An Iterative-Improvement Algorithm for Newton's Forward Difference Interpolation Formula Using Statistical Perspectives of Reduced MSE & Reduced Bias

by Miodrag M Lovric1 & Ashok Sahai.

Abstract: This paper proposes a computerizable iterative algorithm to intermittently improve the efficiency of the interpolation by the well-known simple-n-popular “Newton’s Forward Difference Interpolation Polynomial Formula?, using “Statistical” perspective of “Reduced Mean-Square-Error (MSE)” and that of “Reduced- Bias”. The impugned formula uses the values of the simple forward differences using values of the unknown function “f(x)”at equidistant-points/ knots in the “Interpolation-Interval”, say [x0, xn]. The basic perspective motivating this iterative algorithm is the fuller use of the “information” available in terms of these values of the unknown function “f(x)” at the “n+1” equidistant-points/ knots. This information is used to reduce the Interpolation- Error, which is statistically equivalently measurable in terms of “MSE & BIAS”. The potential of the improvement of the interpolation is tried to be brought forth per an “empirical study” for which the function “f” is assumed to be known in the sense of simulation. The numerical metric of the improvement uses the sum of absolute errors, i.e. the differences between the actual (assumed to be known in the sense of the simulating nature of the empirical study) and the interpolated values at the mid-points of the equidistant-points/ knots in “Interpolation-Interval”, say [0, 1]). This leads to the calibrations of the respective “Percentage Relative (Relative to actual value of the function at that point) Errors (PREs)”, and hence those of the respective “Percentage Relative Gains”, in terms of the reduced values of the PREs, compared to that with the use of the Newton’s original forward difference formula.

Key Words: Interpolation; simulated empirical study; reduced-bias

Authors:
Ashok Sahai, sahai.ashok@gmail.com
Miodrag M Lovric, mlovric@kg.ac.rs

Editor: Abdallah Abdelfattahf, a_afattah@hotmail.com

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