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New Iterative Methods for Interpolation, Numerical Differentiation and Numerical
Integration
M. Ramesh Kumar
Phone No: +91 9840913580
Email address: [email protected]
Home page url: http//ramjan07.page.tl/
Abstract
Through introducing a new iterative formula for divided difference using Neville’s and Aitken’s
algorithms, we study new iterative methods for interpolation, numerical differentiation and numerical integration
formulas with arbitrary order of accuracy for evenly or unevenly spaced data. Basic computer algorithms for new
methods are given.
MSC: 65D05; 65D25; 65D30
Key words: Divided difference; Interpolation; Numerical differentiation; Higher derivatives; Numerical integration;
1. Introduction
Interpolation is used in a wide variety of ways. Originally, it was used to do interpolation in tables defining
common mathematical functions; but that is a far less important use in the present day, due to availability of
computers and calculators. Interpolation is still use in the related problem of extending functions that are known
only at a discrete set of points and such problems occurs frequently when numerically solving differential and
integral equations. Next, Interpolation is used to solve problems from the more general area of interpolation theory.
Interpolation is an important tool in producing computable approximations to commonly used functions. More over,
to numerically integrate or differentiate a function, we often replace the function with simpler approximations and it
is then integrated or differentiated. These simpler expressions are almost always obtained by interpolation [1].
A number of different methods have been developed to construct useful interpolation formulas for evenly
or unevenly spaced points. Newton’s divided difference formula [1,2,3], Lagrange’s formula [1,2,3,10], Neville’s
and Aitken’s iterated interpolation formulas[11,12] are the most popular interpolation formulas for polynomial
interpolation to any arbitrary degree with finite number of points. The Lagrange formula is well suited for many
theoretical uses of interpolation, but it is less desirable when actually computing the value of an interpolating
polynomial [3]. Computation by Lagrange’s method is quite laborious. For any computation the whole data is taken
into calculation. If a new node is added, the computation has to be done afresh. These make Lagrange’s method less
suitable from the practical point of view. In this case, Neville’s and Aitken’s algorithms are very useful to iterate
interpolation formula when a new node is added. Numerical computations by this method are simpler and less
laborious than Lagrange’s method. Also, it have an advantage over the Newton’s interpolation formula is being
very easily programmed for computer.
Numerical approximations to derivatives are used mainly in two ways. First, we are interested in
calculating derivatives of given data that are often obtained empirically. Second, numerical differentiation formulae
are used in deriving numerical methods for solving ordinary and partial differential equations [1]. A number of
different methods have been developed to construct useful formulas for numerical derivatives. Most popular of the
techniques are finite difference type [10], polynomial interpolation type [1,2,3,4], method of undetermined
coefficients [1,2,3,8], and Richardson extrapolation [4,10]. More over, calculations of weights in finite difference
formulas using recursive relations [7], explicit finite difference formulas [9] and few central difference formulas for
finite and infinite data [5,6] are developed to construct useful numerical differentiation. However, when a new node
is added, the computation has to be done afresh. Thus, Iterative formula for numerical differentiation is still to be
developed.
In Section 2, a new iterative formula for divided difference is for evenly or unevenly spaced data. In
Section 3, the new iterative interpolation formulas are presented for both evenly or unequally spaced data derived
with new divided difference table. In Section 4, new iterative methods for higher order numerical differentiation
formulas are presented in recursive approach and also in direct form to any arbitrary order of accuracy for equally or
unequally spaced data. In Section 5, the new numerical integration formulas are derived from differentiation
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formulas, presented in Section 4 and the Taylor formula. In Section 6, is devoted to a brief conclusion.
2. Iterative Formula for divided difference
Definition 2.1
The rth
divided difference of polynomial function )(xP at the points 110 ,,, rxxxx is a polynomial in x,
so we call it as divided difference polynomial of order r of ).(xP It is denoted by 110 ,,, rxxxxP .
Now, Let
],,,,,[][ 1,110,,1, jijjrijjj xxxxxxxPxD (2.1)
For inrrj ,,1, and rni ,,2,1
Equation (2.1) is a divided difference polynomial iterated by the points jijjj xxxx ,,,, 21
],,,,,,,[][ 11,110,,1,1, jiirrrjiirr xxxxxxxxxPxd (2.2)
for niij ,,2,1 and 1,,1, nrri
Equation (2.2) is a divided difference polynomial iterated by the points jiirrr xxxxxx ,,,,,, 121
Note 2.1. As a notation, ,,, cba denotes the smallest interval containing all of real numbers ,,, cba [1].
Theorem 2.1. Let 110 ,, rxxx are ‘r’ numbers and nrr xxx ,,1, are )1( rn distinct numbers in the interval
],[ qp , nr , ],[ qpx and ],[1 qpCf n , then nxxxx ,,, 10
n
ri
i
n
nrrrr xxn
fxDxxxxf )(
)!1(
)(][,,,,
)1(
,,2,1,110
(2.3)
n
ri
i
n
nrrrr xxn
fxdxxxxf )(
)!1(
)(][,,,,
)1(
,,2,1,110
(2.4)
Proof.
Let )(xPn is a polynomial of degree n in ‘x’ that approximates the function f and takes the functional
values )(),(),( 10 nxfxfxf for the arguments nxxx ,, 10 respectively. It can be written as
n
nn xaxaxaaxP 2210)( , All Rsa ' (2.5)
Then, we can write thr order divided difference of )(xPn at the points 110 ,, rxxx in terms of ‘x’ as in the
following form
)(~
,,, 2210110 xPxaxaxaaxxxxP rn
rnrnrn
(Say), All Rsa '
(2.6)
Now )(~
xP rn is a polynomial of degree rn . To interpolate it by Neville’s method of iterated interpolation, we
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3
can use remaining points nrr xxx ,,1, .
xxxD
xxxD
xxxD
jiijjj
jijjj
jjiijjj
][
][1][
,,2,1
1,,1,
,,1,
, inrrj ,,1, and rni ,2,1 (2.7)
After rn iterations, we get thrn )( degree polynomial ].[,,1, xD nrr
i.e, ][,,, ,,1,1210 xDxxxxxP nrrrin (2.8)
Similarly, by Aitken’s method of iterated interpolation,
xxxd
xxxd
xxxd
jjirr
iiirr
ijjiirr
][
][1][
,1,,1,
,1,,1,
,,1,,1,
, niij ,,2,1 and 1,,1, nrri (2.9)
After rn iterations, we get another thrn )( degree polynomial ].[,,1, xd nrr
i.e, ][,,, ,,1,1210 xdxxxxxP nrrrin (2.10)
The above equation is polynomial form of thr divided difference of a polynomial. Since )(xPn approximates the
function )(xf on [p, q], so we have the following two equations (2.11) and (2.12). For some nxxxx ,,, 10
n
i
i
n
n xxn
fxPxf
0
)1(
)()!1(
)()()(
(2.11)
nrrrixxxxxPxxxxxf rinri ,,,2,1,,,,,,,,, 12101210 (2.12)
Using (2.11), we can write
n
ri
i
n
rnr xxn
fxxxxPxxxxf )(
)!1(
)(,,,,,,,,
)1(
110110
(2.13)
Using (2.8) in (2.8), we get (2.3) and (2.10) in (2.13), we get (2.4)
3. Interpolation formulas.
Theorem 3.1. Let 110 ,, rxxx are ‘r’ numbers and nrr xxx ,,1, are )1( rn distinct numbers in the
interval ],[ qp , nr , ],[ qpx and ],[1 qpCf n , for some nxxxx ,,, 10 then
(i).
n
i
i
nr
i
inrrr
i
j
ji
r
i
xxn
fxxxDxxxxxfxf
0
)1(1
0
,,2,1,
1
0
1
1
0
0 )()!1(
)()(][)(],,,[)(
(3.1)
(ii).
n
i
i
nr
i
inrrr
i
j
ji
r
i
xxn
fxxxdxxxxxfxf
0
)1(1
0
,,2,1,
1
0
1
1
0
0 )()!1(
)()(][)(],,,[)(
(3.2)
Proof.
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4
We know that
1
1210210110
],,,[],,,,[],,,,[
r
rrr
xx
xxxxfxxxxfxxxxf
(3.3)
Rearranging this, we get
],,,,[)(],,,[],,,,[ 11011210210 rrrr xxxxfxxxxxxfxxxxf (3.4)
Repeating this, we get
],,,,[)())((
],,,[)())()((
],,[))((],[)()()(
110110
12102210
210101000
rr
rr
xxxxfxxxxxx
xxxxfxxxxxxxx
xxxfxxxxxxfxxxfxf
(3.5)
Using (2.3) in (3.5) and after simplification, we get (3.1), similarly, using (2.4) in (3.5) we get (3.2).
3.1. New Divided Difference Table with Neville’s and Aitken’s scheme
In Newton divided difference table, divided differences of new entries in each column are determined by
divided difference of two neighboring entries in the previous column. But, the procedure of new divided difference
table is different from the Newton divided difference table. For example, consider the argument values ,, 10 xx
62 , xx for the corresponding functional values 6210 ,,,, ffff . As a matter of convenience, we write
)( kk xff . The table 1 is divided into two parts. The first part of the table follows the procedure of New
divided difference table and the second part of the table follows Neville scheme. The first order divided differences
in the third column of the Table 1 are found by the sequence of evaluating ],[ 10 xxf , ],,[ 20 xxf , The second
order divided differences in the fourth column of the Table 1 are found by the sequence of evaluating ],,[ 210 xxxf ,
],,,[ 310 xxxf . Similarly, the sequences ],,,[ 3210 xxxxf , ],,,,[ 4210 xxxxf are evaluated for fifth column.
Then from the sixth column we use Neville’s scheme.
Table 1. New divided difference table with Neville’s Scheme
x y 1
2 3 Neville’s Scheme to construct divided difference polynomial
0x 0f
],[ 10 xxf
1x 1f ],,[ 210 xxxf
],[ 20 xxf ],,,[ 3210 xxxxf
2x 2f ],,[ 310 xxxf ][4,3 xD
],[ 30 xxf ],,,[ 4210 xxxxf ][5,4,3 xD
3x 3f ],,[ 410 xxxf ][5,4 xD ][6,5,4,3 xD
],[ 40 xxf ],,,[ 5210 xxxxf ][6,5,4 xD
4x 4f ],,[ 510 xxxf ][6,5 xD
],[ 50 xxf ],,,[ 6210 xxxxf
5x 5f ],,[ 610 xxxf
],[ 60 xxf
6x 6f
Similar procedure for New divided difference with Aitken’s Scheme
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4. Formulas of Numerical Differentiation
Definition 4.1. Define
xxxx
xxxx
xxxN
jrj
jrj
jj
rjj
1
1
1
)(1,
)(
1
)(
1
1][ ,
xxxN
xxxN
xxxN
jir
jijj
jr
jijj
jji
rjijj
][
][1][
)(,,2,1
)(1,,1,)(
,,1,
(4.1)
xxxx
xf
xxxx
xf
xxxN
jkj
j
jkj
j
jj
kjj
1
1
11
)(1,
)(
)(
)(
)(
1][
~,
xxxN
xxxN
xxxN
jik
jijj
jk
jijj
jji
kjijj
][~
][~
1][
~)(
,,2,1
)(1,,1,)(
,,1,
(4.2)
inj ,,1,0 and ni ,,2,1
][)(
,,1, xNr
jijj and ][~ )(
,,1, xNk
jijj are constructed by Neville’s Algorithm for kr ,,1,0
At the thn iteration, Let )(][
)(,,1,0 xNxN r
rn
, (4.3)
Definition 4.2. Define
xxxx
xxxx
xxxA
jrj
j
rj
)(
1
)(
1
1][
00
0
)(,0
, xxxA
xxxA
xxxA
jr
ji
ir
ii
ij
rjii
][
][1][
)(,1,,1,0
)(,1,1,0)(
,,1,,1,0
, (4.4)
xxxx
xf
xxxx
xf
xxxA
jkj
j
k
j
kj
)(
)(
)(
)(
1][
~0
0
0
0
)(,0 ,
xxxA
xxxA
xxxA
jk
ji
ik
ii
ij
kjii
][~
][~
1][
~)(
,1,,1,0
)(,1,,1,0)(
,,1,,1,0
(4.5)
niij ,,2,1 and 1,,,1,0 nri
][)(
,,1,,1,0 xAr
jii and ][~ )(
,,1,,1,0 xAk
jii are constructed by Aitken’s Algorithm for kr ,,1,0
At the thn iteration , )(][
)(,,1,0 xAxA r
rn (4.6)
Theorem 4.1. Let nxxxx ,,, 10 are )1( n distinct numbers in the interval ],[ qp , Wk and ],[1 qpCf kn
for some nxxxx ,,, 10 then
(i).
n
i
i
knk
nk
kk
xxkn
fxNxN
xfxN
k
xf
k
xf
0
)1()(
,,2,1,01
)1()(
)()!1(
)(][
~)(
!0
)()(
!1
)(
!
)(
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(4.7)
(ii).
n
i
i
nkk
nk
kk
xxnk
fxAxA
xfxA
k
xf
k
xf
0
)1()(
,,2,1,01
)1()(
)()!1(
)(][
~)(
!0
)()(
!1
)(
!
)( (4.8)
Proof.
Let )(xPn is a polynomial of degree n in ‘x’ that approximates the function f. Using Equation (2.3)
n
i
i
kn
n
timesk
xxkn
fxDxxf
0
)1(
,,2,1,0
1
)()!1(
)(][],,[
, (4.9)
Where ][],,,,/,,[ ,,2,1,0210
1
xDxxxxxxP nn
timesk
n
We can write,
xx
xxPxxxP
xxxPi
timesk
n
timesk
in
i
timesk
n
],,[],,,[
],,,[1
)(!1
)(],,,[
)1(1
xxk
xP
xx
xxxP
i
kn
i
timesk
in
2
2
2
)2()1(
)(
],,,[
)(!2
)(
)(!1
)(
xx
xxxP
xxk
xP
xxk
xP
i
timesk
in
i
kn
i
kn
Preceding this, we get
k
i
in
ki
n
i
kn
i
kn
xx
xP
xx
xP
xxk
xP
xxk
xP
)(!0
)(
)(!0
)(
)(!2
)(
)(!1
)(
2
)2()1(
(4.10)
Applying Equation (2.3) for two consecutive data 1, jj xx
1111
1],,[
],,[1
],,,[
jj
timesk
n
jj
timesk
n
jjjj
timesk
n xxxxP
xxxxP
xxxxxxP
(4.11)
Using (4.10) in (4.11), using properties of determinants, we get
][~
][!0
)(][
!2
)(][
!1
)( )(1,
)(1,
)2(1,
)2()1(
1,
)1(
xNxNxP
xNk
xPxN
k
xP kjj
kjj
njj
kn
jj
kn
(4.12)
Similarly, for three consecutive data 21,, jjj xxx
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7
][~
][)(][!2
)(][
!1
)(
],,,,[
)(2,1,
)(2,1,
)2(2,1,
)2()1(
2,1,
)1(
21
1
xNxNxPxNk
xPxN
k
xP
xxxxxP
kjjj
kjjjnjjj
kn
jjj
kn
jjj
timesk
n
(4.13)
Proceeding this, for some i
][~
][)(][!2
)(][
!1
)(
],,,,,,,[
)(,,1,
)(,,1,
)2(,,1,
)2()1(
,,1,
)1(
21
1
xNxNxPxNk
xPxN
k
xP
xxxxxxP
kjijj
kjijjnjijj
kn
jijj
kn
ijjjj
timesk
n
(4.14)
Putting ni and 0j , (i.e at thn iteration)
][~
][)(][!2
)(][
!1
)(
],,,,,,,[
)(,,1,0
)(,,1,0
)2(,,1,0
)2()1(
,,1,0
)1(
210
1
xNxNxPxNk
xPxN
k
xP
xxxxxxP
kn
knnn
kn
n
kn
n
timesk
n
(4.15)
Using Equation (4.3) and (4.9) in (4.15) and nP approximates the function f.
][~
)(!0
)()(
!2
)()(
!1
)(
)()!1(
)(],,[
)(,,1,02
)2(
1
)1(
0
)1(
1
xNxNxf
xNk
xfxN
k
xf
xxkn
fxxf
knk
kk
n
i
i
kn
timesk
(4.16)
After simplification, we get (4.3).
Now, Using (2.4)
n
i
i
kn
n
timesk
xxkn
fxdxxf
0
)1(
,,2,1,0
1
)()!1(
)(][],,[
, (4.17)
Where ][],/,,[ ,,2,1,0,,2,10
1
xdxxxxxxP nn
timesk
n
for some i and j
][~
][)(][!2
)(][
!1
)(
],,,,,,,,[
)(,,,1,0
)(,,,1,0
)2(,,,1,0
)2()1(
,,,1,0
)1(
210
1
xAxAxPxAk
xPxA
k
xP
xxxxxxxP
kji
kjinji
kn
ji
kn
ji
timesk
n
(4.18)
at thn iteration
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][~
][)(][!2
)(][
!1
)(
],,,,,,,[
)(,,1,0
)(,,1,0
)2(,,1,0
)2()1(
,,1,0
)1(
210
1
xAxAxPxAk
xPxA
k
xP
xxxxxxP
kn
knnn
kn
n
kn
n
timesk
n
(4.19)
Using Equation (4.6) and (4.17) in (4.19), nP approximates the function f and after simplification
][~
)(!0
)()(
!2
)()(
!1
)(
)()!1(
)(],,[
)(,,1,02
)2(
1
)1(
0
)1(
1
xAxAxf
xAk
xfxA
k
xf
xxkn
fxxf
knk
kk
n
i
i
kn
timesk
(4.20)
After simplification, we get (4.4),
Theorem 4.2. Let nxxxx ,,, 10 are distinct numbers in the interval ],[ qp , Wt and ],[1 qpCf tn , then
(i).
)!1(
)(
)!(
)(
)!1(
)()(
][~
][~
][~
][~
!
)(
)1(1
)(
10
)1(
0
0
)0(,,2,1,0
)2(,,2,1,02
)1(,,2,1,01
)(,,2,1,00
)(
n
fa
nt
fa
nt
faxx
xNaxNaxNaxNat
xf
tn
t
ntntn
i
i
ntt
nt
nt
n
t
(4.21)
10 a , 1122110 NaNaNaNaa kkkkk , tk ,,2,1 and ni xxxx ,,, 10 , ti ,,2,1,0
(ii).
)!1(
)(
)!(
)(
)!1(
)()(
][~
][~
][~
][~
!
)(
)1(1
)(
10
)1(
0
0
)0(,,2,1,0
)2(,,2,1,02
)1(,,2,1,01
)(,,2,1,00
)(
n
fa
nt
fa
nt
faxx
xAaxAaxAaxAat
xf
tn
t
ntntn
i
i
ntt
nt
nt
n
t
(4.22)
10 a , 1122110 AaAaAaAaa kkkkk , tk ,,2,1 and ni xxxx ,,, 10 , ti ,,2,1,0
Proof:
We can write for some ,2,1,0t using equation (4.3)
)!1(
)(
)!(
)(
)!1(
)()(
][~
][~
][~
][~
!
)(
)1(1
)(
10
)1(
0
0
)(,,2,1,0
)1(,,2,1,02
)1(,,2,1,01
)0(,,2,1,00
)(
n
fa
nt
fa
nt
faxx
xNaxNaxNaxNat
xf
tn
t
ntntn
i
i
kntnnn
t
(4.23)
with all unknown a’s, To these a’s , Put 1)( xf in (4.23), If 0t , then 10 a and for ,4,3,2,1t we have
0022110 NaNaNaNa tttt (4.24)
Using (4.23) and (4.24) , we obtain (4.21).
Similarly, we can write for some ,2,1,0t using equation (4.4)
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)!1(
)(
)!(
)(
)!1(
)()(
][~
][~
][~
][~
!
)(
)1(1
)(
10
)1(
0
0
)0(,,2,1,0
)2(,,2,1,02
)1(,,2,1,01
)(,,2,1,00
)(
n
fa
nt
fa
nt
faxx
xAaxAaxAaxAat
xf
tn
t
ntntn
i
i
ntt
nt
nt
n
t
(4.25)
with all unknown a’s. To these a’s, Put 1)( xf in (4.24), If 0t , then 10 a and for ,4,3,2,1t we have
0022110 AaAaAaAa tttt (4.26)
Using (4.25) and (4.26) , we obtain (4.22).
Algorithm 4.1. For the unevenly spaced points nxxxx ,,,, 210 and known the functional values )( ixf at ix ,
ni ,,2,1,0 then the steps to use (n+1) point formula to estimate tht derivative of f(x) at x are
Step1: For ntok 1 do calculate )(xN k using (4.1) and (4.3)
Step 2: For ntok 1 do calculate ][~ )(
,,2,1,0 xNk
nusing (4.2)
Step3: 10 a , For ntok 1 do 1122110 NaNaNaNaa kkkkk
Step 4: Use (4.21) to find tht derivative at ‘x’.
Algorithm 4.2. For the unevenly spaced points nxxxx ,,,, 210 and known the functional values )( ixf at ix ,
ni ,,2,1,0 then the steps to use (n+1) point formula to estimate tht derivative of f(x) at x are
Step1: For ntok 1 do calculate )(xAk using (4.4) and (4.6)
Step 2: For ntok 1 do calculate ][~ )(
,,2,1,0 xAk
nusing (4.5)
Step3: 10 a , For ntok 1 do 1122110 AaAaAaAaa kkkkk
Step 4: Use (4.22) to find tht derivative at ‘x’.
5. Formulas for Integration
Theorem 5.1. Let nxxxx ,,,, 210 are the distinct numbers in the interval ],[ qp , ],[, qphxx , 0h and if
],[12 qpCf n , with
(i).
)( )!12(
)(
!2
)(
)!1(
)()(
][~
][~
][~
][~
)(
20)12(
11
)2(
0
)1(
0
)(,,2,1,02
)2(,,2,1,01
)1(,,2,1,00
)0(,,2,1,0
nn
nn
nn
nn
i
i
nn
nnnn
hx
x
hOn
f
n
f
n
fxx
xNxNxNxNdxxf
(5.1)
Where 10 a , 1122110 NaNaNaNaa kkkkk , nk ,,2,1
Page 10
10
(ii).
)( )!12(
)(
!2
)(
)!1(
)()(
][~
][~
][~
][~
)(
20)12(
11
)2(
0
)1(
0
)(,,2,1,02
)2(,,2,1,01
)1(,,2,1,00
)0(,,2,1,0
nn
nn
nn
nn
i
i
nn
nnnn
hx
x
hOn
f
n
f
n
fxx
xAxAxAxAdxxf
(5.2)
Where 10 a , 1122110 AaAaAaAaa kkkkk , nk ,,2,1
k121
12
1
1
n
ha
k
ha
k
h n
kn
kk
, nk ,2,1,0 and ni xxxx ,,, 10 , ni ,,2,1,0
Proof.
Using Taylor series on integration
)()!1(
)(!3
)(!2
)()()( 21
)(32
n
nn
hx
x
hOn
hxf
hxf
hxfhxfdxxf (5.3)
Using (4.21) in equation (5.3) and simplifying we obtain,
)()!12(
)(
!2
)(
)!1(
)()(
1][
~][
~][
~][
~
3][
~][
~][
~
2][
~][
~][
~)(
20)12(
11
)2(
0
)1(
0
1)0(
,,2,1,01)1(
,,2,1,01)1(
,,2,1,0)(
,,2,1,0
3
2)0(
,,2,1,01)1(
,,2,1,0)2(
,,2,1,0
2
1)0(
,,2,1,0)1(
,,2,1,0)0(
,,2,1,0
nn
nn
nn
nn
i
i
n
nnnnn
nn
n
nnn
nnn
hx
x
hOn
f
n
f
n
fxx
n
haxNaxNaxNxN
haxNaxNxN
haxNxNhxNdxxf
(5.4)
Rearranging the above equation,
)()!12(
)(
!2
)(
)!1(
)()(
1][
~
143][
~
132][
~
132][
~
20)12(
11
)2(
0
)1(
0
1)(
,,2,1,0
1
2
4
1
3)2(
,,2,1,0
1
1
3
1
2)1(
,,2,1,0
13
2
2
1)0(
,,2,1,0
nn
nn
nn
nn
i
i
nn
n
n
nn
n
nn
n
nn
hOn
f
n
f
n
fxx
n
hxN
n
ha
ha
hxN
n
ha
ha
hxN
n
ha
ha
hahxN
(5.5)
)()!12(
)(
!2
)(
)!1(
)()(
][~
][~
][~
][~
20)12(
11
)2(
0
)1(
0
)(,,2,1,02
)2(,,2,1,01
)1(,,2,1,00
)0(,,2,1,0
nn
nn
nn
nn
i
i
nn
nnnn
hOn
f
n
f
n
fxx
xNxNxNxN
(5.6)
Thus, we obtain equation (5.1). Similarly using (4.22) in (5.3), after simplification, we get (5.2).
Page 11
11
Algorithm 5.1. For the unevenly spaced points nxxxx ,,,, 210 and known the functional values )( ixf at ix ,
ni ,,2,1,0 then the steps to estimate dxxf
hx
x
)( are,
Step1: For ntok 1 do calculate )(xN k using (4.1) and (4.3)
Step 2: For ntok 1 do calculate ][~ )(
,,2,1,0 xNk
nusing (4.2)
Step3: 10 a , For ntok 1 do 1122110 NaNaNaNaa kkkkk
Step 4: For ntok 0 do k121
12
1
1
n
ha
k
ha
k
h n
kn
kk
Step 5: Use (5.1), to find numerically dxxf
hx
x
)(
Algorithm 5.2. For the unevenly spaced points nxxxx ,,,, 210 and known the functional values )( ixf at ix ,
ni ,,2,1,0 then the steps to estimate dxxf
hx
x
)( are,
Step1: For ntok 1 do calculate )(xAk using (4.4) and (4.6)
Step 2: For ntok 1 do calculate ][~ )(
,,2,1,0 xAk
nusing (4.5)
Step3: 10 a , For ntok 1 do 1122110 AaAaAaAaa kkkkk
Step 4: For ntok 0 do k121
12
1
1
n
ha
k
ha
k
h n
kn
kk
Step 5: Use (5.2), to find numerically dxxf
hx
x
)(
6. Conclusion
By introducing a new iterated method for divided difference and new divided difference table, we have
studied iterated methods for interpolation, numerical differentiation and integration formulas with arbitrary order
accuracy for evenly or unevenly spaced data using Neville’s and Aitken’s algorithms. First, we study iterated
interpolation formula which generalizes Newton interpolation formula and Iterated interpolation formula. However
when a new node is added, we have to add one more data to new divided difference table. But new iterated formulas
for higher order derivatives and numerical integration to arbitrary order of accuracy are very handier even when we
add new data for further iteration. Basic computer algorithms are given for new formulas. Through new iterative
method for divided difference, we have studied three major problems of Numerical analysis.
List of References
[1] Kendall E.Atkinson, An Introduction to Numerical Analysis, 2 Ed., John Wiley & Sons, New York, 1989.
[2] Kendall E.Atkinson, Elementary Numerical Analysis, 2 Ed., John Wiley & Sons, NewYork, 1993.
[3] Kendall E.Atkinson, Weimin Han, Elementary Numerical Analysis, 3 Ed., John Wiley & Sons, NewYork,
2004
[4] R.L. Burden, J.D. Faires, Numerical Analysis, seventh ed., Brooks/Cole, Pacific Grove, CA, 2001.
[5] M. Dvornikov, Formulae of Numerical Differentiation, JCAAM, 5, 77-88, (2007) [e- print arXiv:
math.NA/0306092].
[6] M. Dvornikov, Spectral Properties of Numerical Differentiation, JCAAM, 6 (2008), 81-89 [e- print
arXiv: math.NA/0402178].
[7] Bengt Forngberg, Calculation of weights in finite difference formulas, SIAM Rev, Vol 40.No 3,pp 685-691,
Page 12
12
USA, 1998.
[8] C.F. Gerald, P.O. Wheatley, Applied Numerical Analysis, fifth ed., Addison-Wesley Pub. Co., MA, 1994.
[9] J. Li, General Explicit Difference Formulas for Numerical Differentiation, J. Comp. & Appl. Math., 183,
29-52, 2005.
[10] John H. Mathews, Kurtis D.Fink, Numerical methods using MATLAB, 4ed, Pearson Education, USA, 2004.
[11] H. Pandey, Rakesh Kumar Mishra, Computer oriented Numerical Analysis, First Ed, Dominant Publishers &
Distributors, New Delhi, 2003.
[12] S. Rajasekaran, Numerical Methods in Science and Engineering, 2 Ed, S. Chand & Co Ltd, New Delhi, 1999.
