Keywords: evolutionary, iterative, self-adapt, successive over-relaxation, relaxation factors, solution vector, restarting strategy, preconditioned conjugate gradient, generalized minimal residual.

By applying evolutionary computation techniques to existing iterative equation-solving methods, this paper developed a self-adapt Successive Over Relaxation (SOR) solver for both symmetric and non-symmetric linear equation systems. The solver employed two stochastic relaxation factors, evolving them themselves according to the size of their residual vectors, and avoiding the problem of the choice of initial guess or expensive calculation on the optimal relaxation factors a priori.

In this research, an innovative SOR algorithm was developed and coded. Various initial values of relaxation factors were examined and stability analysis indicated that any value outside (0, 2) would be unstable. The results also showed that most matrices tested converged with the final optimal value of the relaxation factor locating in the interval of (1.25, 1.75).

The frequency of parameters exchange is an influential factor in the self-adapt SOR method. This paper found that the more frequently the programme exchanges the relaxation parameters, the less overall computing time it requires. However, if the frequency is too high the current solution vector X is more likely to overflow. Based on the outcome, a rough guide for the lowest frequency is approximately size/300 of the matrix. In other words, the programme is initially set to exchange the relaxation parameters every size/300 loops.

Due to occasional overflow faults and the use of stochastic parameters, the solver was prone to break down in some cases. Thus restarting strategy was designed and made the programme 'smart' and capable to be implemented in a complicated environment.

A number of numerical experiments have been carried out on a number of test matrices produced by artificial matrix generators, 2D and 3D saturated soil dynamic finite element programs. The size of the chosen test matrices ranged from 48 to 1,000,000, symmetric and non-symmetric, positive definite and non-definite, mildly to highly non-definite matrices.

A classical SOR solver and a Preconditioned Conjugate Gradient (PCG) solver were used to compare with the new SOR solver. The numerical tests indicated that this solver outperformed the classical SOR solver for the large matrices tested, while for most symmetric matrices the PCG solver had advantage in terms of running time.

Future work will involve the speed-ups of the SOR solver using grid computing techniques and combination with other solvers, such like PCG, Generalized Minimal Residual (GMRES).

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £75 +P&P)