Polynomial Interpolation using Newton's Divided Difference and ...

E- ISSN: 2249 –1929
Journal of Chemical, Biological and Physical Sciences
An International Peer Review E-3 Journal of Sciences
Available online at www.jcbsc.org
Research Article
Section C: Physical Science
Polynomial Interpolation using Newton’s Divided
Difference and Lagrange’s Methods
R. B. Srivastava * and Purushottam Kumar Srivastava
Department of Mathematics, M. L. K. P. G. College, Balrampur. U. P., India
Received: 21Septembert 2011 Revised: 30 September 2011; Accepted: September 2011
ABSTRACT
Polynomials of degree 10 for function f(x) = tan -1 x have been obtained in the interval [-1, 1]
using Newton’s divided difference and Lagrange’s formulas by dividing the interval in to 200
equal parts. Values of function have been calculated at the points of the interval using these
interpolating polynomials and compared with their exact values for the estimation of errors.
Comparison of the values of interpolating polynomials with exact values show that Newton’s
divided formula is approximately two times better as compared to Lagrange’s interpolation
formula.
Key words: Newton’s divided difference formula, Lagrange’s formula, interpolating polynomial,
difference triangle, approximation.
INTRODUCTION
There are many situations that call for fitting a common function to a collection of data.
Determining a function that agrees precisely with the data, at least to within round-off tolerances, is called
interpolation [1-2] . In the case of an arbitrary collection of data, polynomial interpolation is the most likely
candidate for mainly pragmatic reasons. A polynomial approximation to a collection of data is easy to
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determine in various ways, and polynomials have easily computed derivatives and integrals that might be
useful if the derivative or integral [3-5] of the function underlying the data is needed. However, a
polynomial of degree n-1 is generally required to satisfy a set of n conditions, and polynomials of even
moderately high degree are likely to have a high degree of variation except where explicitly constrained.
This means that in order to obtain stable approximating polynomials either the number of specified
conditions must be kept small, which may not be a reasonable restriction, or, more likely, the conditions
are only required to be satisfied in an approximate way. The problem of approximating [6] a known
function with a simpler function follows a pattern similar to that of fitting a function to a collection of
data. There are many reasons why such an approximation might be required.
The branch of mathematical statistics devoted to the inference of accurate conclusions about the
numerical values of approximately measured quantities, as well as on the errors in the measurements.
Repeated measurements of one and the same constant quantity generally give different results, since
every measurement contains a certain error. There are three basic types of error: systematic, gross and
random. Systematic errors always either overestimate or underestimate the results of measurements and
arise for specific reasons (incorrect set-up of measuring equipment, the effect of environment, etc.), which
systematically affect the measurements and alter them in one direction. The estimation of systematic
errors is achieved using methods which go beyond the confines of mathematical statistics. Gross errors
(often called outliers) arise from miscalculations, incorrect reading of the measuring equipment, etc. The
results of measurements which contain gross errors differ greatly from other results of measurements and
are therefore often easy to identify. Random errors [16-19] arise from various reasons which have an
unforeseen effect on each of the measurements, both in overestimating and in underestimating results.
The main contents of approximation theory concern the approximation of functions. Its
foundations are laid by the work of P. L. Chebyshev (1854–1859) on best uniform approximation of
functions by polynomials and by K. Weierstrass, who in 1885 established that in principle it is possible to
approximate a continuous function on a finite interval by polynomials with arbitrary pre-given error.
Data interpolation [7-10] is one of the most important tasks in geophysical data processing. Its
importance is increasing with the development of 3-D seismics, since most of the modern 3-D acquisition
geometries carry non-uniform spatial distribution of seismic records. Without a careful interpolation,
acquisition irregularities may lead to unwanted artifacts at the imaging step (Gardner and Canning, 1994;
Chemingui and Biondi, 1996). The interpolation problem in geophysics implies interpolating irregularly
sampled data to a regular grid. In general, this problem requires a regularized inversion scheme, such as
the method of inversion to zero offset (Ronen et al., 1991, 1995).
Newton’s forward formula [9-11] is appropriate to apply only when the unknown value lies near the
beginning of the table while Newton’s backward formula is usually applied when the unknown value to
be interpolated lies near the end of the table [25, 26] . Lagrange’s formula for interpolation is applicable for
any type of observation [4] but when the observations are given at equi-spaced values of arguments, the
formulas using various order differences are found to be more convenient and easy to apply rather than
Lagrange’s interpolation formula. In the case, the values of the arguments are unequally spaced; we
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R. B. Srivastava et al.
should use Newton’s divided difference formula to represent the function [12-15, 20-24] .
METHODOLOGY
Lagrange’s formula for equal intervals: Interval [a, b] has been divided into 10 equal parts with the
help of the points
a=x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 =b
and y i = f(x i ); where i=0,1, 2, …. , 10; have been calculated.
In this case, the Lagrange’s formula [4-6]
( x − x )( x − x )....( x − x ) ( x − x )( x − x )( x − x )....( x − x )
f x f x f x
1 2 10 0 2 3 10
( ) = ( 0) +
( 1)
( x0 − x1)( x0 − x 2)....( x0 − x10) ( x1 − x0)( x1 − x2)( x1 − x3)....( x1 − x10)
( x − x0)( x − x1)....( x − x9)
.... f ( x10)
( x10 − x0)( x10 − x1)....( x10 − x9)
+ +
can be solved to give
f(x) = c 0 + c 1 x + c 2 x 2 + ….. + c 10 x 10
Where
c =
j=10
∑
i j j
j=0
b y p(j,10,10-i)
b i =(-1) i /{i! (10-i)! h 10 }
p(i,10,j)=sum of product of j terms in all combinations among x 0 , x 1 ,…,x i-1 ,x i+1 ,…, x 10
We have developed a program in C++ for obtaining the polynomial interpolation of the functions
using Lagrange’s formula.
Newton’s divided difference formula: The Lagrange interpolation formula involves very considerable
computation and its use can be quite risky. It is much more efficient to use the divided differences
method for interpolation [11, 16, 26] .
We have divided the interval [a, b] into 10 equal parts with the help of the points
a=x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 =b
and calculated y i = f(x i ) where i=0,1, 2, …. , 10. We have also drawn difference triangle for the function
y=f(x). Difference triangle has been drawn by using the formula
d n y i = d n-1 y i - d n-1 y i-1 where n, i = 0,1, 2, …. , 10
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec. C. 378-390. 380

Polynomial....
R. B. Srivastava et al.
Now, the Newton’s divided difference formula
f(x) = a 0 + a 1 (x-x 0 ) + a 2 (x-x 0 )(x-x 1 ) + …. + a 10 (x-x 0 )(x-x 1 )….(x-x 9 )
Where a n = (d n y 0 )/(n!h n ); n = 0,1, 2, …. , 10
Becomes
f(x) = c 0 + c 1 x + c 2 x 2 + ….. + c 10 x 10
Where
c0=a0-a1*p(1,1)+a2*p(2,2)-a3*p(3,3)+a4*p(4,4)-a5*p(5,5)+a6*p(6,6)-a7*p(7,7)
+a8*p(8,8) -a9*p(9,9)+a10*p(10,10);
c1=a1-a2*p(2,1)+a3*p(3,2)-a4*p(4,3)+a5*p(5,4)-a6*p(6,5)+a7*p(7,6)-a8*p(8,7)
+a9*p(9,8) -a10*p(10,9);
c2=a2-a3*p(3,1)+a4*p(4,2)-a5*p(5,3)+a6*p(6,4)-a7*p(7,5)+a8*p(8,6)-a9*p(9,7)
+a10*p(10,8);
c3=a3-a4*p(4,1)+a5*p(5,2)-a6*p(6,3)+a7*p(7,4)-a8*p(8,5)+a9*p(9,6)-a10*p(10,7);
c4=a4-a5*p(5,1)+a6*p(6,2)-a7*p(7,3)+a8*p(8,4)-a9*p(9,5)+a10*p(10,6);
c5=a5-a6*p(6,1)+a7*p(7,2)-a8*p(8,3)+a9*p(9,4)-a10*p(10,5);
c6=a6-a7*p(7,1)+a8*p(8,2)-a9*p(9,3)+a10*p(10,4);
c7=a7-a8*p(8,1)+a9*p(9,2)-a10*p(10,3);
c8=a8-a9*p(9,1)+a10*p(10,2);
c9=a9-a10*p(10,1);
c10=a10;
p(i,j)=summation of the product of j elements in all combinations among x 0 ,x 1 ,..,x i-1
We have developed a program in C++ for obtaining the polynomial interpolation of the functions
using Newton’s divided difference formula.
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec.C.378-383. 381

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R. B. Srivastava et al.
RESULT AND DISCUSSION
We have divided the interval into 10 equal parts with the help of the points x 0 ,x 1 ,…..,x 10 such that
x i =x 0 +ih, i=0,1,2,…., 10, h=(b-a)/n, n=10
Polynomial obtained by Newton’s divided difference formula using the developed C++ program
is given belowf(x)=
0.00000002 + 0.99996847 x - 0.00000002 x 2 - 0.33215311 x 3 + 0.00000040 x 4 + 0.18856475 x 5 -
0.00000302 x 6 - 0.09845734 x 7 + 0.00000549 x 8 + 0.02747540 x 9 -0.00000289 x 10
Polynomial obtained by Lagrange’s formula for equal intervals using the developed C++ program
is given belowf(x)=
0.99996847 x -0.00000023 x 2 -0.33215025 x 3 -0.00003796 x 4 + 0.18855174 x 5 -0.00002446 x 6 -
0.09860489 x 7 + 0.00003787 x 8 + 0.02746514 x 9 + 0.00000052x 10
Actual value of f(x)=tan -1 x, Calculated value of f(x)=tan -1 x by Newton’s interpolating
polynomial, Calculated value of f(x)=tan -1 x by Lagrange’s interpolating polynomial, Difference between
actual and calculated values of f(x)=tan -1 x by Newton’s interpolating polynomial, Difference between
actual and calculated values of f(x)=tan -1 x by Lagrange’s interpolating polynomial, Percentage error in
the values of f(x)=tan -1 x calculated by Newton’s interpolating polynomial and Percentage error in the
values of f(x)=tan -1 x calculated by Lagrange’s interpolating polynomial at different values of x are given
in Table-1. Graph between actual values of f(x)=tan -1 x and the values calculated by Newton’s
interpolating polynomial at different points in the interval [-1,1] separated by the distance 0.01 is given in
Graph-1.
1
0.5
0
-0.5
-1
Actual value of f(x)=tan-1 x
Calculated value of f(x)=tan-1 x by Newton’s interpolating polynomial
x
Graph-1: Graph between actual values of f(x)=tan -1 x and the values calculated by Newton’s interpolating
polynomial at different points in the interval [-1,1]
Graph between actual values of f(x)=tan -1 x and the values calculated by Lagrange’s interpolating
polynomial at different points in the interval [-1,1] separated by the distance 0.01 is given in Graph-2.
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec. C. 378-390. 382

Polynomial....
R. B. Srivastava et al.
Difference between actual and calculated values of the function f(x)=tan -1 x by Newton’s interpolating
polynomial is shown in Graph-3. Difference between actual and calculated values of the function
f(x)=tan -1 x by Lagrange’s interpolating polynomial is shown in Graph-4. Maximum percentage error in
the values obtained by Newton’s interpolating polynomial is 0.014126% and by Lagrange’s interpolating
polynomial is 0.03246%. Thus in the case of the function tan -1 x, Newton’s interpolating polynomial is
approximately 2 times better than Lagrange’s interpolating polynomial.
1
f(x)
0
-1
x
Actual value of f(x)=tan-1 x
Calculated value of f(x) =tan-1 x by Lagrange’s interpolating polynomial
Graph-2: Graph between actual values of f(x)=tan -1 x and the values calculated by Lagrange’s interpolating
polynomial at different points in the interval [-1,1]
0.00015
Difference betw een actual and calculated values of f(x)=tan-1 x by New ton’s interpolating
polynomial
Error in f(x)
0.0001
0.00005
0
Difference between actual and calculated values of f(x)=tan-1 x by
Newton’s interpolating polynomial
Graph-3: Difference between actual and calculated values of the function f(x)=tan -1 x by Newton’s
interpolating polynomial
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec.C.378-383. 383

Polynomial....
R. B. Srivastava et al.
Difference between actual and calculated values of f(x)=tan-1 x by Lagrange’s
interpolating polynomial
Error in f(x)
0.0003
0.00025
0.0002
0.00015
0.0001
0.00005
0
Difference between actual and calculated values of f(x)=tan-1 x by Lagrange’s
interpolating polynomial
Graph-4: Difference between actual and calculated values of the function f(x)=tan -1 x by Lagrange’s
interpolating polynomial
CONCLUSION
Newton’s interpolating polynomial is approximately 2 times better than Lagrange’s interpolating
polynomial.
Table-1: Actual value of f(x)=tan -1 x, Calculated value of f(x)=tan -1 x by Newton’s interpolating polynomial,
Calculated value of f(x) =tan -1 x by Lagrange’s interpolating polynomial, Difference between actual and
calculated values of f(x)=tan -1 x by Newton’s interpolating polynomial, Difference between actual and
calculated values of f(x)=tan -1 x by Lagrange’s interpolating polynomial, Percentage error in the values of
f(x)=tan -1 x calculated by Newton’s interpolating polynomial and Percentage error in the values of f(x)=tan -1 x
calculated by Lagrange’s interpolating polynomial at different values of x
x
Actual value
of f(x)=tan -1 x
Calculated
value of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Calculated
value of f(x)
=tan -1 x by
Lagrange’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Lagrange’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Newton’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Lagrange’s
interpolating
polynomial
-1.00 -0.78539819 -0.78539813 -0.78525442 0.00000006 0.00014377 -0.00000764 -0.01830536
-0.99 -0.78037310 -0.78033477 -0.78020257 0.00003833 0.00017053 -0.00491175 -0.02185237
-0.98 -0.77529752 -0.77523106 -0.77510965 0.00006646 0.00018787 -0.00857219 -0.02423199
-0.97 -0.77017093 -0.77008504 -0.76997370 0.00008589 0.00019723 -0.01115207 -0.02560860
-0.96 -0.76499283 -0.76489466 -0.76479250 0.00009817 0.00020033 -0.01283280 -0.02618717
-0.95 -0.75976276 -0.75965792 -0.75956470 0.00010484 0.00019806 -0.01379904 -0.02606866
-0.94 -0.75448024 -0.75437355 -0.75428826 0.00010669 0.00019198 -0.01414086 -0.02544533
-0.93 -0.74914467 -0.74904001 -0.74896199 0.00010466 0.00018268 -0.01397060 -0.02438514
-0.92 -0.74375564 -0.74365568 -0.74358469 0.00009996 0.00017095 -0.01343990 -0.02298470
-0.91 -0.73831260 -0.73821950 -0.73815507 0.00009310 0.00015753 -0.01260983 -0.02133649
-0.90 -0.73281515 -0.73273033 -0.73267180 0.00008482 0.00014335 -0.01157454 -0.01956155
-0.89 -0.72726274 -0.72718722 -0.72713417 0.00007552 0.00012857 -0.01038414 -0.01767862
-0.88 -0.72165489 -0.72158915 -0.72154129 0.00006574 0.00011360 -0.00910962 -0.01574160
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec. C. 378-390. 384

Polynomial....
R. B. Srivastava et al.
x
Actual value
of f(x)=tan -1 x
Calculated
value of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Calculated
value of f(x)
=tan -1 x by
Lagrange’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Lagrange’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Newton’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Lagrange’s
interpolating
polynomial
-0.87 -0.71599120 -0.71593535 -0.71589220 0.00005585 0.00009900 -0.00780038 -0.01382699
-0.86 -0.71027106 -0.71022505 -0.71018612 0.00004601 0.00008494 -0.00647781 -0.01195881
-0.85 -0.70449412 -0.70445752 -0.70442265 0.00003660 0.00007147 -0.00519522 -0.01014487
-0.84 -0.69865990 -0.69863206 -0.69860101 0.00002784 0.00005889 -0.00398477 -0.00842899
-0.83 -0.69276792 -0.69274837 -0.69272047 0.00001955 0.00004745 -0.00282201 -0.00684934
-0.82 -0.68681777 -0.68680555 -0.68678081 0.00001222 0.00003696 -0.00177922 -0.00538134
-0.81 -0.68080896 -0.68080324 -0.68078130 0.00000572 0.00002766 -0.00084018 -0.00406281
-0.80 -0.67474103 -0.67474097 -0.67472160 0.00000006 0.00001943 -0.00000889 -0.00287962
-0.79 -0.66861367 -0.66861856 -0.66860139 0.00000489 0.00001228 -0.00073136 -0.00183664
-0.78 -0.66242641 -0.66243511 -0.66242021 0.00000870 0.00000620 -0.00131335 -0.00093595
-0.77 -0.65617883 -0.65619087 -0.65617776 0.00001204 0.00000107 -0.00183487 -0.00016307
-0.76 -0.64987057 -0.64988494 -0.64987361 0.00001437 0.00000304 -0.00221121 -0.00046779
-0.75 -0.64350128 -0.64351720 -0.64350742 0.00001592 0.00000614 -0.00247397 -0.00095416
-0.74 -0.63707048 -0.63708746 -0.63707924 0.00001698 0.00000876 -0.00266533 -0.00137504
-0.73 -0.63057792 -0.63059539 -0.63058835 0.00001747 0.00001043 -0.00277047 -0.00165404
-0.72 -0.62402320 -0.62404048 -0.62403470 0.00001728 0.00001150 -0.00276913 -0.00184288
-0.71 -0.61740607 -0.61742282 -0.61741811 0.00001675 0.00001204 -0.00271296 -0.00195009
-0.70 -0.61072618 -0.61074191 -0.61073822 0.00001573 0.00001204 -0.00257562 -0.00197142
-0.69 -0.60398316 -0.60399783 -0.60399479 0.00001467 0.00001163 -0.00242888 -0.00192555
-0.68 -0.59717685 -0.59719014 -0.59718782 0.00001329 0.00001097 -0.00222547 -0.00183698
-0.67 -0.59030694 -0.59031856 -0.59031695 0.00001162 0.00001001 -0.00196847 -0.00169573
-0.66 -0.58337325 -0.58338326 -0.58338213 0.00001001 0.00000888 -0.00171588 -0.00152218
-0.65 -0.57637548 -0.57638365 -0.57638299 0.00000817 0.00000751 -0.00141748 -0.00130297
-0.64 -0.56931341 -0.56931984 -0.56931961 0.00000643 0.00000620 -0.00112943 -0.00108903
-0.63 -0.56218702 -0.56219167 -0.56219190 0.00000465 0.00000488 -0.00082713 -0.00086804
-0.62 -0.55499601 -0.55499899 -0.55499935 0.00000298 0.00000334 -0.00053694 -0.00060181
-0.61 -0.54774028 -0.54774177 -0.54774237 0.00000149 0.00000209 -0.00027203 -0.00038157
-0.60 -0.54041976 -0.54041970 -0.54042077 0.00000006 0.00000101 -0.00001110 -0.00018689
-0.59 -0.53303438 -0.53303313 -0.53303421 0.00000125 0.00000017 -0.00023451 -0.00003189
-0.58 -0.52558410 -0.52558166 -0.52558291 0.00000244 0.00000119 -0.00046425 -0.00022641
-0.57 -0.51806885 -0.51806539 -0.51806670 0.00000346 0.00000215 -0.00066786 -0.00041500
-0.56 -0.51048863 -0.51048446 -0.51048595 0.00000417 0.00000268 -0.00081686 -0.00052499
-0.55 -0.50284356 -0.50283879 -0.50284016 0.00000477 0.00000340 -0.00094861 -0.00067615
-0.54 -0.49513361 -0.49512839 -0.49512982 0.00000522 0.00000379 -0.00105426 -0.00076545
-0.53 -0.48735893 -0.48735341 -0.48735493 0.00000552 0.00000400 -0.00113264 -0.00082075
-0.52 -0.47951967 -0.47951406 -0.47951555 0.00000561 0.00000412 -0.00116992 -0.00085919
-0.51 -0.47161594 -0.47161037 -0.47161183 0.00000557 0.00000411 -0.00118105 -0.00087147
-0.50 -0.46364799 -0.46364257 -0.46364406 0.00000542 0.00000393 -0.00116899 -0.00084763
-0.49 -0.45561606 -0.45561087 -0.45561236 0.00000519 0.00000370 -0.00113912 -0.00081209
-0.48 -0.44752038 -0.44751561 -0.44751704 0.00000477 0.00000334 -0.00106587 -0.00074633
-0.47 -0.43936130 -0.43935704 -0.43935838 0.00000426 0.00000292 -0.00096959 -0.00066460
-0.46 -0.43113917 -0.43113542 -0.43113670 0.00000375 0.00000247 -0.00086979 -0.00057290
-0.45 -0.42285436 -0.42285120 -0.42285246 0.00000316 0.00000190 -0.00074730 -0.00044933
-0.44 -0.41450733 -0.41450480 -0.41450596 0.00000253 0.00000137 -0.00061036 -0.00033051
-0.43 -0.40609851 -0.40609667 -0.40609771 0.00000184 0.00000080 -0.00045309 -0.00019700
-0.42 -0.39762846 -0.39762723 -0.39762831 0.00000123 0.00000015 -0.00030933 -0.00003772
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec.C.378-383. 385

Polynomial....
R. B. Srivastava et al.
x
Actual value
of f(x)=tan -1 x
Calculated
value of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Calculated
value of f(x)
=tan -1 x by
Lagrange’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Lagrange’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Newton’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Lagrange’s
interpolating
polynomial
-0.41 -0.38909772 -0.38909712 -0.38909808 0.00000060 0.00000036 -0.00015420 -0.00009252
-0.40 -0.38050687 -0.38050687 -0.38050777 0.00000000 0.00000090 0.00000000 -0.00023653
-0.39 -0.37185657 -0.37185711 -0.37185791 0.00000054 0.00000134 -0.00014522 -0.00036035
-0.38 -0.36314753 -0.36314857 -0.36314934 0.00000104 0.00000181 -0.00028638 -0.00049842
-0.37 -0.35438046 -0.35438201 -0.35438275 0.00000155 0.00000229 -0.00043738 -0.00064620
-0.36 -0.34555611 -0.34555802 -0.34555873 0.00000191 0.00000262 -0.00055273 -0.00075820
-0.35 -0.33667538 -0.33667764 -0.33667830 0.00000226 0.00000292 -0.00067127 -0.00086730
-0.34 -0.32773906 -0.32774159 -0.32774213 0.00000253 0.00000307 -0.00077196 -0.00093672
-0.33 -0.31874815 -0.31875083 -0.31875136 0.00000268 0.00000321 -0.00084079 -0.00100706
-0.32 -0.30970353 -0.30970636 -0.30970678 0.00000283 0.00000325 -0.00091378 -0.00104939
-0.31 -0.30060628 -0.30060908 -0.30060956 0.00000280 0.00000328 -0.00093145 -0.00109113
-0.30 -0.29145741 -0.29146019 -0.29146060 0.00000278 0.00000319 -0.00095383 -0.00109450
-0.29 -0.28225803 -0.28226078 -0.28226107 0.00000275 0.00000304 -0.00097429 -0.00107703
-0.28 -0.27300933 -0.27301189 -0.27301225 0.00000256 0.00000292 -0.00093770 -0.00106956
-0.27 -0.26371250 -0.26371482 -0.26371509 0.00000232 0.00000259 -0.00087975 -0.00098213
-0.26 -0.25436872 -0.25437081 -0.25437105 0.00000209 0.00000233 -0.00082164 -0.00091599
-0.25 -0.24497934 -0.24498112 -0.24498132 0.00000178 0.00000198 -0.00072659 -0.00080823
-0.24 -0.23554565 -0.23554711 -0.23554727 0.00000146 0.00000162 -0.00061984 -0.00068776
-0.23 -0.22606906 -0.22607015 -0.22607030 0.00000109 0.00000124 -0.00048215 -0.00054850
-0.22 -0.21655098 -0.21655169 -0.21655183 0.00000071 0.00000085 -0.00032787 -0.00039252
-0.21 -0.20699286 -0.20699319 -0.20699334 0.00000033 0.00000048 -0.00015943 -0.00023189
-0.20 -0.19739622 -0.19739620 -0.19739631 0.00000002 0.00000009 -0.00001013 -0.00004559
-0.19 -0.18776260 -0.18776223 -0.18776233 0.00000037 0.00000027 -0.00019706 -0.00014380
-0.18 -0.17809360 -0.17809288 -0.17809296 0.00000072 0.00000064 -0.00040428 -0.00035936
-0.17 -0.16839081 -0.16838978 -0.16838984 0.00000103 0.00000097 -0.00061167 -0.00057604
-0.16 -0.15865591 -0.15865459 -0.15865465 0.00000132 0.00000126 -0.00083199 -0.00079417
-0.15 -0.14889060 -0.14888903 -0.14888908 0.00000157 0.00000152 -0.00105447 -0.00102088
-0.14 -0.13909659 -0.13909481 -0.13909487 0.00000178 0.00000172 -0.00127969 -0.00123655
-0.13 -0.12927565 -0.12927373 -0.12927377 0.00000192 0.00000188 -0.00148520 -0.00145426
-0.12 -0.11942957 -0.11942754 -0.11942757 0.00000203 0.00000200 -0.00169975 -0.00167463
-0.11 -0.10956018 -0.10955808 -0.10955811 0.00000210 0.00000207 -0.00191675 -0.00188937
-0.10 -0.09966930 -0.09966720 -0.09966724 0.00000210 0.00000206 -0.00210697 -0.00206684
-0.09 -0.08975883 -0.08975676 -0.08975678 0.00000207 0.00000205 -0.00230618 -0.00228390
-0.08 -0.07983065 -0.07982867 -0.07982869 0.00000198 0.00000196 -0.00248025 -0.00245520
-0.07 -0.06988666 -0.06988482 -0.06988484 0.00000184 0.00000182 -0.00263283 -0.00260422
-0.06 -0.05992882 -0.05992715 -0.05992717 0.00000167 0.00000165 -0.00278664 -0.00275327
-0.05 -0.04995905 -0.04995760 -0.04995762 0.00000145 0.00000143 -0.00290238 -0.00286234
-0.04 -0.03997935 -0.03997814 -0.03997815 0.00000121 0.00000120 -0.00302656 -0.00300155
-0.03 -0.02999166 -0.02999073 -0.02999075 0.00000093 0.00000091 -0.00310086 -0.00303418
-0.02 -0.01999799 -0.01999735 -0.01999737 0.00000064 0.00000062 -0.00320032 -0.00310031
-0.01 -0.01000033 -0.00999999 -0.01000001 0.00000034 0.00000032 -0.00339989 -0.00319989
0.00 -0.00000066 -0.00000064 -0.00000066 0.00000002 0.00000000 -3.03030303 0.00000000
0.01 0.00999901 0.00999871 0.00999869 0.00000030 0.00000032 0.00300030 0.00320032
0.02 0.01999667 0.01999607 0.01999605 0.00000060 0.00000062 0.00300050 0.00310052
0.03 0.02999035 0.02998945 0.02998943 0.00000090 0.00000092 0.00300097 0.00306765
0.04 0.03997803 0.03997686 0.03997684 0.00000117 0.00000119 0.00292661 0.00297663
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec. C. 378-390. 386

Polynomial....
R. B. Srivastava et al.
x
Actual value
of f(x)=tan -1 x
Calculated
value of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Calculated
value of f(x)
=tan -1 x by
Lagrange’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Lagrange’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Newton’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Lagrange’s
interpolating
polynomial
0.05 0.04995774 0.04995633 0.04995631 0.00000141 0.00000143 0.00282239 0.00286242
0.06 0.05992750 0.05992587 0.05992585 0.00000163 0.00000165 0.00271995 0.00275333
0.07 0.06988535 0.06988355 0.06988353 0.00000180 0.00000182 0.00257565 0.00260427
0.08 0.07982934 0.07982740 0.07982738 0.00000194 0.00000196 0.00243018 0.00245524
0.09 0.08975752 0.08975549 0.08975548 0.00000203 0.00000204 0.00226165 0.00227279
0.10 0.09966800 0.09966593 0.09966591 0.00000207 0.00000209 0.00207690 0.00209696
0.11 0.10955887 0.10955682 0.10955679 0.00000205 0.00000208 0.00187114 0.00189852
0.12 0.11942827 0.11942628 0.11942625 0.00000199 0.00000202 0.00166627 0.00169139
0.13 0.12927435 0.12927248 0.12927245 0.00000187 0.00000190 0.00144654 0.00146974
0.14 0.13909531 0.13909358 0.13909355 0.00000173 0.00000176 0.00124375 0.00126532
0.15 0.14888932 0.14888780 0.14888777 0.00000152 0.00000155 0.00102089 0.00104104
0.16 0.15865463 0.15865336 0.15865330 0.00000127 0.00000133 0.00080048 0.00083830
0.17 0.16838953 0.16838855 0.16838850 0.00000098 0.00000103 0.00058198 0.00061168
0.18 0.17809233 0.17809165 0.17809157 0.00000068 0.00000076 0.00038182 0.00042674
0.19 0.18776134 0.18776101 0.18776095 0.00000033 0.00000039 0.00017576 0.00020771
0.20 0.19739497 0.19739497 0.19739491 0.00000000 0.00000006 0.00000000 0.00003040
0.21 0.20699160 0.20699199 0.20699191 0.00000039 0.00000031 0.00018841 0.00014976
0.22 0.21654972 0.21655050 0.21655038 0.00000078 0.00000066 0.00036019 0.00030478
0.23 0.22606781 0.22606896 0.22606882 0.00000115 0.00000101 0.00050870 0.00044677
0.24 0.23554441 0.23554590 0.23554577 0.00000149 0.00000136 0.00063258 0.00057739
0.25 0.24497810 0.24497993 0.24497977 0.00000183 0.00000167 0.00074701 0.00068169
0.26 0.25436750 0.25436962 0.25436941 0.00000212 0.00000191 0.00083344 0.00075088
0.27 0.26371127 0.26371363 0.26371339 0.00000236 0.00000212 0.00089492 0.00080391
0.28 0.27300814 0.27301070 0.27301046 0.00000256 0.00000232 0.00093770 0.00084979
0.29 0.28225684 0.28225958 0.28225929 0.00000274 0.00000245 0.00097075 0.00086800
0.30 0.29145619 0.29145902 0.29145870 0.00000283 0.00000251 0.00097099 0.00086119
0.31 0.30060509 0.30060795 0.30060753 0.00000286 0.00000244 0.00095141 0.00081170
0.32 0.30970234 0.30970517 0.30970469 0.00000283 0.00000235 0.00091378 0.00075879
0.33 0.31874695 0.31874967 0.31874919 0.00000272 0.00000224 0.00085334 0.00070275
0.34 0.32773790 0.32774046 0.32773983 0.00000256 0.00000193 0.00078111 0.00058889
0.35 0.33667421 0.33667648 0.33667579 0.00000227 0.00000158 0.00067424 0.00046930
0.36 0.34555495 0.34555694 0.34555614 0.00000199 0.00000119 0.00057589 0.00034437
0.37 0.35437930 0.35438088 0.35437998 0.00000158 0.00000068 0.00044585 0.00019188
0.38 0.36314639 0.36314750 0.36314648 0.00000111 0.00000009 0.00030566 0.00002478
0.39 0.37185544 0.37185600 0.37185490 0.00000056 0.00000054 0.00015060 0.00014522
0.40 0.38050574 0.38050577 0.38050449 0.00000003 0.00000125 0.00000788 0.00032851
0.41 0.38909659 0.38909602 0.38909456 0.00000057 0.00000203 0.00014649 0.00052172
0.42 0.39762735 0.39762610 0.39762446 0.00000125 0.00000289 0.00031436 0.00072681
0.43 0.40609741 0.40609553 0.40609372 0.00000188 0.00000369 0.00046294 0.00090865
0.44 0.41450623 0.41450375 0.41450164 0.00000248 0.00000459 0.00059830 0.00110734
0.45 0.42285326 0.42285016 0.42284784 0.00000310 0.00000542 0.00073311 0.00128177
0.46 0.43113810 0.43113437 0.43113181 0.00000373 0.00000629 0.00086515 0.00145893
0.47 0.43936023 0.43935600 0.43935314 0.00000423 0.00000709 0.00096276 0.00161371
0.48 0.44751930 0.44751462 0.44751143 0.00000468 0.00000787 0.00104576 0.00175858
0.49 0.45561498 0.45560992 0.45560637 0.00000506 0.00000861 0.00111059 0.00188975
0.50 0.46364695 0.46364158 0.46363765 0.00000537 0.00000930 0.00115821 0.00200584
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec.C.378-383. 387

Polynomial....
R. B. Srivastava et al.
x
Actual value
of f(x)=tan -1 x
Calculated
value of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Calculated
value of f(x)
=tan -1 x by
Lagrange’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Lagrange’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Newton’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Lagrange’s
interpolating
polynomial
0.51 0.47161490 0.47160938 0.47160503 0.00000552 0.00000987 0.00117045 0.00209281
0.52 0.47951862 0.47951302 0.47950828 0.00000560 0.00001034 0.00116784 0.00215633
0.53 0.48735791 0.48735246 0.48734725 0.00000545 0.00001066 0.00111827 0.00218730
0.54 0.49513260 0.49512738 0.49512157 0.00000522 0.00001103 0.00105426 0.00222769
0.55 0.50284255 0.50283778 0.50283134 0.00000477 0.00001121 0.00094861 0.00222933
0.56 0.51048762 0.51048350 0.51047653 0.00000412 0.00001109 0.00080707 0.00217243
0.57 0.51806784 0.51806456 0.51805675 0.00000328 0.00001109 0.00063312 0.00214065
0.58 0.52558309 0.52558070 0.52557224 0.00000239 0.00001085 0.00045473 0.00206437
0.59 0.53303343 0.53303212 0.53302288 0.00000131 0.00001055 0.00024576 0.00197924
0.60 0.54041880 0.54041892 0.54040873 0.00000012 0.00001007 0.00002221 0.00186337
0.61 0.54773933 0.54774082 0.54772967 0.00000149 0.00000966 0.00027203 0.00176361
0.62 0.55499506 0.55499804 0.55498606 0.00000298 0.00000900 0.00053694 0.00162164
0.63 0.56218606 0.56219071 0.56217754 0.00000465 0.00000852 0.00082713 0.00151551
0.64 0.56931251 0.56931895 0.56930447 0.00000644 0.00000804 0.00113119 0.00141223
0.65 0.57637453 0.57638264 0.57636702 0.00000811 0.00000751 0.00140707 0.00130297
0.66 0.58337229 0.58338225 0.58336514 0.00000996 0.00000715 0.00170731 0.00122563
0.67 0.59030604 0.59031767 0.59029925 0.00001163 0.00000679 0.00197016 0.00115025
0.68 0.59717596 0.59718925 0.59716904 0.00001329 0.00000692 0.00222547 0.00115879
0.69 0.60398227 0.60399699 0.60397518 0.00001472 0.00000709 0.00243716 0.00117388
0.70 0.61072528 0.61074114 0.61071748 0.00001586 0.00000780 0.00259691 0.00127717
0.71 0.61740518 0.61742198 0.61739624 0.00001680 0.00000894 0.00272107 0.00144800
0.72 0.62402236 0.62403965 0.62401187 0.00001729 0.00001049 0.00277073 0.00168103
0.73 0.63057709 0.63059437 0.63056439 0.00001728 0.00001270 0.00274035 0.00201403
0.74 0.63706964 0.63708663 0.63705426 0.00001699 0.00001538 0.00266690 0.00241418
0.75 0.64350045 0.64351642 0.64348143 0.00001597 0.00001902 0.00248174 0.00295571
0.76 0.64986974 0.64988399 0.64984643 0.00001425 0.00002331 0.00219275 0.00358687
0.77 0.65617806 0.65618992 0.65614927 0.00001186 0.00002879 0.00180744 0.00438753
0.78 0.66242564 0.66243452 0.66239053 0.00000888 0.00003511 0.00134053 0.00530022
0.79 0.66861290 0.66861767 0.66857052 0.00000477 0.00004238 0.00071342 0.00633850
0.80 0.67474025 0.67474020 0.67468947 0.00000005 0.00005078 0.00000741 0.00752586
0.81 0.68080813 0.68080252 0.68074781 0.00000561 0.00006032 0.00082402 0.00886006
0.82 0.68681699 0.68680477 0.68674594 0.00001222 0.00007105 0.00177922 0.01034482
0.83 0.69276714 0.69274747 0.69268453 0.00001967 0.00008261 0.00283934 0.01192464
0.84 0.69865912 0.69863147 0.69856375 0.00002765 0.00009537 0.00395758 0.01365043
0.85 0.70449340 0.70445681 0.70438427 0.00003659 0.00010913 0.00519380 0.01549056
0.86 0.71027035 0.71022433 0.71014667 0.00004602 0.00012368 0.00647922 0.01741309
0.87 0.71599042 0.71593457 0.71585155 0.00005585 0.00013887 0.00780038 0.01939551
0.88 0.72165418 0.72158849 0.72149932 0.00006569 0.00015486 0.00910270 0.02145903
0.89 0.72726202 0.72718656 0.72709119 0.00007546 0.00017083 0.01037590 0.02348947
0.90 0.73281443 0.73272967 0.73262793 0.00008476 0.00018650 0.01156637 0.02544983
0.91 0.73831189 0.73821878 0.73811007 0.00009311 0.00020182 0.01261120 0.02733533
0.92 0.74375492 0.74365497 0.74353892 0.00009995 0.00021600 0.01343857 0.02904182
0.93 0.74914396 0.74903929 0.74891549 0.00010467 0.00022847 0.01397195 0.03049748
0.94 0.75447953 0.75437295 0.75424087 0.00010658 0.00023866 0.01412629 0.03163240
0.95 0.75976211 0.75965738 0.75951642 0.00010473 0.00024569 0.01378458 0.03233775
0.96 0.76499218 0.76489395 0.76474386 0.00009823 0.00024832 0.01284065 0.03246046
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec. C. 378-390. 388

Polynomial....
R. B. Srivastava et al.
x
Actual value
of f(x)=tan -1 x
Calculated
value of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Calculated
value of f(x)
=tan -1 x by
Lagrange’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Newton’s
interpolating
polynomial
Difference
between actual
and calculated
values of
f(x)=tan -1 x by
Lagrange’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Newton’s
interpolating
polynomial
Percentage
error in the
values of
f(x)=tan -1 x
calculated by
Lagrange’s
interpolating
polynomial
0.97 0.77017027 0.77008438 0.76992458 0.00008589 0.00024569 0.01115208 0.03190074
0.98 0.77529687 0.77523035 0.77506047 0.00006652 0.00023640 0.00857994 0.03049155
0.99 0.78037244 0.78033406 0.78015321 0.00003838 0.00021923 0.00491816 0.02809300
1.00 0.78539753 0.78539747 0.78520530 0.00000006 0.00019223 0.00000764 0.02447550
Maximum percentage error 0.014126 0.03246
REFERENCES
1. E. J. McShane, American Mathematical Monthly, 53(5), 1946, 259.
2. P. M. Hummel, American Mathematical Monthly, 54(4), 1947, 218.
3. B. Fischer, Lothar Reichel Mathematics of Computation, 53(187), 1989, 265.
4. Monatsh, Math., 131, 2000, 215.
5. Michael I. Ganzburg, Bull. Austral. Math. Soc., 70, 2004, 475.
6. Kunkle, Thomas Journal of approximation theory, 71(1), 1992, 94.
7. Egecioglu, Omer, E. Gallopoulos, Koc, K. Cetin, Journal of complexity, 5(4), 1989, 417.
8. Goodyear, H. William, J. Astronaut. Sci., 34(3), 1986, 287.
9. A. McCurdy, K. C. Ng, B. N. Parlett, Mathematics of Computation, 168, 1998, 501.
10. R. Howell, J. Mathews, The AMATYC Review, 14(1), 1992, 20.
11. Al-Hussainan, A. Adel, Al-Eideh, M. Basel, Al-Zalzalah, S. H.Yousef, Int. J. Appl. Math.,
7(3), 2001, 325.
12. R. J. Pan, Journal- Fujian Teachers University Natural Science Edition, 17(2), 2001, 28.
13. M. Higashi, K. Amada, Journal- Japan Society for Precision Engineering, 67(5), 2001,
749.
14. Argyros, K. Ioannis, Catinas, Emil, Pavaloiu, Ion Adv. Nonlinear Var. Inequal., 3(1), 2000,
7.
15. Rabut, Christophe, SIAM J. Numer. Anal., 38(4), 2000, 1294.
16. A. E. Al-Ayyoub, Comput. & Structures 58(4), 1996, 689.
17. Kannappan, Pl. C. R. Math. Rep. Acad. Sci. Canada, 16(5), 1994, 187.
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec.C.378-383. 389

Polynomial....
R. B. Srivastava et al.
18. Schwaiger, J. Aequationes mathematicae, 48(2/3), 1994, 317.
19. M. Neamtu, SIAM Journal on Numerical Analysis, 29(5), 1992, 1435.
20. M.P. Cullinan, IMA Journal of numerical analysis, 10(4), 1990, 583.
21. E. T. Y. Lee, American Mathematical Monthly, 96(7), 1989, 618.
22. A. McCurdy, K. C. Ng, B. N. Parlett, Mathematics of Computation, 43(168), 1984, 501.
23. J. I. Maeztu, SIAM Journal on Numerical Analysis, 19(5), 1982, 1032.
24. F.T. Krogh, Mathematics of Computation, 33(148), 1979, 1265.
25. G. Mühlbach, Numer. Math., 32(4), 1979, 393.
26. Herbert E. Salzer, Proceedings of the American Mathematical Society, 13(2), 1962, 210.
AUTHORS UNCORRECTED COPY
*Correspondence Author: R. B. Srivastava; Department of Mathematics,
M. L. K. P. G. College, Balrampur, U. P., India
J. Chem. Bio. Phy. Sci. 2011, Vol.1, N0.2, Sec. C. 378-390. 390