An evaluation of four reordering algorithms to reduce the computational cost of the Jacobi-preconditioned Conjugate Gradient Method using high-precision arithmetic

Abstract

In this work, four heuristics for bandwidth and profile reductions are evaluated. Specifically, the results of a recent proposed heuristic for bandwidth and profile reductions of symmetric and asymmetric matrices using a one-dimensional self-organising map is evaluated against the results obtained from the variable neighbourhood search for bandwidth reduction heuristic, the original reverse Cuthill-McKee method, and the reverse Cuthill-McKee method with starting pseudo-peripheral vertex given by the George-Liu algorithm. These four heuristics were applied to three datasets of linear systems composed of sparse symmetric positive-definite matrices arising from discretisations of the heat conduction and Laplace equations by finite volumes. The linear systems are solved by the Jacobi-preconditioned conjugate gradient method when using high-precision numerical computations. The best heuristic in the simulations performed with one of the datasets used was the Cuthill-McKee method with starting pseudo-peripheral vertex given by the George-Liu algorithm. On the other hand, no gain was obtained in relation to the computational cost of the linear system solver when a heuristic for bandwidth and profile reduction is applied to instances contained in two of the datasets used.