Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications

Constantin Papaodysseus, Dimitris Arabadjis, Fotios Giannopoulos, Athanasios Rafail Mamatsis and Constantinos Chalatsis
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Constantin Papaodysseus: School of Electrical and Computer Engineering, National Technical University of Athens, Iroon Polytechneiou 9, 15780 Athens, Greece
Dimitris Arabadjis: School of Engineering, University of West Attica, Petrou Ralli & Thivon 250 Egaleo, 12241 Athens, Greece
Fotios Giannopoulos: School of Electrical and Computer Engineering, National Technical University of Athens, Iroon Polytechneiou 9, 15780 Athens, Greece
Athanasios Rafail Mamatsis: School of Electrical and Computer Engineering, National Technical University of Athens, Iroon Polytechneiou 9, 15780 Athens, Greece
Constantinos Chalatsis: School of Electrical and Computer Engineering, National Technical University of Athens, Iroon Polytechneiou 9, 15780 Athens, Greece

Mathematics, 2021, vol. 9, issue 11, 1-34

Abstract: 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.

Keywords: 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; loss of significance during multiplication (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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