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Coarse-grain parallelizations of interval algorithms decomposing dense matrices and solving triangular systems on multicomputers

Grobkörnige Parallelisierungen von Intervallalgorithmen zur Faktorisierung vollbesetzter Matrizen und zur Lösung von Dreieckssystemen auf Multicomputern

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Abstract

The evaluation of scalar products with maximum accuracy plays an important role in computing inclusions for the solutions of linear systems. In this paper, we discuss this operation within the context of parallel algorithms for distributed-memory systems (multicomputers). We describe new variants for solving triangular systems of linear equations and for computing the LU factorization of matrices under the assumption that scalar products are implemented as single, indivisible operations and that no processor works on different scalar products simultaneously. All algorithms work in the real and interval case; the theoretical results are supplemented by measurements obtained from a transputer network.

Zusammenfassung

Die Auswertung von Skalarprodukten mit maximaler Genauigkeit spielt eine wichtige Rolle bei der Berechnung von Lösungseinschließungen für lineare Gleichungssysteme. In dieser Arbeit diskutieren wir diese Operation im Zusammenhang mit parallelen Algorithmen für speicherentkoppelte Systeme (Multicomputer). Wir beschreiben neue Varianten zur Lösung linearer Dreieckssysteme und zur Berechnung der LU-Zerlegung unter der Annahme, daß Skalarprodukte als unteilbare Operationen implementiert sind und daß kein Prozessor an mehreren Skalarprodukten gleichzeitig arbeitet. Alle Algorithmen sind für Punkt- und Intervallprobleme anwendbar; die theoretischen Resultate werden durch Messungen auf einem Transputernetzwerk bestätigt.

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  1. Institut für Informatik, Universität Basel, Mittlere Strasse 142, CH-4056, Basel

    R. Reith &  C. P. Ullrich

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  1. R. Reith
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  2. C. P. Ullrich
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Reith, R., Ullrich, C.P. Coarse-grain parallelizations of interval algorithms decomposing dense matrices and solving triangular systems on multicomputers. Computing 53, 243–257 (1994). https://doi.org/10.1007/BF02307377

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