The convergence of the iteration can be improved by the method of Gauss-Seidel iteration. Its basic idea is to use the new iterated values immediately when they are available, and not to postpone their usage until the next iteration step. Consider the calculation of in the normal iteration:

During the calculation of , values have already been calculated, so they can be used instead of their previous value, modifying the iteration, thus:

(recall that in the radiosity equation).

A trick, called **successive relaxation**,
can further improve the speed of convergence. Suppose that during the
*m*th step of the iteration the radiosity vector was
computed. The difference from the previous estimate is:

showing the magnitude of difference, as well as the direction of the
improvement in *N* dimensional space. According to practical
experience, the direction is quite accurate, but the magnitude is
underestimated, requiring the correction by a relaxation factor :

The determination of is a crucial problem. If it is too small, the convergence will be slow; if it is too great, the system will be unstable and divergent. For many special matrices, the optimal relaxation factors have already been determined, but concerning our radiosity matrix, only practical experiences can be relied on. Cohen [CGIB86] suggests that relaxation factor 1.1 is usually satisfactory.

Mon Oct 21 14:07:41 METDST 1996