An Eccentric Barycentric fixed Point Algorithm

One of the newest areas of mathematical programming, the calculation of fixed points, permits solution of problems unsolvable by earlier methods. The algorithm of this paper, moreover, details two theoretical improvements over previous fixed point algorithms. First, previous algorithms utilize a homotopy structure in which an additional dimension is added to the original problem. This algorithm eliminates the “additional dimension,” thereby simplifying the analysis and easing understanding. Second, previous algorithms require the points generated to conform to a fixed predetermined grid. This algorithm, however, allows the points to be selected flexibly dependent upon the actual function values encountered. The algorithm is therefore adaptive. Hopefully, this adaptive feature will speed convergence. It also suggests a new area of research, that of determining “optimum” adaption functions. To achieve all this, the algorithm utilizes an unconventional modification of the barycentric subdivision in which the points may be selected “off center” (hence the name eccentric). Finally, the algorithm corrects an erroneous procedure often “discovered” by new students, and since it builds from that, this paper may thus also serve as a useful introduction to the field for students.