Abstract
In 1946, von Neumann and his collaborators used a special distribution of random matrices as a model for estimatinga priori the machine precision needed to solve large linear systems. The present paper identifiesisotropy as a group-theoretic property of this distribution, shows that its matrices are almost never ill-conditioned, and explains how to use other isotropically distributed random matrices for testing the accuracy of numerical methods for solving linear systems and associated error diagnostics.
Zusammenfassung
Die zur Lösung linearer Gleichungssysteme benötigte Genauigkeit wurde schon 1946 durch von Neumann und seine Mitarbeiter mittels speziell verteilter Zufallsmatrizen geschätzt. In der vorliegenden Arbeit erscheint dieIsotropie als gruppentheoretische Eigenschaft dieser Verteilung. Ferner wird gezeigt, dass die Zufallsmatrizen fast nie schlecht konditioniert sind. Schliesslich diskutieren die Autoren die Verwendung anderer isotrop verteilter Zufallsmatrizen zur Prüfung von Genauigkeit und a-priori-Fehlerschranken bei Algorithmen zur Lösung linearer Gleichungssysteme.
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Dedicated to Eduard Stiefel
The main results of this paper were reported in [3], and communicated in 1976 to those working on the LINPACK project.
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Birkhoff, G., Gulati, S. Isotropic distributions of test matrices. Journal of Applied Mathematics and Physics (ZAMP) 30, 148–158 (1979). https://doi.org/10.1007/BF01601929
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DOI: https://doi.org/10.1007/BF01601929