An Iterative-Improvement Algorithm for Newton's Forward Difference Formula Using Statistical Perspective

by Shanaz Ansari Wahid, Ashok Sahai & Vrijesh Tripathi.

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 Formula’, using ‘Statistical’ perspective 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 equivalent to the well-known concept of 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]).

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

Authors:
Shanaz Ansari Wahid, shanazw@hotmail.com
Ashok Sahai, sahai.ashok@gmail.com
Vrijesh Tripathi, tripathi@sta.uwi.edu

Editor: Ahmed H. Youssef, ahyoussef@hotmail.com

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