inversion formulas
zbMATH Keywords: Ammar-Gader formula
zbMATH Keywords: circulant algebras
zbMATH Keywords: least squares approximation
zbMATH Keywords: preconditioner
zbMATH Keywords: unconstrained minimization
zbMATH Keywords: BFGS-type algorithm
reviewed by: Otu Vaarmann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915806