Notes on computation of Kakutani fixed points

摘要

This paper explores the matrix constructions concerning a vector-labeling simplicial homotopy algorithm for Kakutani fixed points of upper semicontinuous set-valued self-mappings of the unit simplex Sn. It is shown that the pivoting of the computations is always implemented by postmultiplying the current matrix by a special matrix A = (ai, j), with a1, 1 = 1, ai, j = −1 for i ≥ j ≥ 2, and ai, j = 0 otherwise.The without-exception feasibility of the algorithm as well as the convergence of the computations are set up. In so doing, the J4 triangulation of (0, 1] × Sn with continuous refinement of grid size is developed in detail.