Synthesis of a class of potential function algorithms

The family of one-dimensional mean iterative processes is modeled by the algorithm schema zi+1 = zi + ??i (yi-zi), where {yi} and {zi} are sequences in a linear vector space and {??i} is a sequence in E. Based on general properties of this schema it is possible to define the initial phases of a step-by-step approach to the synthesis of a class of fixed point algorithms. Relatively simple conditions have been derived that guide the practitioner through each of these phases. The benefit of this procedure is that each phase consists of demonstrating a particular necessary property of a convergent algorithm. For the class of fixed point algorithms that are also potential function algorithms, we present additional phases of the synthesis process, and the corresponding conditions, that if satisfied, demonstrate algorithm convergence. Hence, for potential function algorithms the task of synthesis is subdivided into manageable pieces: the failure of a proposed algorithm to satisfy any specific condition is then easily addressed, either through modification or through choice of a different algorithm.