Structured perturbations and symmetric matrices

Link:
Autor/in:
Verlag/Körperschaft:
Hamburg University of Technology
Erscheinungsjahr:
2000
Medientyp:
Text
Schlagworte:
  • Condition number
  • Structured perturbations
  • Symmetric matrices
  • 510: Mathematik
Beschreibung:
  • For a given n × n matrix the ratio between the componentwise distance to the nearest singular matrix and the inverse of the optimal Bauer-Skeel condition number cannot be larger than (3 + 2√2)n. In this note a symmetric matrix is presented where the described ratio is equal to n for the choice of most interest in numerical computation, for relative perturbations of the individual matrix components. It is shown that a symmetric linear system can be arbitrarily ill-conditioned, while any symmetric and entrywise relative perturbation of the matrix of less than 100% does not produce a singular matrix. That means that the inverse of the condition number and the distance to the nearest ill-posed problem can be arbitrarily far apart. Finally we prove that restricting structured perturbations to symmetric (entrywise) perturbations cannot change the condition number by more than a factor (3 + 2\√2)n. © 1998 Elsevier Science Inc. All rights reserved.
Beziehungen:
DOI 10.1016/S0024-3795(97)10078-7
Quellsystem:
TUHH Open Research
Interne Metadaten
Quelldatensatz
oai:tore.tuhh.de:11420/9440