S. A. WAHID

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proposed vis-à-vis the original (Primal) Bernstein’s poly-

nomial operator in each example case of n-value.

Essentially, the empirical study is a simulation one

wherein we would assume that approximated function,

namely “

x”, is known to us.

We have confined to illustrations of relative gain in

efficiency by Iterative Improvement for the following

four illustrative-functions:

exp; ln2; sin2,and 10

fx xxx.

To illustrate the POTENTIAL of improvement with

our proposed Dual-Fusion Operator

PDFBV ;'

xn

,

we have TWO numerical values of quantities ~ two per-

centage relative errors (PREs) corresponding to original

(Primal) Bernstein’s Operator

P

B;

xn: Say;

PRE_PFB ;

xn verses that of the proposed Dual-

Fusion Operator i.e.;

PRE_PDFB V;

xn. We cal-

culated Percentage Relative Gains (PRGs) in using our

“Dual-Fusion” variant of Bernstein Polynomial in place

of Original “Primal” variant of Bernstein Polynomial

PRG_UPDFB ;

x

n. These quantities are defined:

PRE_PFB;fxn100.

0.33 0.33

0

abs. PFB ; d d

0

xnf xxfxx

&PRE_PDFBV

;1fx n00.

0.33 0.33

0

abs. PDFBV ; d d

0

xnf xxfxx

.

Hence, PRG_ PDFBV

; 100fxn.

PRE_PFB;

PRE_PDFBV ;PRE_PFB ;

fx n

xn fxn

PREs for Original-Primal/Variant Primal-Dual Bern-

stein polynomial respectively for each of example # of

approximation Knots/Intervals.

PRGs by using Proposed Dual-Fusion Polynomials

with the n intervals in

0,1 2 over using the Original

Primal-Bernstein Polynomial for approximation of func-

tion, “

x” are tabulated in APPENDIX in Tables 1-

4.

5. Conclusions

For all the FOUR illustrative functions, namely

exp9; ln2; sin2,and 10

fxx xx, the PRGs

are above 99.9% for 3, 6,and9n

. It is very signifi-

cant to note that the PRGs are (almost) 100% for 6n

for all example-functions, i.e. for only SEVEN “Knots”!

6. References

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