Applied Mathematics, 2011, 2, 1535-1538
doi:10.4236/am.2011.212218 Published Online December 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
An Efficient Combinatorial-Probabilistic Dual-Fusion
Modification of Bernstein’s Polynomial Appr oximation
Operator
Shanaz Ansari Wahid
Department of Mathematics & Statistics, Faculty of Science & Agriculture,
St. Augustine Campus of the University of the West Indies, Debe, Trinidad & Tobago
E-mail: Shanaz.Wahid@sta.uwi.edu, shanazw@hotmail.com
Received March 29, 2011; revised November 14, 2011; accepted November 22, 2011
Abstract
The celebrated Weierstrass Approximation Theorem (1885) heralded intermittent interest in polynomial ap-
proximation, which continues unabated even as of today. The great Russian mathematician Bernstein, in
1912, not only provided an interesting proof of the Weierstrass’ theorem, but also displayed a sequence of
the polynomials which approximate the given function() [0,1]fx C
. An efficient “Combinatorial-Probabili-
stic Dual-Fusion” version of the modification of Bernstein’s Polynomial Operator is proposed. The potential
of the aforesaid improvement is tried to be brought forth and illustrated through an empirical study, for
which the function is assumed to be known in the sense of simulation.
Keywords: Approximation, Bernstein Operator, Dual-Fusion, Simulated Empirical Study
1. Introduction
The problem of approximation arises in many contexts of
“Numerical Analyses and Computing” [1-4]. Weirstrass
[5] proved his celebrated approximation theorem: “…If,
,
f
Cab, then for every , a polynomial “p
such that “
δ0
fpδ”. In other words, result estab-
lished the existence of an algebraic polynomial in con-
cerned variable capable of approximating the unknown
function in that variable, as closely as we please!
This result was a big beginning of the Mathematicians’
interest in “Polynomial Approximation” [4,6-8] of an
unknown function using its values generated, experi-
mentally or otherwise, at certain equidistant knots in the
impugned interval of the relevant variable. The Great
Russian mathematician Bernstein proved the Weirstrass
theorem in a style, which was very thought-provoking
and curious in many ways. He first noted a simple though
a very significant feature of this theorem, namely that if
it holds for
0,1C, it does hold for
,Cab also vice-
versa. In fact,
0,1C and
,Cab are essentially iden-
tical, for all practical purposes, inasmuch as they are
linearly “isometric” as normed spaces, order isomorphic
as lattices, and isomorphic as algebras (rings) [9].
Also, the most important contribution in the Bern-
stein’s proof of the Weirstrass’ theorem consisted in the
fact that Bernstein actually displayed a sequence of poly-
nomials that approximate a given function
0,1fC.
If,
f
x is any bounded function on
0,1C, the
sequence of “Bernstein Polynomials” [6] for
f
x is
defined by:


 


( )
0
Bn. . (1). /
C0,1,Say
kn knk
k
n
f
xxxf
k
xEfx






kn
(1)
The aim of this paper is to propose a more efficient
polynomial approximation operator exploiting the “com-
binatorial structure” of the “Bernstein’s Polynomial Ap-
proximation operator”, and the fact that the unknown
function might, without any loss of the generality, be
assumed to be 1
0, 2
fC
, as in [10-14].
2. The Proposition of the Variant of the
Bernstein Polynomial Approximator
In context of the aforementioned sequence of “Bernstein
Polynomials” for
f
x, a significant observation which
must be taken note of is that the use is made of the values
S. A. WAHID
1536
of the unknown function “
f
x” at the equidistant-
knots “;0(1)
kk
nn”, assumed to be knowable through
the experiment in the relevant scientific field of in-
vestigation or known otherwise.
(s)
In any approximating polynomial operator use is made
of the “Knots” and of the corresponding “Weights”.
In our proposition of a variant of the “Bernstein’s
Polynomial” we propose to systematically introduce new
corresponding weights, without essentially changing the
location of the “equi-distant” “knots”, except for the fact
that the impugned interval is 1
0,C
2

, rather than
0,1C, thanks to “isometric” spaces noted earlier. We
propose a variant of the “Bernstein Polynomial” which is
having a better combinatorial structure in favor of the in-
terval for 1
0, 2
fC

, and that is the main strategy for
making it better than the original/usual “Bernstein’s
Polynomial”! We consider the following PRIMAL vari-
ant of the Bernstein’s Polynomial:
Say,




P
0
( )
.(
)nk
n
k
B;0.67 )
(0.33 2*
kn k
k
fx n x
fkn





3. The Combinatorial-Probability &
Dual-Fusion Variant for Bernstein’s
Polynomial
The original interval
0,1C isometric to the impugn-
ned interval 1
0, 2
C


could be thought of in terms of
its two parts, namely “0.33
x
” and “0.67
x
for
0,1xC with “k” and “” “knots” sitting in each,
respectively. The expected number of points sitting these
two parts. Respectively, would be
nk
(0.33 )
x
n and (0.67 )
x
n
.
The “combinatorial probability” of description would be:
binomial






*0.33, binomial*, /binomial,
binomialn0.33, biialn0.67+,
nxkxnk n
xn xnk


0.67+
nom
n
k

n
Hence the PRIMAL-Variant of the “Bernstein’s Poly-
nomial” in (2.1) comes off to be as below.
Say,
 



P
0
BV;[]binomial
7+,
*0.33,
* binomial *0.6. 2*
kn
k
f
xnnx k
nkfkn
nx
The correspondingly DUAL (-Weights) variant of Bern-
stein Polynomial would be:
Say,
 





P
0
BV;[]binomial *0.33,
* binomial *0.67, 2*
kn
k
fx nnxk
nxnkfk


n
We define the “PRIMAL-DUAL-Fusion-Weights” va-
riant of the Bernstein Polynomial as
Follows: Say,

 
PD
PDFBV ;BV;BV;2fx nfxfx

To make comprehensive the combinatorial systema-
ticness of PRIMAL-DUAL Fusion variant of Bernstein
polynomial say
PDFBV ;
f
xn, we note that it will
work for an approximation polynomial focusing interval
(1 3)2,(1 3x) 2x
around “0.33”, which will
~0,13for 13x.
Impugned interval will be wider, the greater the value
of “13x
”! For example, in the approximating poly-
nomial in “x” for values of
0,1 3;x interval will be
symmetrically, centered on
“0.165”,
e.g. ~0.035,0.295for0.26.x
To balance the “Pull”, systematically, the weights
“(1/2)” and “(1/2)” are assigned to the relevant weights
in BP
;
f
xn & BD
;
f
xn. These weights are also,
respectively, “DUAL” to each-other, again!
The aforesaid (PRIMAL-DUAL Fusion) variant of the
Bernstein Polynomial, namely, PDFBV

;
f
xn will,
apparently induce a “(Systematic)Bias” in the approxi-
mating “Polynomial”, which is amenable more system-
atically than that in the original “Bernstein’s Polyno-
mial”.
Similar to what was noted in Sahai (2004) [10], in the
absence of any conclusive analytical study [The deriv-
able “Upper” bounds on the error of approximation (as
noted in the paper by Sahai (2004) [8]) are not of much
use. In fact, a smaller/lower “Upper Bound” does not
guarantee a better approximation and the extent of the
resultant “GAIN” is unavailable, too! Hence, we go for
an empirical simulation study to illustrate the potential
“GAIN” through our PRIMAL-DUAL Fusion variant of
the Bernstein Polynomial, namely, PDFBV

;
f
xn.
4. The Empirical Simulation Study
To illustrate gain in efficiency by using our proposed
“Dual-Fusion” variant of Bernstein Polynomial Appro-
ximation, we have carried an empirical study. We have
taken example-cases of n = 3, 6, and 9 (i.e. n + 1 = 4, 7,
and 10 knots) in the empirical study.
To numerically illustrate the relative gain in efficiency
in using “Dual-Fusion” variant of Bernstein Polynomial
Copyright © 2011 SciRes. AM
S. A. WAHID
Copyright © 2011 SciRes. AM
1537
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 “

f
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
x
fx xxx.
To illustrate the POTENTIAL of improvement with
our proposed Dual-Fusion Operator

PDFBV ;'
f
xn

,
we have TWO numerical values of quantities ~ two per-
centage relative errors (PREs) corresponding to original
(Primal) Bernstein’s Operator
P
B;
f
xn: Say;

PRE_PFB ;
f
xn verses that of the proposed Dual-
Fusion Operator i.e.;

PRE_PDFB V;
f
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 ;
f
x

n. These quantities are defined:
PRE_PFB;fxn100.





0.33 0.33
0
abs. PFB ; d d
0
f
xnf xxfxx


&PRE_PDFBV

;1fx n00.





0.33 0.33
0
abs. PDFBV ; d d
0
f
xnf xxfxx








.
Hence, PRG_ PDFBV

; 100fxn.





PRE_PFB;
PRE_PDFBV ;PRE_PFB ;
fx n
f
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, “

f
x” are tabulated in APPENDIX in Tables 1-
4.
5. Conclusions
For all the FOUR illustrative functions, namely
 
exp9; ln2; sin2,and 10
x
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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Computing,” Brooks/Cole Publishing Company, Belmont,
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[2] P. J. Heartley, A. Wynn-Evans, “A Structured Introduc-
tion to Numerical Mathematics,” Stanley Thornes, Bel-
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[3] B. F. Polybon, “Applied Numerical Analysis,” PWS-
Kent, Boston, 1992.
[4] A. Sheilds, “Polynomial Approximation,” The Math In-
telligencer, Vol. 9, No. 3, 1987, pp. 5-7.
[5] K. Weierstrass, “Uber die analytische Darstellbarkeit
sogenannter willkurlicher Functionen einer reellen Ve-
randerlichen Sitzungsberichteder,” Koniglich Preus-
sischen Akademie der Wissenschcaften zu Berlin, 1885,
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[6] N. L. Carothers, “A Short Course on Approximation The-
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[7] E. R. Hedrick, “The Significance of Weirstrass Theo-
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[8] G. G. Lorentz, “Approximation of Functions,” Chelsea,
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[9] N. L. Carothers, “Real Analysis,” Cambridge University
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[10] A. Sahai, “An Iterative Algorithm for Improved Ap-
proximation by Bernstein’s Operator Using Statistical
Perspective,” Applied Mathematics and Computation,
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doi:10.1016/S0096-3003(03)00081-X
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S. A. WAHID
1538
APPENDIX
Table 1. (Iterative) algorithmic (In %) relative (absolute)
efficiency/gain for f(x) = exp(x).
Items n3 6 9
PRE_PFB (f; x)[n] 7.7098 7.97506 8.0614
PRE_PDFBV (f; x)[n] 0.0004 0.00000 0.0000
PRG_PDFBV (f; x)[n] 99.994 100.000 99.999
Table 2. (Iterative) algorithmic (In %) relative (absolute)
efficiency/gain for f(x) = ln(2 + x).
Items n3 6 9
PRE_PFB (f; x)[n] 5.0970 5.0217 4.9961
PRE_PDFBV (f; x)[n] 0.0001 0.0000 0.0000
PRG_PDFBV (f; x)[n] 99.996 100.00 99.999
Table 3. (Iterative) algorithmic (In %) relative (absolute)
efficiency/gain for f(x) = sin(2 + x).
Items n3 6 9
PRE_PFB (f; x)[n] 5.0404 5.3105 5.4020
PRE_PDFBV (f; x)[n] 0.0004 0.0000 0.0000
PRG_PDFBV (f; x)[n] 99.991 99.999 99.999
Table 4. (Iterative) algorithmic (In %) relative (absolute)
efficiency/gain for f(x) = 10x.
Items n3 6 9
PRE_PFB (f; x)[n] 16.112 17.498 17.935
PRE_PDFBV (f; x)[n] 0.0120 0.0000 0.0000
PRG_PDFBV (f; x)[n] 99.925 99.999 99.999
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