Computation of orthonormal factors for fundamental solution matrices
type: Item
instance of: scholarly article
title: Computation of orthonormal factors for fundamental solution matrices
zbMATH Open document ID: 0941.65066
author: Luca Dieci
author: E. Van Vleck
published in: Numerische Mathematik
publication date: 2000-07-20T00:00:00Z
review text: The paper is concerned with the linear system \[ Y'(t)=A(t)Y(t), \quad Y(0)=Y_0, \quad t\geq 0, \tag{1} \] where \(A:t\in[0,t_f]\to A(t)\in {\mathbb R}^{n\times n}\) is a \(C^{k-1}\) function, \(k\geq 1,\) so that \(Y\in C^k\), \(Y_0\) is a full rank matrix and \(Y(t), Y_0\in {\mathbb R}^{n\times p}, n\geq p.\) The purpose is to compute the \(QR\) factorization of some or all columns of a fundamental solution matrix \(Y(t)\) with the emphasis on the integration of the nonlinear polynomial equation for \(Q.\) The authors consider the situation where \(Q\) can be written as a product of elementary Householder or Givens transformations. In general, these transformations do not vary smoothly on the whole interval. Therefore, in order to obtain a well-defined process the authors use a reimbedding procedure obtaining as a result a piece-wise smooth factorization of \(Q\). In order to obtain, if needed, globally smooth factors, a post-processing based on monitoring signs on the diagonal of \(R\) can be implemented. Numerical results are considered to compare the new methods with the existing techniques.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 1398755
zbMATH Keywords: linear differential equations
zbMATH Keywords: \(QR\) factorization
zbMATH Keywords: Householder transformation
zbMATH Keywords: Givens transformation
zbMATH Keywords: linear system
zbMATH Keywords: fundamental solution matrix
zbMATH Keywords: numerical results
reviewed by: Yuriy V. Rogovchenko
MaRDI profile type: Publication
full work available at URL: s002119900083
OpenAlex ID: W2031202067
DOI: 10.1007/S002119900083
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description: scientific article; zbMATH DE number 1398755