The preconditioned conjugate gradient method on the connection machine. (English) Zbl 0725.65033

Tong, Charles

Int. J. High Speed Comput. 1, No. 2, 263-288 (1989).

This is a report on numerical experiments where more than ten preconditioners were compared in their performance on a quarter of the connection machine (which means \(2^{14}\) processors, connected in hypercube mode). The problem is the solution of the 5-point approximation to the Poisson and the anisotropic Poisson equation in the unit square; by computing time the simple 4-step Jacobi preconditioner turned out to be best. Interestingly, the unpreconditioned conjugate gradient method needed less than twice as much time.
There is a broad discussion of several aspects of the experiments, and a list of computing times measured for basic arithmetical operations (showing rather unexpected time relations, from a sequential point of view).

Reviewer: G.Stoyan (Budapest)

MSC:

65F10	Iterative numerical methods for linear systems
65F35	Numerical computation of matrix norms, conditioning, scaling
65N22	Numerical solution of discretized equations for boundary value problems involving PDEs
65Y05	Parallel numerical computation
65N06	Finite difference methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

incomplete LU factorization; symmetric successive overrelaxation; diagonal natural ordering; parallel red/black ordering; parallel computation; parallel operations; communications; numerical experiments; performance; connection machine; Poisson equation; preconditioner; conjugate gradient method; computing times