Locating and computing in parallel all the simple roots of special functions using PVM

摘要

An algorithm is proposed for locating and computing in parallel and with certainty all the simple roots of any twice continuously differentiable function in any specific interval. To compute with certainty all the roots, the proposed method is heavily based on the knowledge of the total number of roots within the given interval. To obtain this information we use results from topological degree theory and, in particular, the Kronecker–Picard approach. This theory gives a formula for the computation of the total number of roots of a system of equations within a given region, which can be computed in parallel. With this tool in hand, we construct a parallel procedure for the localization and isolation of all the roots by dividing the given region successively and applying the above formula to these subregions until the final domains contain at the most one root. The subregions with no roots are discarded, while for the rest a modification of the well-known bisection method is employed for the computation of the contained root. The new aspect of the present contribution is that the computation of the total number of zeros using the Kronecker–Picard integral as well as the localization and computation of all the roots is performed in parallel using the parallel virtual machine (PVM). PVM is an integrated set of software tools and libraries that emulates a general-purpose, flexible, heterogeneous concurrent computing framework on interconnected computers of varied architectures. The proposed algorithm has large granularity and low synchronization, and is robust. It has been implemented and tested and our experience is that it can massively compute with certainty all the roots in a certain interval. Performance information from massive computations related to a recently proposed conjecture due to Elbert (this issue, J. Comput. Appl. Math. 133 (2001) 65–83) is reported.