On a two-phase iterative-sandwich algorithm for improved polynomial approximation by a Modified Lupas Operator using statistical perspectives

摘要

This paper aims at constructing a two-phase iterative computerizable numerical algorithm for an improved approximation by 'Modified Lupas' operator. The algorithm uses the 'statistical perspectives' for exploiting the information about the unknown function 'f' available in terms of its known values at the 'equidistant-knots' in C[0,1] more fully. The improvement, achieved by an aposteriori use of this information, happens iteratively. Any typical iteration uses the concepts of 'Mean Square Error (MSE)' and 'Bias' ; the application of the former being preceded by that of the latter in the algorithm. At any iteration, the statistical concept of 'MSE' is used in "Phase II", after that of the 'Bias' in ?Phase I?. Like a 'Sandwich', the top and bottom-breads are the operations of 'Bias-Reduction' per the ?Phase I? of our algorithm, and the operation of 'MSE-Reduction' per the ?Phase II? is the stuffing in the sandwich. The algorithm is an iterative one amounting to a desired-height 'Docked-Pile' of sandwiches with the bottom?bread of the first iteration serving as the top-bread for the second-iteration sandwich, and so-on-and-so forth. The potential of the achievable improvements through the proposed 'computerizable numerical iterative algorithm' is illustrated per an 'empirical study' for which the function 'f' is assumed to be known in the sense of simulation. The illustration has been confined to ?Three Iterations? only, for the sake of simplicity of the illustration.