Parallel all-row preconditioned interval linear solver for nonlinear equations on multiprocessors

Abstract

Interval Newton methods in conjunction with generalized bisection form the basis of algorithms that find all real roots within a specified X ⊂ R n of a system of nonlinear equations F( X) = 0 with mathematical certainty, even in finite-precision arithmetic. One of the major computational cost in such methods is solving a correspondent linear interval system. This paper proposes parallel implementations of the all-row preconditioned interval linear solver on two multiprocessor architectures: shared bus multiprocessor and hypercube connected multiprocessor. Efficient parallel algorithms that are specifically tailored for these architectures are presented. The algorithms effectively use the hardware features to minimize communication overheads. Performance evaluation is carried out by means of actual measurements of the algorithms running on real parallel computers. Our numerical results show that data partitioning, data allocation and processor scheduling play important roles in the performance of the parallel computation. In both systems, significant speedup can be obtained by properly overlapping computation and communication.