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

Amodio, Pierluigi; Brugnano, Luigi; Iavernaro, Felice; Mazzia, Francesca

doi:10.1007/978-3-030-40616-5_22

“…and analogously for the division X/Y . The same subclass of grossnumbers has been used in [49,36] to implement a dynamic precision floating point arithmetic. For the moment, the Matlab class has been implemented without using the vectorization facility, thus requiring a loop for vectorial functions.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

Computation of higher order Lie derivatives on the Infinity Computer

Iavernaro

¹

,

Mazzia

²

,

Mukhametzhanov

³

et al. 2021

Journal of Computational and Applied Mathematics

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“…In Sect. 4 we will illustrate this procedure applied to the accurate determination of zeros of functions [a further example may be found in Amodio et al (2020)].…”

Section: Stepsmentioning

confidence: 99%

“…One interesting application is the possibility of handling ill-conditioned problems or even of implementing algorithms which are labeled as unstable in standard floating-point arithmetic. 1 One example in this direction has been illustrated in Amodio et al (2020). It consists in the use of the iterative refinement to improve the accuracy of a computed solution to an ill-conditioned linear system until a prescribed input accuracy is achieved.…”

mentioning

confidence: 99%

On the use of the Infinity Computer architecture to set up a dynamic precision floating-point arithmetic

Amodio

¹

,

Brugnano

²

,

Iavernaro

³

et al. 2020

Self Cite

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We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix ①. The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented. Keywords Infinity Computer • Dynamic precision floating-point arithmetic • Conditioning 1 Introduction The Arithmetic of Infinity was introduced by Y.D. Sergeyev with the aim of devising a new coherent computational environment able to handle finite, infinite and infinitesimal quantities, and to execute arithmetical operations with them. It is based on a positional numeral system with the infinite radix ①, called grossone and representing, by definition, the number of elements of the set of natural numbers N (see, for example, Sergeyev (2008, 2009) and the survey paper Sergeyev (2017)). Similar to the standard positional notation Communicated by Yaroslav D. Sergeyev.

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“…Among the many fields of research this new methodology has been successfully applied, we mention numerical differentiation and optimization [5,14,22], numerical solution of differential equations [18,2,19,12,7], models for percolation and biological processes [20,9], cellular automata [10,4]. 1 The aim of the present study is to devise a dynamic precision floating-point arithmetic by exploiting the computational platform provided by the Infinity Computer. In contrast with standard variable precision arithmetics, here not only may the accuracy be dynamically changed during the execution of a given algorithm, but variables stored with different accuracies may be combined through the usual algebraic operations.…”

Section: Introductionmentioning

confidence: 99%

“…One interesting application is the possibility of handling ill-conditioned problems or even of implementing algorithms which are labeled as unstable in standard floating-point arithmetic. 2 One example in this direction has been illustrated in [1]. It consists in the use of the iterative refinement to improve the accuracy of a computed solution to an ill-conditioned linear system until a prescribed input accuracy is achieved.…”

Section: Introductionmentioning

confidence: 99%

On the use of the Infinity Computer architecture to set up a dynamic precision floating-point arithmetic

Amodio,

Brugnano,

Iavernaro

et al. 2020

Preprint

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0

View full text Add to dashboard Cite

We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities symbolized by the positive and negative finite powers of the radix ①. The computational features offered by the Infinity Computer allows us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.

show abstract

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

Abstract: La pubblicazione è resa disponibile sotto le norme e i termini della licenza di deposito, secondo quanto stabilito dalla Policy per l'accesso aperto dell'Università degli Studi di Firenze (https://www.sba.unifi.it/upload/policy-oa-2016-1.pdf)

Cited by 4 publications

References 14 publications

Computation of higher order Lie derivatives on the Infinity Computer

Computation of higher order Lie derivatives on the Infinity Computer

On the use of the Infinity Computer architecture to set up a dynamic precision floating-point arithmetic

On the use of the Infinity Computer architecture to set up a dynamic precision floating-point arithmetic

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