Numerical Algorithmic Science and Engineering within Computer Science: Rationale, Foundations and Organization

Abstract: A re-calibration is proposed for "numerical analysis" as it arises specifically within the broader, embracing field of modern computer science (CS). This would facilitate research into theoretical and practicable models of real-number computation at the foundations of CS, and it would also advance the instructional objectives of the CS field. Our approach is premised on the key observation that the great "watershed" in numerical computation is much more between finite- and infinite-dimensional numerical problems than it is between discrete and continuous numerical problems. A revitalized discipline for numerical computation within modern CS can more accurately be defined as "numerical algorithmic science & engineering (NAS&E), or more compactly, as "numerical algorithmics," its focus being the algorithmic solution of numerical problems that are either discrete, or continuous over a space of finite dimension, or a combination of the two. It is the counterpart within modern CS of the numerical analysis discipline, whose primary focus is the algorithmic solution of continuous, infinite-dimensional numerical problems and their finite-dimensional approximates, and whose specialists today have largely been repatriated to departments of mathematics. Our detailed overview of NAS&E from the viewpoints of rationale, foundations, and organization is preceded by a recounting of the role played by numerical analysts in the evolution of academic departments of computer science, in order to provide background for NAS&E and place the newly-emerging discipline within its larger historical context.