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

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.


Introduction
Szasz (1950) proposed the following generalization of the well-known Bernstein polynomial approximation operator extending it to an infinite interval.
Heinz-Gerd Lehnhoff (1981) proposed the "Modified Szasz-Mirakjan Operator" as follows: .∀x C [0, 1] and f C [0,1] (2) In which, k! for all non-negative integer values of k .Motivated by the above, we propose the following modification of the Szasz operator: . ∀x C [0, 1] and f C [0, 1] (3) 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 "T k (x)".In fact, the T k (x)s could be interpreted as "proportional to probabilities" ["T k (x) ≥ 0"].Using this interpretation, therefore, Incidentally, as we could use a suitable transformation (translation and change of scale) of the variable x, we could assume, without any loss of generality, that we are interested in the approximation of a function in n could be generated enabling us to construct the "Bias Polynomial Function", say 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: Now, we embark upon the "Phase II" of our proposed 'Iterative Algorithm'.The concept "Minimum Mean Square Error Estimator (MMSEE)"of Searles(1964) is seminal to this phase of our algorithm.As per (1.4), our "Modified Szasz Polynomial" estimator is analogous to the sample-mean ' x'.Searles(1964) considered the class of estimators k. x , and chose the "Optimal" value say "k 0 " by minimizing the MSE (k x ) to lead to the MMSEE (k 0 .x).Similarly, we consider the perturbed 'Polynomial', say b. O n ( f ; x), and hence determine the estimated values of the unknown function ' f (x)' at the knots '(k/n)', say Et f (k/n), k = 0(1)n, vis-à-vis known values of the unknown function " f (x)", namely ' f (k/n)'; k = 0(1)n.Hence the "Knot-Wise Squared-Error", sayE2r n could be generated to lead to the construction of the "Squared-Error Polynomial Function", This will be a "Quadratic Polynomial in b", say To complete the "FIRST Iteration" we again apply the details of the 'Phase I' to treat our 'Reduced-MSE Polynomial' estimator "b 0 .O n ( f ; x)", to achieve the improved [at Iteration # 1] 'Modified Szasz Polynomial' Operator/Estimator: Thus operations defining "FIRST Iteration" could well be characterized as a "Sandwich"!The top and bottom slices 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 stacked pile of sandwiches of desired height with the bottom slice of the first iteration serving as the top slice for the second-iteration sandwich, and so on.
At any iteration, the improvements will begin and end with the "Phase I" operation of the 'Bias-Reduction' of the improvement algorithm, sandwiching its "Phase II" operation of 'MSE-Reduction'.As such, at any iteration, we will have two improvement-operations only, namely that of 'Phase II' followed by that of "Phase I" borrowing the last operation of the preceding iteration.Only the "First Iteration" will, therefore be an exception using three improvement-operations Phase I -Phase II -Phase I.

The empirical simulation study
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.We have taken the cases of n = 2, 3, and 4(i.e.n + 1 = 3, 4, and 5, knots) in the empirical study to numerically illustrate the relative gain in efficiency in using the Algorithm vis-à-vis the Original Modified Szasz Polynomial Operator in each example-case of the n-values.
Essentially, the empirical study is a simulation one in which we assume that the function to be approximated, namely f (x), is known to us.Once again we have confined ourselves to illustrating the relative gain in efficiency by the Iterative Improvement for the following four functions: The PRE using the Improvement Iteration (I#1, or2, or3) on Bernstein (Polynomial) using n intervals in [0, 1], i.e. [(k − 1)/n, k/n]; k = 1(1)n : × 100; where # = 1, 2 or 3 The PREs respective to the Original Modified Szasz Polynomial Operator and respective to the First, Second, and the Third Sandwich-Algorithmic Improvement Iteration Polynomials, respectively, for each of for each of the example # s of approximation Knots/Intervals.And the Percentage Relative Gains (PRGs), defined exactly analogously to PRE, by using the proposed Sandwich-Algorithmic Improvement Iteration: I#(e.g.1, 2 or3) Polynomials with the n intervals in [0, 1] over using the (Original) Modified Szasz Polynomial Operator for the approximation of the (Targeted) function, f are tabulated in the following four tables (Tables 1 to Table 4) in the Appendix.

Conclusion
These seven numerical quantities have been computed using Maple Release 12, for all the four illustrative functions (exp(x), ln(1 + x), sin(1 + x), and 5 x ) mentioned in Section 3, and for three values of n (n = 2, 3, 4), and are, respectively, tabulated in Tables 1-4 [Appendix].The PREs for our Algorithmic Sandwich-Iterative Polynomial Approximations are PROGRESSIVELY lower on each subsequent iteration, as compared to that for the Original Modified Szasz Polynomial Approximation, for all the illustrative functions.The PRGs due to the use of our proposed Algorithmic Sandwich-Iterative Polynomial Approximations vis-a-vis the Original Modified Szasz Polynomial Approximation are also PRO-GRESSIVELY increasing on each subsequent iteration, for all the illustrative functions.Lastly, it is very heartening to note that when we use (n = 4) intervals, i.e.FIVE KNOTS for the polynomial approximation, the PRG becomes almost 100% for the third iteration in the function ln(1+x).Otherwise also, the speed of convergence is highly accelerated by the Sandwich-Iterative Algorithmic improvement in the Modified Szasz Polynomial using the Statistical perspective.
In fact, the improvement attained by the third iteration is not very great.
In this section we propose the "Two-Phase Iterative Improvement Algorithm for Modified Szasz Operators MS n ( f ; x)" using the 'TWIN' statistical perspectives of 'Bias' & 'MSE'.In the statistical sense, MS n ( f ; x) is an estimate of the unknown function f (x)'.Now, we use our 'estimator' (the modified Szasz Operators MS n ( f ; x)) to 'estimate' the values of the unknown function f (x) at the knots (k/n),say Et f (k/n), k = 0(1)n, and compare these values with the values of the unknown function " f (x)", namely ' f (k/n)'; k = 0(1)n.Hence the error at the nth knot ", say Er f 5x .To illustrate the potential improvement with our proposed Algorithm, we have considered THREE Iterations, and the nu- merical values of seven quantities -three Percentage Relative Errors (PREs) corresponding to Improvement Iteration (# = 1, or 2, or 3) (PRE I(#)MS n( f ; x)[n]), Original Modified Szasz Polynomial Operator (PRE MS n( f ; x)[n]), and the three corresponding Percentage Relative Gains (PRG) in using our Iterative Algorithmic Modified Szasz Polynomial Operators in place of the Original Modified Szasz Polynomial Operators MS n(f ; x)[n], namely PRG I(#)MS n( f ; x)[n]; # = 1(1)3).These quantities are defined as follows.The PRE using (original) Modified Szasz (Polynomial) using n intervals in [0, 1], i.e. [(k − 1)/n, k/n]; k = 1(1)n :

Table 1 .
It could also be noted that this perspective of the Sandwich-Iterative Improvement could be applied to any Polynomial Approximator, other than Modified Szasz Polynomial; more particularly to those belonging to the class of Positive Linear Operators, as they admit to the Probabilistic/Statistical perspective rather more readily!APPENDIX (Iterative) Algorithmic (In%) Relative (Absolute) Efficiency/Gain for f (x) = exp (x) .