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Asymptotic distribution of global errors in the numerical computations of dynamical systems

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Title: Asymptotic distribution of global errors in the numerical computations of dynamical systems
Authors: Turchetti, G.1, Vaienti, S.2, Zanlungo, F.3,4 francesco.zanlungo@gmail.com
Source: Physica A. Nov2010, Vol. 389 Issue 21, p4994-5006. 13p.
Subjects: Asymptotic distribution, Error analysis in mathematics, Numerical analysis, Discrete-time systems, Perturbation theory, Mathematical models
Abstract: Abstract: We propose an analysis of the effects introduced by finite-accuracy and round-off arithmetic on numerical computations of discrete dynamical systems. Our method, which uses the statistical tool of the decay of fidelity, computes the error by directly comparing the numerical orbit with the exact one (or, more precisely, with another numerical orbit computed with a much higher accuracy). Furthermore, as a model of the effects of round-off arithmetic on the map, we also consider a random perturbation of the exact orbit with an additive noise, for which exact results can be obtained for some prototype maps. We investigate the decay laws of fidelity and their relationship with the error probability distribution for regular and chaotic maps, for both additive and numerical noise. In particular, for regular maps we find an exponential decay for additive noise, and a power-law decay for numerical noise. For chaotic maps, numerical noise is equivalent to additive noise, and our method is suitable for identifying a threshold for the reliability of numerical results, i.e., the number of iterations below which global errors can be ignored. This threshold grows linearly with the number of bits used to represent real numbers. [ABSTRACT FROM AUTHOR]
Copyright of Physica A is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Asymptotic distribution of global errors in the numerical computations of dynamical systems
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  Data: <searchLink fieldCode="AR" term="%22Turchetti%2C+G%2E%22">Turchetti, G.</searchLink><relatesTo>1</relatesTo><br /><searchLink fieldCode="AR" term="%22Vaienti%2C+S%2E%22">Vaienti, S.</searchLink><relatesTo>2</relatesTo><br /><searchLink fieldCode="AR" term="%22Zanlungo%2C+F%2E%22">Zanlungo, F.</searchLink><relatesTo>3,4</relatesTo><i> francesco.zanlungo@gmail.com</i>
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  Data: <searchLink fieldCode="JN" term="%22Physica+A%22">Physica A</searchLink>. Nov2010, Vol. 389 Issue 21, p4994-5006. 13p.
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  Data: <searchLink fieldCode="DE" term="%22Asymptotic+distribution%22">Asymptotic distribution</searchLink><br /><searchLink fieldCode="DE" term="%22Error+analysis+in+mathematics%22">Error analysis in mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Discrete-time+systems%22">Discrete-time systems</searchLink><br /><searchLink fieldCode="DE" term="%22Perturbation+theory%22">Perturbation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+models%22">Mathematical models</searchLink>
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  Data: Abstract: We propose an analysis of the effects introduced by finite-accuracy and round-off arithmetic on numerical computations of discrete dynamical systems. Our method, which uses the statistical tool of the decay of fidelity, computes the error by directly comparing the numerical orbit with the exact one (or, more precisely, with another numerical orbit computed with a much higher accuracy). Furthermore, as a model of the effects of round-off arithmetic on the map, we also consider a random perturbation of the exact orbit with an additive noise, for which exact results can be obtained for some prototype maps. We investigate the decay laws of fidelity and their relationship with the error probability distribution for regular and chaotic maps, for both additive and numerical noise. In particular, for regular maps we find an exponential decay for additive noise, and a power-law decay for numerical noise. For chaotic maps, numerical noise is equivalent to additive noise, and our method is suitable for identifying a threshold for the reliability of numerical results, i.e., the number of iterations below which global errors can be ignored. This threshold grows linearly with the number of bits used to represent real numbers. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Physica A is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.physa.2010.06.060
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        Text: English
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        Type: general
      – SubjectFull: Error analysis in mathematics
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      – SubjectFull: Numerical analysis
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      – SubjectFull: Discrete-time systems
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      – SubjectFull: Mathematical models
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              Text: Nov2010
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