Improving the convergence of fixed-point algorithms

Abstract

We consider methods to improve the convergence of fixed-point algorithms using complementary pivoting in triangulations. In terms of measures for triangulation introduced in an earlier paper, we obtain results suggesting that orthogonal projections of regular triangulations of R ⁿ may be preferable to the standard triangulations of an n-simplex. Two special features of the restart method are that the starting point can be controlled and that the computational work of an iteration depends on whether the new vertex lies in the artificial level or not. Refinements of the measures referred to above allow comparisons of triangulations with these features taken into account. The results suggest that it may be worthwhile to use one level of a homotopy-type triangulation in a restart method. We also propose new triangulations taking advantage of the known starting point. An approximate inverse Jacobian is an automatic by-product of the algorithms; this matrix can improve their performance considerably in a number of ways. Finally, we report some computational experience that reinforces most of our conclusions.

This research was supported by National Science Foundation Grant GK-42092, and by a CORE fellowship.