Adaptable Arithmetic: A New Approach to Numerical Computations

Title: Adaptable Arithmetic: A New Approach to Numerical Computations

Research Question: How can we develop a new approach to numerical computations that dynamically changes the precision of each intermediate operation to deliver the target accuracy, while ensuring the accuracy and reliability of the results?

Methodology: The authors present a new adaptable numeric core inspired both from floating point expansions and on-line arithmetic. This core is cut down to four tools: a tool for arithmetic operations, a tool for comparisons, a tool for function evaluations, and a tool for root finding. The authors prove the correctness of the tool that contains arithmetic operations, and these proofs are formally checked by the Coq assistant.

Results: The authors develop a new adaptable arithmetic approach that dynamically changes the precision of each intermediate operation to deliver the target accuracy. They provide examples from four fields of application to illustrate the new properties and goals. They also prove the correctness of their approach and formally verify many results published in the literature.

Implications: This work may allow users to develop application-specific adaptable libraries based on the toolset or write new formal proofs based on the set of validated facts. It may also lead to the development of new algorithms that reach the accuracy aimed by the user without the need for a priori precision setting. Furthermore, it may contribute to the improvement of the accuracy and reliability of numerical computations in various fields.

Link to Article: https://arxiv.org/abs/0107025v2 Authors: arXiv ID: 0107025v2