Numerical Non-Archimedean Calculus: Applications, Software Implementation and Preliminary Hardware Design

The purpose of this work is to pioneer the use of non-Archimedean models in application-oriented numerical computations and, in particular, in optimization. Roughly speaking, non-Archimedean models are those mathematical theories where ordered fields containing infinite, finite and infinitesimal quantities exist. Moving from theory to practice requires an investigation which recalls and somehow goes beyond the one which involved real numbers. First, it demands to clearly identify the non-Archimedean model to target (as opposed to reals which are unique), and to propose a finite-length binary encoding for its numbers to be used within computers (reals have many alternatives given by IEEE754 standard and not only, e.g., the binary32 floating point numbers). Then, it is mandatory to develop a software library to numerically simulate operations between non-Archimedean values, as well as to design the circuitry, i.e., a non-Archimedean CPU and FPU needed to execute them in hardware in case the approach would become pervasive. Finally, the whole investigation must be justified by the identification of pros in using non-Archimedean numbers in place of reals, and the existence of concrete applications, e.g., in the industrial domain which may benefit from these positive features. The goal of this work is precisely to provide a preliminary answer to each of the previous interrogatives, also conducting experimental studies.