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A Dynamic Precision Floating-Point Arithmetic Based on the Infinity Computer Framework

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Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

Abstract

We introduce a dynamic precision floating-point arithmetic based on the Infinity Computer. This latter is a computational platform which can handle both infinite and infinitesimal quantities epitomized by the positive and negative finite powers of the symbol , which acts as a radix in a corresponding positional numeral system. The idea is to use the positional numeral system from the Infinity Computer to devise a variable precision representation of finite floating-point numbers and to execute arithmetical operations between them using the Infinity Computer Arithmetics. Here, numerals with negative finite powers of will act as infinitesimal-like quantities whose aim is to dynamically improve the accuracy of representation only when needed during the execution of a computation. An application to the iterative refinement technique to solve linear systems is also presented.

This work was funded by the INdAM-GNCS 2018 Research Project “Numerical methods in optimization and ODEs”.

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Notes

  1. 1.

    For further references and applications see the survey [15].

  2. 2.

    \(\doteq \) denotes the identification operation, so the meaning of remains unaltered.

  3. 3.

    For example, one could take the values \(X^{(k+1)}\), \(Y^{(k+1)}\) and \(Z^{(k+1)}\) as a reference solution with respect to \(X^{(k)}\), \(Y^{(k)}\) and \(Z^{(k)}\), and change the procedure accordingly.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bari, Bari, Italy

    Pierluigi Amodio & Felice Iavernaro

  2. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Firenze, Italy

    Luigi Brugnano

  3. Dipartimento di Informatica, Università di Bari, Bari, Italy

    Francesca Mazzia

Authors
  1. Pierluigi Amodio
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  2. Luigi Brugnano
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  3. Felice Iavernaro
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  4. Francesca Mazzia
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Corresponding author

Correspondence to Felice Iavernaro .

Editor information

Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

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Amodio, P., Brugnano, L., Iavernaro, F., Mazzia, F. (2020). A Dynamic Precision Floating-Point Arithmetic Based on the Infinity Computer Framework. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_22

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  • DOI: https://doi.org/10.1007/978-3-030-40616-5_22

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