Abstract
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 {textcircled {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.
Highlights
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))
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
Summary
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)). 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).
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