Kronecker-product approximations for some function-related matrices.
type: Item
instance of: scholarly article
title: Kronecker-product approximations for some function-related matrices.
zbMATH Open document ID: 1046.65033
published in: Linear Algebra and its Applications
publication date: 2004-03-14T00:00:00Z
review text: The paper focusses on the Kronecker-product approximations for some function-related matrices. The purpose is the formulation and the proof of ``existence theorems'', with estimates on the Kronecker rank and the corresponding approximation error for reasonable wide classes of practically interesting matrices. The first section presents an introduction concerning the approximation of matrices by sums of Kronecker-product of smaller-size matrices. The motivation of the work is given in the second section which begins with some numerical examples of Kronecker-product approximations for the case of uniform and non-uniform respectively grids \(z_1,z_2,\dots, z_n\) of \(n\) nodes in \(\Omega= [0,1]\times [0,1]\) and for a matrix \(A= [a_{ij}]\) associated with a function \(f= f (z',z)\) so that \(a_{ij}= f (z_i,z_j)\), \(i, j =1,\dots,n\) which depends upon the distance \(\rho\) between \(z'\) and \(z\). The third section deals with Kronecker-products and low-rank matrices, stating that a sum of \(r\) Kronecker-products is mapped to a sum of \(r\) rank-one matrices. Section four presents the main theoretical results concerning functions, grids and matrices. One considers the general situation \(f (z', z)= F(u, v)\), \(z'= (x', y')\), \(z= (x, y)\), \(u= x'-x\), \(v= y'-y\) where \(F(u, v)\) is such that any mixed derivative \(D^m F= {\partial^k\partial^l\over(\partial u^k)(\partial v^l)} F\), \(m= k+l\) satisfies the inequality \(| D^m F|\leq cd^m m!\rho^{g- m}\), \(\rho= \sqrt{u^2+ v^2}\) with some constants \(c,d > 0\) and \(g\). \(F\) is called a complete asymptotically smooth function. In section five there are some further results for a formally different case of \(A= [ f (x_i,x_j)]\), \(0\leq i\), \(j\leq n\), \(0< x_1<\cdots< x_n< 1\) where \(f(x',x)= F(x'-x)\) such that \[ \Biggl|{\partial^m F\over\partial x^m}\Biggr|\leq cd^m m!| x|^{g-m}\qquad (\forall)m\in \mathbb{N}. \] Some possible generalizations of the presented results are discussed such as: practically e…
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A12
zbMATH DE Number: 2056699
zbMATH Keywords: Kronecker product approximation
zbMATH Keywords: Low-rank matrices
zbMATH Keywords: Asymptotically smooth functions
zbMATH Keywords: function-related matrices
zbMATH Keywords: numerical examples
author: E. E. Tyrtyshnikov
reviewed by: Romulus Militaru
MaRDI profile type: Publication
full work available at URL: j.laa.2003.08.013
OpenAlex ID: W2000071783
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DOI: 10.1016/J.LAA.2003.08.013
description: scientific article; zbMATH DE number 2056699