A investigation on solving dense systems of linear equations with high performance verified computing on shared memory architectures

In Computer Science the guarantee that an algorithm has been verified to be correct does not guarantees that the numerical result obtained by it is correct. Hence, automatic result verification is an important additional tool to reduce the impact of arithmetic errors in numerical computation. As in traditional computation, one fundamental problem in verified computing is bounding the solution set of systems of linear equations. However, even using highly optimized libraries, solving interval linear systems, that is, those where input data are interval instead of point numbers, stills representing a considerable computational cost when dealing with large dense interval systems. Thus, the use of high performance computing techniques appears as a solution. Several works have been developed along the years using parallel computing techniques on clusters with message passing paradigm. However, many changes have been occurring in high performance computing. Given the number of cores on multicore chips expected to reach tens in a few years, efficient implementations of numerical libraries using shared memory programming models is of urgent interest. In this context, this work presents a revision of high performance verified computing basis, like interval arithmetic, and investigates implementation of the widely used libraries and tools for verified computing. Finally, the paper discusses some issues about implementing an system of linear equations solver using verified computing on shared memory architectures.