This Presentation helps u to understand newton's forward backward interpolation

1. Presentation
on
Interpolation and forward ,backward ,
central method
In partial fulfillment of the subject
CVNM
Submitted by:
Mitesh Patel (130120119155) / Mechanical / 4C1
Mitul Patel (130120119156) / Mechanical / 4C1
Neel Patel (130120119157) / Mechanical / 4C1
(2140001)
GANDHINAGAR INSTITUTE OF TECHNOLOGY

2. INTERPOLATION AND EXTRAPOLATION
The process of finding the values inside the interval 𝑥0<𝑥< 𝑥 𝑛 is known as
interpolation.
The process of finding the values outside the interval 𝑥0<𝑥< 𝑥 𝑛 is known as
extrapolation.
Interpolation
Forward
interpolation
Backward
interpolation

3. POLYNOMIAL INTERPOLATION
For a two point data a first order(linear)polynomial connecting two
points is used.
For a three point data a second order (quadratic) polynomial connecting
three point used
For four point data a third order (cubic) polynomial connecting three
point.

5. FINITE DIFFERENCES
FINITE
DIFFRENCES
FORWARD
DIFFERENCES
CENTRAL
DIFFERENCES
BACKWARD
DIFFERENCES
 Finite differences are of three types :-

6. RULES OF INTERPOLATION
Interpolation formulas can be used only when the values of the
argument 𝑥 are equidistant.
The point 𝑥0 should be selected very close to the point at which
interpolation is required.
Usually in the forward interpolation the very first value of 𝑥 is taken
equal to 𝑥0.
Backward interpolation is suitable for interpolation near the end of
tabulated values in the backward interpolation.
In backward interpolation the last value of 𝑥 is taken equal to 𝑥 𝑛.

7. FIRST FORWARD DIFFERENCES
The 𝑦1- 𝑦0 , 𝑦2 - 𝑦1 , 𝑦𝑛 - 𝑦 𝑛−1.differences are called the first forward
differences of the function.
y = f (x) and we denote these difference by
∆𝑦0 , ∆𝑦1 , ∆𝑦2……….., ∆𝑦𝑛respectively, where Δ is called the
descending or forward difference operator.
In general, the first forward differences is defined by
Δ𝑦𝑥= 𝑦 𝑥+1– 𝑦𝑥.
where Δ is called first forward difference operator.

8. SECOND FORWARD DIFFRENCE
OPERATOR
The differences of first forward differences are called second forward
differences.
∆ 𝟐
𝑦0=∆𝑦1- ∆𝑦0.
∆ 𝟐 𝑦1 =∆𝑦2 - ∆𝑦1.
∆ 𝟐
𝒚 𝒏−𝟏= ∆𝑦𝑛 - ∆𝑦 𝑛−1.
∆ 𝟐 𝑦0 , ∆ 𝟐 𝑦1,………..,∆ 𝟐 𝒚 𝒏−𝟏are called second forward differences.
where ∆ 𝟐is called second forward difference order.

9. TABLE
Argument
x
Entry
y = f(x)
First
Differences
∆
Second
Differences
∆ 𝟐
Third
Differences
∆ 𝟑
Fourth
Differences
∆ 𝟒
𝑥0
𝑥1
𝑥2
𝑥3
𝑥4
𝑦0
𝑦1
𝑦2
𝑦3
𝑦4
∆𝑦0
∆𝑦1
∆𝑦2
∆𝑦3
∆ 𝟐
𝑦0
∆ 𝟐
𝑦1
∆ 𝟐 𝑦2
∆ 𝟑
𝑦0
∆ 𝟑 𝑦1
∆ 𝟒 𝑦0

12. FIRST BACKWARD DIFFRENCES
The 𝑦1- 𝑦0 , 𝑦2 - 𝑦1 , 𝑦𝑛 - 𝑦 𝑛−1.differences are called the first forward
differences of the function.
y = f (x) and we denote these difference by
𝛁𝑦1 , 𝛁𝑦2 , 𝛁𝑦3………..,𝛁𝑦𝑛respectively, where 𝛁is called the descending
or forward difference operator.
In general, the first forward differences is defined by
𝛁𝑦𝑛= 𝑦𝑛 – 𝑦 𝑛−1.
where is called first backward difference operator.

13. SECOND BACKWARD DIFFRENCE
OPERATOR
The differences of first forward differences are called second backward
differences.
𝛁 𝟐
𝑦1=∆𝑦1- ∆𝑦0.
𝛁 𝟐 𝑦2 =∆𝑦2 - ∆𝑦1.
𝛁 𝟐
𝒚 𝒏= ∆𝑦𝑛 - ∆𝑦 𝑛−1.
𝛁 𝟐 𝑦1 , ∆ 𝟐 𝑦2,………..,∆ 𝟐 𝒚 𝒏are called second forward
differences.
where 𝛁is called second backward difference operator.

14. TABLE
Argument
x
Entry
y = f(x)
First
Differences
𝛁
Second
Differences
𝛻 𝟐
Third
Differences
𝛻 𝟑
Fourth
Differences
𝛻 𝟒
𝑥0
𝑥1
𝑥2
𝑥3
𝑥4
𝑦0
𝑦1
𝑦2
𝑦3
𝑦4
𝛁𝑦0
𝛁𝑦1
𝛁𝑦2
𝛁𝑦3
𝛁 𝟐
𝑦0
𝛁 𝟐 𝑦1
𝛁 𝟐
𝑦2
𝛁 𝟑
𝑦0
𝛁 𝟑
𝑦1
𝛁 𝟒 𝑦0

16. THE DIFFERENT
TYPES OF
OPERATORS.

17. CENTRAL DIFFERNCES (δ)
If we denote the differences 𝛿𝑦1/2 , δ𝑦3/2 ,…..., δ𝑦 𝑛−1/2 respectively,
then we have
𝛿𝑦1/2=𝑦1- 𝑦0 , δ𝑦3/2=𝑦2 - 𝑦1 , ……..,
δ𝑦 𝑛−1/2=𝑦𝑛 - 𝑦 𝑛−1.
Where δ is called first central difference operator.
Where 𝛿𝑦1/2 , δ𝑦3/2 ,……….., δ𝑦 𝑛−1/2 are called first central
differences.

18. GENERAL 𝑁 𝑇𝐻
TERM FOR
CENTRAL DIFFRENCES
In the general , the 𝑛 𝑡ℎ central differences can be written as:-
𝜹 𝒏
𝒚𝒊−(
𝟏
𝟐
)
=𝜹 𝒏−𝟏
𝒚𝒊 - 𝜹 𝒏−𝟏
𝒚𝒊−𝟏 .
where n = 1,2,3………n.
following table shows how the central difference can be written.

19. TABLE
Argument
x
Entry
y = f(x)
First
Differences
𝛅
Second
Differences
𝛅 𝟐
Third
Differences
𝛅 𝟑
Fourth
Differences
𝛅 𝟒
𝑥0
𝑥1
𝑥2
𝑥3
𝑥4
𝑦0
𝑦1
𝑦2
𝑦3
𝑦4
𝛿𝑦1/2
δ𝑦3/2
δ𝑦5/2
δ𝑦7/2
𝛅 𝟐y1
𝛅 𝟐 𝑦2
𝛅 𝟐 𝑦3
𝛅 𝟑 𝑦3/2
𝛅 𝟑
𝑦5/2
𝛅 𝟒 𝑦2

20. TYPES OF OPERATORS
Operators
Shifting operator Unit operator
Inverse
operator
Differential
operator
Forward
difference
operators
Backward
difference
operator

21. FORWARD AND BACKWARD
DIFFERNCES OPERATORS EQUATIONS
∆𝑓(𝑥) = 𝑓(𝑥 + ℎ)- 𝑓(𝑥).
This equation is known as forward difference operators equation.
∇𝑓(𝑥) = 𝑓(𝑥 ) - 𝑓(𝑥 − ℎ).
This equation is known as backward difference operators equation.

22. SHIFTING OPERATOR (∈)
E𝑓(𝑥) = 𝑓(𝑥 + h).
E2
𝑓(𝑥) = 𝑓(𝑥+ 2h).
E3
𝑓(𝑥) = 𝑓(𝑥+ 3h).
⋮ ⋮
E 𝑛
𝑓(𝑥)= 𝑓(𝑥 + nh).
E is also known as displacement or translation operator.

23. INVERSE OPERATOR
𝐸−1 𝑓(𝑥+ h) = 𝑓(𝑥 – h).
𝐸−2 𝑓(𝑥+ h) = 𝑓(𝑥 – 2h).
𝐸−3 𝑓(𝑥+ h) = 𝑓(𝑥 – 3h).
⋮ ⋮
𝐸−𝑛
𝑓(𝑥+ h) = 𝑓(𝑥 – nh).
where 𝜖−1 is known as inverse operator.

24. DIFFERNTIAL OPERATOR
𝐷𝑓(𝑥) =
𝑑
𝑑𝑥
𝑓(𝑥).
𝐷2 𝑓 𝑥 =
𝑑2
𝑑𝑥2 𝑓(𝑥).
⋮ ⋮
𝐷 𝑛
𝑓 𝑥 =
𝑑 𝑛
𝑑𝑥 𝑛
𝑓 𝑥 .
where 𝐷 is known as differential operator.

25. UNIT OPERATOR
The unit operator 1 is defined as 1.f(x)= f(x).

26. RELATION BETWEEN FORWARD AND
SHIFTING OPERATOR
 ∆=E-1.
By definition ,
∆𝑓(𝑥)=𝑓 𝑥 + h − 𝑓(𝑥).
∆𝑓(𝑥)=E 𝑓 𝑥 − 1. 𝑓(𝑥).
∆𝑓(𝑥)=(E-1)𝑓(𝑥).
∆= 𝐸 − 1.

27. RELATION BETWEEN THE BACKWARD
AND INVERSE OPERATOR
∇=1-𝐸−1
.
By definition ,
∇𝑓(𝑥)=𝑓(𝑥)-𝑓(𝑥 − h).
∇𝑓(𝑥)=1.𝑓(𝑥)-𝐸−1
𝑓(𝑥).
∇𝑓 𝑥 =(1-𝐸−1
)𝑓(𝑥).
∇=(1-𝐸−1
).

28. RELATION BETWEEN THE CENTRE
AND INVERSE OPERATOR
 δ=𝐸−1/2∆.
By definition ,
δ𝑓(𝑥)=𝑓(𝑥 +
h
2
)-𝑓 𝑥 −
h
2
.
δ𝑓(𝑥)=𝐸1/2 𝑓(𝑥)-𝐸−
1
2 𝑓 𝑥 .
δ𝑓 𝑥 = (𝐸1/2-𝐸−1/2)𝑓 𝑥 .
δ= 𝐸−1/2(E-1).
δ= 𝐸−1/2
∆.

29. THANK
YOU