Abstract: The development of Computer Science has enhanced widely the research capacities in almost all fields in Science. Scientific Computing is an early motivation for the development of digital computers in which the reliability of numerical calculations is absolutely concerned with the finiteness of machine-level computer arithmetic and memory.

The limits of the IEEE754 floating point standard, adopted by most of processor manufacturers, has being recently reviewed (IEEE754-2008), but it has not included a standard for exact real and rational arithmetic
yet. In this context, bad numerical analysis as well as bizarre occurrences of data can jeopardize the reliability of the results in the implementation of a model in a digital computer. Some examples of Floating Point catastrophes are analyzed.

This research work explores the role of three paradigms to develop special purpose processors for scientific computing tasks in integrable spaces (Lp and Sobolev). These paradigms are Type-2 Theory of Effectivity (from Computable Analysis), algebraic models of processors (formal VLSI design methods) and on-line arithmetic (variable precision arithmetic). As a case of study the convolution operation is presented, and a recursive approach is identified with the numerical methods for the solution of ordinary differential equations in Sobolev Spaces.

In this talk, some aspects of the prototype (software and hardware) are presented and its evaluation is developed according to the formal paradigms introduced. Finally, the evaluation of precision improvements has been carried out comparing them with the IEEE754 floating point approach applied to some basic procedures from Physical Oceanography proposed by UNESCO.