Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications
In the present paper, a novel approach is introduced for the study, estimation and exact tracking of the finite precision error generated and accumulated during any number of multiplications. It is shown that, as a rule, this operation is very “toxic”, in the sense that it may force the finite preci...
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| Format: | Article |
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MDPI AG
2021-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/11/1199 |
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| author | Constantin Papaodysseus Dimitris Arabadjis Fotios Giannopoulos Athanasios Rafail Mamatsis Constantinos Chalatsis |
| author_facet | Constantin Papaodysseus Dimitris Arabadjis Fotios Giannopoulos Athanasios Rafail Mamatsis Constantinos Chalatsis |
| author_sort | Constantin Papaodysseus |
| collection | DOAJ |
| description | In the present paper, a novel approach is introduced for the study, estimation and exact tracking of the finite precision error generated and accumulated during any number of multiplications. It is shown that, as a rule, this operation is very “toxic”, in the sense that it may force the finite precision error accumulation to grow arbitrarily large, under specific conditions that are fully described here. First, an ensemble of definitions of general applicability is given for the rigorous determination of the number of erroneous digits accumulated in any quantity of an arbitrary algorithm. Next, the exact number of erroneous digits produced in a single multiplication is given as a function of the involved operands, together with formulae offering the corresponding probabilities. In case the statistical properties of these operands are known, exact evaluation of the aforementioned probabilities takes place. Subsequently, the statistical properties of the accumulated finite precision error during any number of successive multiplications are explicitly analyzed. A method for exact tracking of this accumulated error is presented, together with associated theorems. Moreover, numerous dedicated experiments are developed and the corresponding results that fully support the theoretical analysis are given. Eventually, a number of important, probable and possible applications is proposed, where all of them are based on the methodology and the results introduced in the present work. The proposed methodology is expandable, so as to tackle the round-off error analysis in all arithmetic operations. |
| first_indexed | 2024-03-10T11:03:26Z |
| format | Article |
| id | doaj.art-686c8de3d0644948ac1570731bfee69e |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T11:03:26Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-686c8de3d0644948ac1570731bfee69e2023-11-21T21:19:29ZengMDPI AGMathematics2227-73902021-05-01911119910.3390/math9111199Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of MultiplicationsConstantin Papaodysseus0Dimitris Arabadjis1Fotios Giannopoulos2Athanasios Rafail Mamatsis3Constantinos Chalatsis4School of Electrical and Computer Engineering, National Technical University of Athens, Iroon Polytechneiou 9, 15780 Athens, GreeceSchool of Engineering, University of West Attica, Petrou Ralli & Thivon 250 Egaleo, 12241 Athens, GreeceSchool of Electrical and Computer Engineering, National Technical University of Athens, Iroon Polytechneiou 9, 15780 Athens, GreeceSchool of Electrical and Computer Engineering, National Technical University of Athens, Iroon Polytechneiou 9, 15780 Athens, GreeceSchool of Electrical and Computer Engineering, National Technical University of Athens, Iroon Polytechneiou 9, 15780 Athens, GreeceIn the present paper, a novel approach is introduced for the study, estimation and exact tracking of the finite precision error generated and accumulated during any number of multiplications. It is shown that, as a rule, this operation is very “toxic”, in the sense that it may force the finite precision error accumulation to grow arbitrarily large, under specific conditions that are fully described here. First, an ensemble of definitions of general applicability is given for the rigorous determination of the number of erroneous digits accumulated in any quantity of an arbitrary algorithm. Next, the exact number of erroneous digits produced in a single multiplication is given as a function of the involved operands, together with formulae offering the corresponding probabilities. In case the statistical properties of these operands are known, exact evaluation of the aforementioned probabilities takes place. Subsequently, the statistical properties of the accumulated finite precision error during any number of successive multiplications are explicitly analyzed. A method for exact tracking of this accumulated error is presented, together with associated theorems. Moreover, numerous dedicated experiments are developed and the corresponding results that fully support the theoretical analysis are given. Eventually, a number of important, probable and possible applications is proposed, where all of them are based on the methodology and the results introduced in the present work. The proposed methodology is expandable, so as to tackle the round-off error analysis in all arithmetic operations.https://www.mdpi.com/2227-7390/9/11/1199finite precision error in a single multiplicationfinite precision error in successive multiplicationsexact tracking of round-off errorfinite precision errormultiplication with finite word lengthstatistical properties of finite precision error |
| spellingShingle | Constantin Papaodysseus Dimitris Arabadjis Fotios Giannopoulos Athanasios Rafail Mamatsis Constantinos Chalatsis Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications Mathematics finite precision error in a single multiplication finite precision error in successive multiplications exact tracking of round-off error finite precision error multiplication with finite word length statistical properties of finite precision error |
| title | Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications |
| title_full | Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications |
| title_fullStr | Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications |
| title_full_unstemmed | Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications |
| title_short | Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications |
| title_sort | analysis evaluation and exact tracking of the finite precision error generated in arbitrary number of multiplications |
| topic | finite precision error in a single multiplication finite precision error in successive multiplications exact tracking of round-off error finite precision error multiplication with finite word length statistical properties of finite precision error |
| url | https://www.mdpi.com/2227-7390/9/11/1199 |
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