A Two-phase Iterative Algorithm for Improved Approximation by Szasz Operator Using Statistical Perspectives

Ashok Sahai,Robin Antoine,Peter S Chami

doi:10.5539/jmr.v2n2p123

Abstract

This paper aims at constructing a two-phase iterative numerical algorithm for the improved approximation of a continuous function by the ‘Modified Szasz’ operator. The algorithm uses a ‘statistical perspective’ to more fully expoit the information about the unknown function f . The improvement occurs iteratively. A typical iteration uses the twin statistical concepts of ‘Mean Square Error’ (MSE) and ‘Bias’; the application of the latter concept being preceded by that of the former in the algorithm. At any iteration, the statistical concept of ‘MSE’ is used in “Phase II”, after that of the ‘Bias’ in “Phase I”. The procedure is like a sandwich. The top and bottom slices are the operations of ‘Bias-Reduction’ in “Phase I” of the algorithm, and the operation of ‘MSE-Reduction’ in “Phase II” is the stuffing in the sandwich. The improvement acheived by this algorithm is evaluated by means of a simulation study using known functions. The simulation has been confined to three iterations only, for the sake of simplicity.

Highlights

Szasz(1950) proposed the following generalization of the well-known Bernstein polynomial approximation operator extending it to an infinite interval
To illustrate the gain in efficiency of the “Modified Szasz Operators” by using our proposed “Sandwich-Iterative Algorithm of Improvement of Polynomial Approximation, we have carried out an empirical study
The Percentage Relative Gains (PRGs) due to the use of our proposed Algorithmic SandwichIterative Polynomial Approximations vis-a-vis the Original Modified Szasz Polynomial Approximation are PROGRESSIVELY increasing on each subsequent iteration, for all the illustrative functions

Summary

Introduction

Szasz(1950) proposed the following generalization of the well-known Bernstein polynomial approximation operator extending it to an infinite interval. We propose the following modification of the Szasz operator: MS n ( f ; x) = This modification is a more appropriate one inasmuch as“MS n( f ; x)” may be interpreted as the weighted average of the (n + 1) known values of the unknown function f (x) , namely ‘ f (k/n) ;k = 0(1)n; “Weights” being “Tk(x)”. On the other hand the “Modified Szasz Polynomial” approximation/estimator of the unknown function ‘ f (x)’ is “MS n( f ; x)”, as per the equation (3) in the preceding section This enables us to achieve per our “Phase I” of the iterative algorithm, the “Reduced-Bias Polynomial” approximation / estimator of the unknown function “ f (x)” just by subtracting the “Estimated Bias Polynomial” per (5) above to get: On ( f ; x) = MS n ( f ; x) − ErS n ( f ; x). The “First Iteration” will, be an exception using three improvement-operations Phase I –Phase II – Phase I

The empirical simulation study

Conclusion