Zur Kondition des linearen Ausgleichsproblems mit linearen Gleichungen als Nebenbedingungen
type: Item
instance of: scholarly article
title: Zur Kondition des linearen Ausgleichsproblems mit linearen Gleichungen als Nebenbedingungen
zbMATH Open document ID: 0554.65030
DOI: 10.1007/BF01389762
author: Werner Sautter
published in: Numerische Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/132925
review text: The author studies the condition for the linear least squares problem (LS) with linear equality constraints (LSE): Compute \(x\in R^ n\) with \(A_ 1x=b_ 1\), \(\min \| A_ 2x-b_ 2\|,\quad A=(A_ 1,A_ 2)^ T,\) \(rank(A)=n\). He characterizes the solution by pseudo-inverses and related projections and shows some perturbation results. Then he demonstrates, that the condition number \(\| A\| \cdot \| X\|\) (with X left inverse of A) gives too pessimistic error estimations. He proposes instead three different condition numbers, which give better results. It is noted, that component-wise estimations are possible in the same manner, if component-wise bounds for the data errors are available. Then the question of optimal scaling and minimal condition arises, but this question is not pursued further by the author. If the problems LS and LSE have identical solutions, the condition of LS may be better than that of LSE or vice versa. The different cases are demonstrated with an interesting example.
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65C99
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 62J05
zbMATH DE Number: 3883510
zbMATH Keywords: linear equality constraints
zbMATH Keywords: perturbation
zbMATH Keywords: linear least squares problem
zbMATH Keywords: pseudo-inverses
zbMATH Keywords: condition number
zbMATH Keywords: optimal scaling
MaRDI profile type: Publication
cites work: Optimally scaled matrices
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OpenAlex ID: W122856953
description: scientific article; zbMATH DE number 3883510