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Numerical Algorithms for the Parametric Continuation of Stiff ODEs Deriving from the Modeling of Combustion with Detailed Chemical Mechanisms Read chapter Read first chapter

2020 | OriginalPaper | Chapter

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

Authors : Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro, Francesca Mazzia

Published in: Numerical Computations: Theory and Algorithms

Publisher: Springer International Publishing

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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 https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-40616-5_22/495240_1_En_22_IEq1_HTML.gif , 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 https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-40616-5_22/495240_1_En_22_IEq2_HTML.gif 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.

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Footnotes
1
For further references and applications see the survey [15].
 
2
\(\doteq \) denotes the identification operation, so the meaning of https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-40616-5_22/495240_1_En_22_IEq24_HTML.gif remains unaltered.
 
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.
 
Literature
1.
Amodio, P., Iavernaro, F., Mazzia, F., Mukhametzhanov, M.S., Sergeyev, Y.D.: A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic. Math. Comput. Simul. 141, 24–39 (2016)MathSciNetCrossRef
2.
Bailey, D.H., Borwein, J.M.: High-precision arithmetic in mathematical physics. Mathematics 3(2), 337–367 (2015)CrossRef
3.
4.
De Cosmis, S., Leone, R.D.: The use of grossone in mathematical programming and operations research. Appl. Math. Comput. 218(16), 8029–8038 (2012)MathSciNetMATH
5.
Gaudioso, M., Giallombardo, G., Mukhametzhanov, M.S.: Numerical infinitesimals in a variable metric method for convex nonsmooth optimization. Appl. Math. Comput. 318, 312–320 (2018)MATH
6.
7.
Iavernaro, F., Mazzia, F., Trigiante, D.: Stability and conditioning in numerical analysis. J. Numer. Anal. Ind. Appl. Math. 1(1), 91–112 (2006)MathSciNetMATH
8.
Moler, C.B.: Iterative refinement in floating point. J. ACM (JACM) 14(2), 316–321 (1967)CrossRef
9.
Sergeyev, Y.D.: Solving ordinary differential equations by working with infinitesimals numerically on the infinity computer. Appl. Math. Comput. 219(22), 10668–10681 (2013)MathSciNetMATH
10.
Sergeyev, Y.D., Mukhametzhanov, M.S., Mazzia, F., Iavernaro, F., Amodio, P.: Numerical methods for solving initial value problems on the infinity computer. Int. J. Unconvent. Comput. 12(1), 3–23 (2016)
11.
Sergeyev, Y.D.: A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica 19(4), 567–596 (2008)MathSciNetMATH
12.
Sergeyev, Y.D.: Higher order numerical differentiation on the infinity computer. Optim. Lett. 5(4), 575–585 (2011)MathSciNetCrossRef
13.
Sergeyev, Y.D., Kvasov, D., Mukhametzhanov, M.S.: On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales. Commun. Nonlinear Sci. Numer. Simul. 59, 319–330 (2018)MathSciNetCrossRef
14.
Sergeyev, Y.D.: Numerical computations and mathematical modelling with infinite and infinitesimal numbers. J. Appl. Math. Comput. 29(1–2), 177–195 (2009)MathSciNetCrossRef
15.
Sergeyev, Y.D.: Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems. EMS Surv. Math. Sci. 4(2), 219–320 (2017)MathSciNetCrossRef
16.
Sergeyev, Y.D.: Computer system for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them. USA patent 7,860,914 (2010)
17.
Žilinskas, A.: On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions. Appl. Math. Comput. 218(16), 8131–8136 (2012)MathSciNetMATH
Metadata
Title
A Dynamic Precision Floating-Point Arithmetic Based on the Infinity Computer Framework
Authors
Pierluigi Amodio
Luigi Brugnano
Felice Iavernaro
Francesca Mazzia
Copyright Year
2020
Book
Numerical Computations: Theory and Algorithms

Print ISBN: 978-3-030-40615-8

Electronic ISBN: 978-3-030-40616-5

Copyright Year: 2020

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

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