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Accurate arithmetic results for decimal data on non-decimal computers

Genaue arithmetische Ergebnisse für dezimale Daten auf nicht-dezimalen Computern

Abstract

Recently, techniques have been devised and implemented which permit the computation of smallest enclosing machine number interval for the exact results of a good number of highly composite operations. These exact results refer, however, to the data as they are represented in the computer. This note shows how the conversion of decimal data into non-decimal representations may be joined with the mathematical operation on the data into one high-accuracy algorithm. Such an algorithm is explicitly presented for the solution of systems of linear equations.

Zusammenfassung

In letzter Zeit wurde eine Methodik entwickelt und implementiert, die die Erzeugung kleinstmöglicher Maschinenzahl-Intervalle für die genauen Ergebnisse verschiedenster vielstufiger mathematischer Operationen gestattet. Diese genauen Ergebnisse beziehen sich jedoch auf die Daten so wie sie im Computer dargestellt sind. In der vorliegenden Note wird gezeigt, wie die Konversion dezimaler Daten in nicht-dezimale Darstellungen mit der mathematischen Operation an den Daten in einen einzigen hoch-genauen Algorithmus zusammengefügt werden kann. Für die Lösung von Systemen linearer Gleichungen wird ein solcher Algorithmus im Detail vorgestellt.

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References

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Dedicated to Professor R. Albrecht on the occasion of his 60th birthday

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Auzinger, W., Stetter, H.J. Accurate arithmetic results for decimal data on non-decimal computers. Computing 35, 141–151 (1985). https://doi.org/10.1007/BF02260501

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Keywords

  • Linear Equation
  • Computational Mathematic
  • Exact Result
  • Good Number
  • Mathematical Operation