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1. SARDAR PATEL COLLEGE
OF
ENGINEERING,BAKROL
HUMANITY & SCIENCE DEPARTMENT

2. TOPIC:-INTERPOLATION
SUBJECT:-N.S.M(2130606)
GUIDED by:- A.K.D SIR
SR. NO. NAME ENROLLMENT
NO.
1 PRAJAPATI TIRATH A. 151240106066
BRANCH : CIVIL-B

3. TYPES OF INTERPOLATIONS
• NEWTON’S FORWARD INTERPOLATION
• NEWTON’S BACKWARD INTERPOLATION
• GAUSS FORWARD INTERPOLATION
• GAUSS BACKWARD INTERPOLATION
• STIRLING’S INTERPOLATION
• LAGRANGE’S INTERPOLATION
• NEWTON DIVIDED DIFERENTIAL INTERPOLATION

4. NEWTON’S FORWARD INTERPOLATION
• 𝑦𝑛 𝑥 = 𝑦0 + 𝑝𝚫𝑦0 +
𝑝(𝑝−1)
2!
𝚫2𝑦0 +
𝑝 𝑝−1 𝑝−2
3!
𝚫3𝑦0 +
•
𝑝 𝑝−1 𝑝−2 (𝑝−3)
4!
𝚫4𝑦0

5. NEWTON’S BACKWARD INTERPOLATION
𝑦𝑛 𝑥 = 𝑦0 + 𝑝𝛁𝑦𝑛 +
𝑝(𝑝 − 1)
2!
𝛁2𝑦𝑛 +
𝑝 𝑝 − 1 𝑝 − 2
3!
𝛁3𝑦𝑛 +
𝑝 𝑝 − 1 𝑝 − 2 (𝑝 − 3)
4!
𝛁4𝑦𝑛

6. LAGRANGE’S INTERPOLATION FORMULA
• LAGRANGE’S FORMULA IS APPLICABLE TO PROBLEMS WHERE THE
INDEPENDENCE VARIABLE OCCUR WITH EQUAL AND UNEQUAL
INTERVALS, BUT PREFERABLY THIS FORMULA IS APPLIED IN A SITUATION
WHERE THERE ARE UNEQUAL INTERVALS IN THE GIVEN INDEPENDENCE
SERIES. LET THE VALUES OF THE INDEPENDENCE VARIABLES (X) ARE
GIVEN AS A,B,C,D,.... ETC., AND THE CORRESPONDING VALUES OF THE
FUNCTION (DEPENDENT VARIABLE) AS F(A),F(B),F(C),F(D),...

7. FORMULA

8. GAUSS FORWARD INTER POLATION FORMULA:-
8
...
)
(
3
)
1
)(
1
(
)
1
(
!
2
)
1
(
)
0
(
)
0
(
)
(
2
,
1
,
0
3
2
1
,
0
1











 d
f
x
x
x
f
x
x
f
x
f
x
f
GAUSS BACKWARD INTERPOLATION FORMULA:-
...
2
)
2
(
)
1
(
!
3
)
1
(
)
1
(
2
2
)
1
(
)
0
(
)
0
(
)
(
3
3
2
2
2

















f
f
x
x
f
x
f
f
x
f
x
f

9. • IS THE SIMPLEST FORM OF INTERPOLATION,
CONNECTING TWO DATA POINTS WITH A STRAIGHT
LINE.
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
0
0
0
1
0
1
0
0
1
0
0
1
x
x
x
x
x
f
x
f
x
f
x
f
x
x
x
f
x
f
x
x
x
f
x
f










NEWTON’S DIVIDED DIFFERENTIAL INTERPOLATION

10. Find The value Of 𝒕𝒂𝒏 0.12
𝑥 0.10 0.15 0.20 0.25 0.30
𝑦 = 𝑡𝑎𝑛 𝑥 0.1003 0.1511 0.2027 0.2553 0.3093
EXAMPLES

11. SOLUTION
X Y Δ Δ2 Δ3 Δ4
0.10 0.1003
0.0508
0.15 0.1511 0.0008
0.0516 0.0002
0.20 0.2027 0.0010 0.0002
0.0526 0.0004
0.25 0.2553 0.0014
0.0540
0.30 0.3093

12. APPLYING NEWTON’S FORWARD DIFFERENCE INTERPOLATION FORMULA.
𝑦𝑛 𝑥 = 𝑦0 + 𝑝𝚫𝑦0 +
𝑝(𝑝 − 1)
2!
𝚫2𝑦0 +
𝑝 𝑝 − 1 𝑝 − 2
3!
𝚫3𝑦0 +
𝑝 𝑝 − 1 𝑝 − 2 (𝑝 − 3)
4!
𝚫4𝑦0
HERE 𝑦𝑛 𝑥 = TAN(0.12)
∴
𝑝 =
𝑥−𝑥0
ℎ
=
0.12−0.10
0.05
=
0.02
0.05
= 0.4
∴ 𝑦𝑛 𝑥 = 0.1003 + 0.4 0.0508 +
0.4 0.4−1
2
0.0008 +
0.4 0.4−1 0.4−2
6
0.0002 +
0.4 0.4−1 0.4−2 (0.4−3)
24
0.0002
𝑦𝑛 𝑥 = 0.1205

13. CONSIDER FOLLOWING TABULAR VALUES
DETERMINE Y (300)
𝑥 50 100 150 200 250
𝑦 618 724 805 906 1032

14. Solution
X Y 𝛁 𝛁2 𝛁3 𝛁4
50 618
106
100 724 -25
81 45
150 805 20 -40
101 5
200 906 5
126
250 1032

15. Applying Newton’s Backward Difference interpolation Formula.
𝑦𝑛 𝑥 = 𝑦0 + 𝑝𝛁𝑦𝑛 +
𝑝(𝑝 − 1)
2!
𝛁2𝑦𝑛 +
𝑝 𝑝 − 1 𝑝 − 2
3!
𝛁3𝑦𝑛 +
𝑝 𝑝 − 1 𝑝 − 2 (𝑝 − 3)
4!
𝛁4𝑦𝑛
Here:- 𝑦𝑛 𝑥 = 𝑦𝑛 300
∴ 𝑝 =
𝑥−𝑥𝑛
ℎ
=
300−250
50
= 1
∴ 𝑦𝑛 𝑥 = 1032 + 126 +
1(1+1)
2!
25 +
1 1+1 1+2
3!
5 +
1 1+1 1+2 1+3
4!
(−40)
= 1032 + 126 + 25 + 5 − 4
𝑦𝑛 300 = 1148

17. Find the unique Polynomial of degree 2 such that
P(1)=1, P(3)=27, P(4)=64
Use Lagrange’s method of Interpolation [2002-2003]
Sol: Here x = 1 3 4
P(x) = 1 27 64
Now by Lagrange’s formula

18. 18
)
3
4
)(
1
4
(
)
3
)(
1
)(
4
(
)
4
3
)(
1
3
(
)
4
)(
1
)(
3
(
)
4
1
)(
3
1
(
)
4
)(
3
)(
1
(
)
(















x
x
P
x
x
P
x
x
P
x
P
)
1
)(
3
(
)
3
)(
1
(
64
)
1
)(
2
(
)
4
)(
1
(
27
)
3
)(
2
(
)
4
)(
3
(
1
)
(












x
x
x
x
x
x
x
P
3
)
3
4
(
64
2
)
4
5
(
27
6
12
7
)
(
2
2
2










x
x
x
x
x
x
x
P

19. 19
48
19
8
)
288
114
48
(
6
1
)
384
512
128
108
405
81
12
7
(
6
1
2
2
2
2
2















x
x
x
x
x
x
x
x
x
x

20. •
•THANK YOU