Newton's Forward/Backward Difference Interpolation

Newton's Forward/Backward Difference Interpolation

1. Equal Spacing: Newton's Forward and Backward
Dierence Interpolation
Dr. Varun Kumar
Dr. Varun Kumar(IIIT Surat) Unit 2 / Lecture-3 1 / 9

2. Outlines
1
Equal Spacing: Newton's Forward Dierence Formulation
Example
2
Equal Spacing: Newton's Backward Dierence Formulation
Example
Dr. Varun Kumar(IIIT Surat) Unit 2 / Lecture-3 2 / 9

3. Equal Spacing: Newton's Forward Dierence Formula
Important points
)As per the Newton's divide dierence interpolation formula
f(x)=f0+ (xx0)f[x0;x1] + (xx0)(xx1)f[x0;x1;x2] +:::+
(xx0)(xx1)::::(xxn1)f[x0; :::;xn]
(1)
or
f[x0; ::::;xk] =
f[x1; :::::;xk]f[x0; ::::;xk1]
xkx0
)Above expression is valid for arbitrarily spaced nodes.
)In most of the practical experimentation, it can be adopted.
)For some instances, the interval may be equi-spaced, i.e,
x0;x1=x0+h;x2=x0+ 2h; :::::;xn=x0+nh (2)
Dr. Varun Kumar(IIIT Surat) Unit 2 / Lecture-3 3 / 9

4. Continued{
)First forward dierence offatxjby
4fj=fj+1fj
)Second forward dierence offatxjby
4
2
fj=4fj+1 4fj
)Thek
th
forward dierence offatxjby
4
k
fj=4
k1
fj+1 4
k1
fj8k= 1;2; :::
)In case of regular spacing in input (x), then
f[x0;x1; :::::;xk] =
1
k!h
k
4
k
f0 (3)
Dr. Varun Kumar(IIIT Surat) Unit 2 / Lecture-3 4 / 9

5. Continued{
)Fork= 1, from (3)
f[x0;x1] =
f1f0
x1x0
=
1
h
(f1f0) =
1
1!h
4f0 (4)
)Letxk+1=x0+ (k+ 1)h
f[x0; ::::::;xk+1] =
f[x1; :::;xk+1]f[x0; :::;xk]
xk+1x0
=
1
(k+ 1)h
h
1
k!h
k
4
k
f1
1
k!h
k
4
k
f0
i
=
1
(k+ 1)!h
k+1
4
k+1
f0
(5)
Dr. Varun Kumar(IIIT Surat) Unit 2 / Lecture-3 5 / 9

6. Continued{
)Applying Newton's forward dierence interpolation formula
f(x)pn(x) =f0+r4f0+
r(r1)
2!
4
2
f0+:::+
r(r1):::(rn+ 1)
n!
4
n
f0
(6)
wherer=
xx0
h
Q :56. Utilize the four values in the following table and
estimate the error.
Dr. Varun Kumar(IIIT Surat) Unit 2 / Lecture-3 6 / 9

7. Equal Spacing: Newton's Backward Dierence Formulation
)First backward dierence offatxjby
rfj=fjfj1
)Second backward dierence offatxjby
r
2
fj=rfj rfj1
)k
th
backward dierence offatxjby
r
k
fj=r
k1
fj r
k1
fj1 (7)
)As per Newton's backward dierence interpolation formula
f(x)pn(x) =f0+rrf0+
r(r+ 1)
2!
r
2
f0+:::+
r(r+ 1):::(r+n1)
n!
r
n
f0
(8)
wherer=
xx0
h
Dr. Varun Kumar(IIIT Surat) Unit 2 / Lecture-3 7 / 9

8. Example
Q J0(x) forx= 1:72 from
the four values in the following table, using
(a)
(b)
Dr. Varun Kumar(IIIT Surat) Unit 2 / Lecture-3 8 / 9

9. Exercise Problem
1
Calculate the Lagrange polynomialp2(x) for the values
(1:00) = 1:0000, (1:02) = 0:9888, (1:04) = 0:9784 of the gamma
function and from it approximate (1:01) and (1:03).
2
Finde
0:25
ande
0:75
by linear interpolation ofe
x
withx0= 0,
x1= 0:5, andx0= 0:5,x1= 0, respectively. Then ndp2(x) by
quadratic interpolation ofe
x
withx0= 0,x1= 0:5, andx2= 1 and
from ite
0:25
ande
0:75
compare the errors.
3
Solve the above problem using Newton's divide and dierence
interpolation method.
Dr. Varun Kumar(IIIT Surat) Unit 2 / Lecture-3 9 / 9