Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications
| Autor: | Dimitris Arabadjis, Fotios Giannopoulos, Constantin Papaodysseus, Constantinos Chalatsis, Athanasios Rafail Mamatsis |
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| Rok vydání: | 2021 |
| Předmět: |
Work (thermodynamics)
Computer science General Mathematics 02 engineering and technology Tracking (particle physics) Operand 01 natural sciences finite precision error in successive multiplications 0203 mechanical engineering Error analysis QA1-939 Computer Science (miscellaneous) exact tracking of round-off error 0101 mathematics Engineering (miscellaneous) finite precision error in a single multiplication multiplication with finite word length Function (mathematics) finite precision error statistical properties of finite precision error 010101 applied mathematics Arbitrarily large 020303 mechanical engineering & transports Multiplication loss of significance during multiplication Algorithm Mathematics |
| Zdroj: | Mathematics Volume 9 Issue 11 Mathematics, Vol 9, Iss 1199, p 1199 (2021) |
| ISSN: | 2227-7390 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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