Abstract. Keywords: Interpolation; simulated empirical study; reduced-bias. AMS Classification Number: 32E30; 47A57; 65D05.

Transcription

1 An Iterative-Improvement Algorithm for Newton's Forward Difference Formula Using Statistical Perspective Shanaz Ansari Wahid,*Ashok Sahai & Vrijesh Tripathi Department of Mathematics & Computer Science; Faculty of Science & Agriculture St. Augustine Campus; The University of The West Indies. Trinidad & Tobago; West Indies. 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 [x 0, x n ]. 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]). This leads to the calibrations of the respective Percentage Relative (Relative to actual value of the function at that point) Errors (PREs), and hence that of the respective Percentage Relative Gains, in Terms of the reduced values of the PRE, compared to that with the use of the Newton s original forward difference formula. Keywords: Interpolation; simulated empirical study; reduced-bias. AMS Classification Number: E0; A; D0. *Corresponding Author s ~ sahai.ashok@gmail.com 1 P a g e

2 Introduction Newton s forward difference formula is significant in many contexts {one may refer [1] to []}. It is a finite difference identity capable of giving an interpolated value between the tabulated points {f k } in terms of the first value f 0 and powers of the forward difference Δ. For k ε [0. 1]; the formula states f k = f 0 + k * Δ + (k * (k-1) /!) * Δ + (k * (k-1) * (k-) /!) * Δ + (1.1) This could also be translated-n-rewritten in the form f (x + k) = f (x) + (k) 1 Δ f (x) + (k) Δ f (x) + (k) Δ f (x) + (1.) Wherein (k) n the falling factorial, formula looks apparently like an analog of a Taylor Series expansion. This correspondence had been one of the motivating facts behind the development of Umbral Calculus. Another alternative form of the above-mentioned equation (1.), using the popular Binomial Coefficient happens to be as below. f (x + k) = f (x) + k C 1 Δ f (x) + k C Δ f (x) + k C Δ f (x) + (1.) Wherein the Binomial Coefficient k C n represents a polynomial of degree n in k. It would be important to note that we have taken the Interpolation-Interval as [0, 1], rather than say [a, b], without any loss of generality. In fact, C [0, 1] and C [a, b] are essentially identical, for all practical purposes, inasmuch as they are linearly isometric as normed spaces, order isomorphic as lattices, and isomorphic as algebras (rings)! Now, let us consider a finite version of the Newton's forward difference formula with n replacing in (1.1), (1.), and (1.) above, as the number Knots is finite. P a g e

3 Hence, we assume that we are given the values of the unknown function f(x) at the n+1 equidistant-knots k/n; k = 0 (1) n. The impugned formula uses the values of the simple forward differences using values of the unknown function f(x) at the equidistant-points/ knots in the Interpolation-Interval, say [0, 1]. 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 to reduce the Interpolation Error, i.e. equivalently statistically the 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 is developed using the sum of the absolute (absolute differences between the actual (knowable, as the function is 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 the Interpolation-Interval, say [0, 1]). For each of the Iteration, this leads to the calibrations of the Percentage Relative (Relative to the actual value of the function at the relevant point) Errors (PREs), for various illustrative values of n for the example-function f (x). Similarly, the respective Percentage Relative Gains (PRGs) in terms of the reduced values of the PRE with the use of the Newton s original forward difference formula are also calibrated. P a g e

4 The following section details the algorithm enabling the use of the seminal perspective of 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. As the same perspective (Of Reduced Error/ Bias) is available to be used at the end of a particular iteration, beginning from the first iteration, the proposed algorithm is an iterative one.. The Iterative Improvement Algorithm for More Efficient Interpolation. At the very outset of this section, we begin by taking note of a very significant fact that we do know the n+1 values f (k/n); k = 0 (1) n of the unknown function f(x) to lead us to the n values of the various powers (1 to n) of the forward difference Δ to be used in (1.1) or (1.) [with x= 0; k = x] to get the interpolated value, as per (1.), as also to get the Interpolation Function, say IF N[n] f(x). 1 1 The second important fact to be noted is that this Newton s Forward Difference Interpolation Formula is a polynomial of degree n in x [a = 0], which could also be used 1 1 to get the estimated value of f(x), as per this polynomial; called by Et N n f k n say, for the point x = k/n ; k = 0 (1) n. = Et (k/n), 1 1 The third important fact, in our context, is that these estimated values namely Et (k/n) s would, quite probably, be different from the given values f (k/n) s. 0 Let us call by Er (k/n) the Error-Values Et (k/n) f (k/n) ; k = 0 (1) n. (.1) P a g e

5 We use these knowable/ calibratable values f (k/n) s, as per (1.), in place of the given values f (k/n) to develop the values of the various powers (1 to n) of the forward difference Δ to be used in (1.1) or (1.), to generate say the function Er (x) for interpolating the Error-in-interpolation / Bias function At the First Iteration, now, we use this aforesaid Error Function Er (x) to improve the Interpolation-Function (IF) say IF N[n] f (x), as below: IF N[n] f(x) - Er (x) = IF N[n] (1) f(x); Say [The IF N[n] after Iteration # 1 ] (.). Apparently, with the correction in (.), the function IF N[n] (1) f(x) has smaller BIAS/Error than IF N[n] f(x), because of the Bias/ Error correction/reduction to the extent of Er (x). Also now, apparently at the second iteration, we could do the same calibrations as in the first iteration with IF N[n] (1) f(x) in the shoes of IF N[n] f(x), and the Bias/Error function Er (1) (x) in the shoes of Er (x), respectively. We would note that Er (1) (x) could be generated analogously to Er (x), with IF N[n] (1) (x) in shoes of IF N[n] f(x) to lead to the calibration of Error-Values, as per (.1). Thence we could be led to the achievement of the Bias/ Error Function Er (1) (x). Hence, at the Second Iteration, we use this aforesaid Error Function Er (1) (x) to improve the interpolation function say IF N[n] (1) f (x), as below: IF N[n] (1) f(x) Er (1) (x) = IF N[n] () f(x); Say [~ IF N[n] after Iteration # ] (.) So-on-and-so-forth, we could continue improving iteratively, per this proposed iterative-improvement algorithm, till we please! P a g e

6 Typically, at I-th. Iteration we use Bias/ Error Function Er (I-1) (x) to improve the interpolation function say IF (I-1) N[n] f (x), as below: IF (I-1) N[n] f(x) Er (I-1) (x) = IF (I) N[n] f(x); Say [after Iteration # I ] (.). The following section aims at bringing forth the potential of the possible improvement of the interpolation as per an empirical study for which the function f is assumed to be known in the sense of simulation.. The empirical simulation study To illustrate the gain in efficiency by using our proposed Iterative Algorithm of Improvement of Interpolation Using Newton s Forward Difference Interpolation Formula, we have carried an empirical study. We have taken the example-cases of n =,,, and (i.e. n + 1 =,,, and, knots) in the empirical study to numerically illustrate the relative gain in efficiency in using the Algorithm vis-à-vis the Original Newton s Forward Difference Interpolation Formula in each example case of the n-value. Essentially, the empirical study is a simulation one wherein we would have to assume that the function, being tried to be approximated, namely f (x) is known to us. We have confined to the illustrations of the relative gain in efficiency by the Iterative Improvement for the following four illustrative functions: f (x) = exp(x), ln ( + x), sin ( +x), and x. 1 0 To illustrate the POTENTIAL of improvement with our proposed Iterative Algorithm, we have considered only THREE Iterations. P a g e

7 Numerical values of SEVEN quantities are computed; namely THREE Percentage Relative Errors [(PRE_I (#) N[n] s) corresponding to our Iteration (# = 1, or, or )], ONE Percentage Relative Error (PRE_N[n] s) corresponding to the Original Newton s Forward Difference Interpolation Formula, and the THREE Percentage Relative Gains [(PRG_I(#)N[n] s; # = 1(1)] by using our Iterative Algorithm rather than the Original Newton s Forward Difference Interpolation Formula. These values, calculated using MAPLE RELEASE, are defined below. The PRE_N[n] of Original Newton s Forward Difference Interpolation Formula : PRE_N[n] = [ k=n k=1 abs.{(et N[n] f ( k 1 ) f ( k 1 n n ))} k=n { abs.[ f ( k 1 k=1 n )] } ]*0% (.1) The PRE_ I (#) N[n] of the Interpolation Formula IF (I) N[n] f(x) after using I th. Improvement Iteration (I # 1, or, or ) on Original Newton s Forward Difference Interpolation Formula using n intervals in [0, 1], {i.e. [(k- 1)/n, k/n]; k= 1(1) n}: PRE_ I (#) N[n] = [ k=n k=0 abs.{(et (I(#)) N[n] f ( k 1 ) f ( k 1 n n ))} k=n { abs.[ f ( k 1 k=0 n )] } ]*0%; I (#) = 1// (.) And, hence, the Percentage Relative Gain (PRG), defined exactly analogously to PRE, by using the current Improvement Iteration [: I (#) = 1, or, or ] Iterpolation Polynomial with the n intervals in [0, 1] over using the Original Newton s Forward Difference Interpolation Polynomial could be defined as below. PRG_ I (#) N[n] = [ (PRE_N[n] PRE I # N[n]) (PRE_N[n]) ]*0%; I (#) = 1// (.) 1 P a g e

8 P a g e For the relevant (Targeted) four functions, f (x), these seven values, namely for each I (#) [I (#) = 1; or ; or ] the PRE_ I (#) N[n] s, the PRG_ I (#) N[n] s, and the PRE_N[n] are tabulated in the following four tables ~ Tables 1 to Table ; respectively. Table 1: (Iterative) Algorithmic (In %) Relative (Absolute) Efficiency/Gain for f (x) = exp(x). Items n PRE_N[n] PRE_I (1) N[n] PRE_I () N[n] PRE_I () N[n] PRG_I (1) N[n] PRG_I () N[n] PRG_I () N[n] Table : (Iterative) Algorithmic (In %) Relative (Absolute) Efficiency/Gain for f (x) = ln (+x). Items n PRE_N[n] PRE_I (1) N[n] PRE_I () N[n] PRE_I () N[n] 0.0 PRG_I (1) N[n] PRG_I () N[n] PRG_I () N[n] Table : (Iterative) Algorithmic (In %) Relative (Absolute) Efficiency/Gain for f (x) = sin (+x). Items n PRE_N[n] PRE_I (1) N[n] PRE_I () N[n] PRE_I () N[n] PRG_I (1) N[n] PRG_I () N[n] PRG_I () N[n]

9 Table : (Iterative) Algorithmic (In %) Relative (Absolute) Efficiency/Gain for f (x) = x. Items n PRE_N[n] PRE_I (1) N[n] PRE_I () N[n] PRE_I () N[n] PRG_I (1) N[n] PRG_I () N[n] PRG_I () N[n] Conclusion The aforesaid SEVEN numerical quantities have been computed using Maple Release, for all the four illustrative functions (exp(x), ln ( + x), sin ( + x), and x ) mentioned in Section, tabulated in Tables 1. The PRE s for our proposed Iterative Algorithm are PROGRESSIVELY lower on each of the subsequent iterations, as compared to that for Original Newton s Forward Difference Interpolation Polynomial, for all the illustrative functions. The PRG s due to the use of our proposed Iterative Algorithm vis-a-vis Original Newton s Forward Difference Interpolation Polynomial are also increasing PROGRESSIVELY on each of the subsequent iterations, for all the illustrative functions. Lastly, it is very important to take note of the fact that the PRG s are particularly more significant when the number of the knots on which the values of the unknown function f(x) could be generated experimentally is rather small. Quite possibly, it could, as well, be the situation due to expensive (in terms of resources available) experimental set-up for generating these values. Hence, also in such realistic conditions on ground the interpolations could be more accurately achievable through using the Original Newton s Forward Difference Interpolation Polynomial strengthened by the proposed Iterative Algorithm! Similarly, other interpolation-formulae could also be iteratively improved. P a g e

10 1 REFERENCES: [1] Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, th printing. New York: Dover, p. 0, 1. [] Beyer, W. H. CRC Standard Mathematical Tables, th ed. Boca Raton, FL: CRC Press, p., 1. [] Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, nd ed. Reading, MA: Addison-Wesley, 1. [] Jordan, C. Calculus of Finite Differences, rd ed. New York: Chelsea, 1. [] Nörlund, N. E. Vorlesungen über Differenzenrechnung. New York: Chelsea, 1. [] Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 10. [] Whittaker, E. T. and Robinson, G. "The Gregory-Newton Formula of Interpolation" and "An Alternative Form of the Gregory-Newton Formula." - in The Calculus of Observations: A Treatise on Numerical Mathematics, th ed. New York: Dover, pp. -1, 1. 1 P a g e