newton interpolation

1. Difference Operators

2. Introduction
Types of Operators:
 Forward Difference Operator
 Backward Difference Operator
 Central Difference Operator
 Shift Operator
 Averaging Operator
Difference Operators

3. Types of operators
Difference Operators

Forward
Difference
Operator

4. Forward Difference Operator
Definition:
Forward Difference operator,
denoted by  ,as
f(x)=f(x+h)-f(x)
The expression f(x+h)-f(x)
gives the Forward Difference of
f(x) and the operator  is called the
First Forward Difference
Operator. Given the step size h this
formula uses the values at x and
x+h the point at the next step
As it is moving in the forward
direction, it is called the Forward
Difference Operator
Difference Operators

5. Backward Difference Operator
Definition:
Backward Difference operator,
denoted by , as
 f(x)=f(x)-f(x-h)
Given the step size h note that this
formula uses the values at x and
x-h, the point at the previous step.
As it moves in the backward
direction, it is called the Backward
Difference Operator
Difference Operators

6. Central Difference Operator
Definition:
Central Difference operator,
denoted by  , is defined by
 f(x) = f(x +
𝒉
𝟐
)-f(x -
𝒉
𝟐
)
Thus ,
𝜹𝟐
f(x) = f(x+h) – 2f(x) + f(x-h)
Difference Operators

7. Shift Operator
Definition:
Shift operator, denoted by E is the
operator which shifts the value at
the next point with step h, i.e.
E f(x)=f(x+h)
Thus,
E𝒚𝒊 = 𝒚𝒊+𝟏 , 𝐄𝟐𝒚𝒊 = 𝒚𝒊+𝟐
and
𝑬𝒌
𝒚𝒊 = 𝒚𝒊+𝒌
Difference Operators
Definition:
Shift operator, denoted by E is the
operator which shifts the value at
the next point with step h, i.e.
E f(x)=f(x+h)
Thus,
E𝒚𝒊 = 𝒚𝒊+𝟏 , 𝐄𝟐𝒚𝒊 = 𝒚𝒊+𝟐
and
𝑬𝒌𝒚𝒊 = 𝒚𝒊+𝒌

8. Averaging Operator
Definition:
The Averaging Operator, denoted
by μ , gives the average value
between two central points, i.e.,
μ f(x) =
𝟏
𝟐
[f(x +
𝒉
𝟐
) - f(x -
𝒉
𝟐
)]
Thus,
μ 𝒚𝒊 =
𝟏
𝟐
(𝒚𝒊 +
𝟏
𝟐
+ 𝒚𝒊 −
𝟏
𝟐
)
𝝁𝟐
𝒚𝒊 =
𝟏
𝟐
(𝝁 𝒚𝒊 +
𝟏
𝟐
+ 𝝁 𝒚𝒊 −
𝟏
𝟐
)
=
𝟏
𝟒
(𝒚𝒊 +𝟏 + 2𝒚𝒊 + 𝒚𝒊 −𝟏)
Difference Operators

9. Relation Between Operators
Different Relations:
 E = ∆ + 1
 𝐸−1
= 1-∇
 E ∇ =∇ E = ∆
 δ = 𝐸
1
2 - 𝐸
−1
2
 ∆ = δ 𝐸
1
2
 E = 𝑒ℎ𝐷
 (1+ ∆)(1-∇) = 1
µ
Difference Operators

10. Forward Difference Table
Argument
X
Entry
Y=f(x)
First
Differences
∆y
Second
Differences
∆2y
Third Differences
∆3y
𝑥𝑜
𝑥𝑜+ h
𝑥𝑜+ 3h
𝑥𝑜+ 2h
𝑦𝑜
𝑦1
𝑦2
𝑦3
𝑦1 − 𝑦𝑜= ∆𝑦𝑜
𝑦2 −𝑦1 =∆𝑦1
𝑦3 −𝑦2 =∆ 𝑦2
∆𝑦2−∆𝑦1=∆2
𝑦1
∆𝑦1 − ∆𝑦𝑜 =∆2𝑦𝑜
∆2
𝑦1 − ∆2
𝑦𝑜=∆3
𝑦𝑜
Difference Operators

11. Backward Difference Table
Argument
X
Entry
Y=f(x)
First
Differences
𝛻 y
Second
Differences
𝛻2y
Third Differences
𝛻3
y
𝒙𝒐
𝒙𝒐+ h
𝒙𝒐+ 3h
𝒙𝒐+ 2h
𝒚𝒐
𝒚𝟏
𝒚𝟐
𝒚𝟑
𝒚𝟏 − 𝒚𝒐= 𝜵 𝒚𝒐
𝒚𝟐 − 𝒚𝟏 = 𝜵𝒚𝟏
𝒚𝟑 − 𝒚𝟐 = 𝜵𝟐
𝜵𝒚𝟐 − 𝜵𝒚𝟏 =
𝜵𝟐
𝒚𝟏
𝜵𝒚𝟏 − 𝜵𝒚𝒐 =
𝜵𝟐
𝒚𝒐
𝜵𝟐
𝒚𝟏 − 𝜵𝟐
𝒚𝒐= 𝜵𝟑
𝒚𝒐
Difference Operators
𝒚𝒐
𝒚𝟏

12. -
𝒙𝒐
𝒙𝒐+ h
𝒙𝒐+ 3h
𝒙𝒐+ 2h
Backward Difference Table
Argument
X
Entry
Y=f(x)
First
Differences
𝛻 y
Second
Differences
𝛻2y
Third Differences
𝛻3
y
𝜵𝒚𝟐 − 𝜵𝒚𝟏 =
𝜵𝟐
𝒚𝟏
𝜵𝒚𝟏 − 𝜵𝒚𝒐 =
𝜵𝟐
𝒚𝒐
𝜵𝟐
𝒚𝟏 − 𝜵𝟐
𝒚𝒐= 𝜵𝟑
𝒚𝒐
Difference Operators
= 𝜵 𝒚𝒐
𝜵 𝒚𝟏
𝒚𝟑
𝒚𝟑
𝒚𝟐
𝒚𝟐
𝒚𝟐
𝒚𝟏
𝒚𝟏
𝒚𝟏
𝒚𝒐
𝒚𝒐
𝜵 𝒚𝟐

13. 𝜵 𝒚𝒐
𝜵 𝒚𝟏
𝜵 𝒚𝟐
-
𝒙𝒐
𝒙𝒐+ h
𝒙𝒐+ 3h
𝒙𝒐+ 2h
Backward Difference Table
Argument
X
Entry
Y=f(x)
First
Differences
𝛻 y
Second
Differences
𝛻2y
Third Differences
𝛻3
y
𝜵𝒚𝟐 − 𝜵𝒚𝟏 =
𝜵𝟐
𝒚𝟏
𝜵𝒚𝟏 − 𝜵𝒚𝒐 =
𝜵𝟐
𝒚𝒐
𝜵𝟐
𝒚𝟏 − 𝜵𝟐
𝒚𝒐= 𝜵𝟑
𝒚𝒐
Difference Operators
= 𝜵 𝒚𝒐
𝜵 𝒚𝟏
𝒚𝟑
𝒚𝟑
𝒚𝟐
𝒚𝟐
𝒚𝟐
𝒚𝟏
𝒚𝟏
𝒚𝟏
𝒚𝒐
𝒚𝒐
𝜵 𝒚𝟐

14. 𝜵𝟐
𝒚𝟏 − 𝜵𝟐
𝒚𝒐= 𝜵𝟑
𝒚𝒐
𝜵𝟐
𝒚𝟏 − 𝜵𝟐
𝒚𝒐= 𝜵𝟑
𝒚𝒐

15. Central Difference table
Difference Operators
Argument
X
Entry
Y=f(x)
First
Differences
δy
Second
Differences
δ2y
Third Differences
δ3
y
𝑥𝑜
𝑥1
𝑥3
𝑥2
𝑦𝑜
𝑦1
𝑦2
𝑦3
δ𝑦1
2
δ𝑦3
2
δ𝑦5
2
δ2𝑦2
δ2
𝑦1
δ3
𝑦3
2

16. Interpolation

17. Introduction
 Topics to be covered:
• Newton’s Interpolation Method
• Gauss’s Interpolation Method
• Lagrange’s Interpolation Method
• Stirling’s formula for Central Differences
• Inverse Interpolation
Interpolation

18. Definition
The process of finding the curve passing through the points (𝑥0, 𝑦0),
(𝑥1, 𝑦1), (𝑥2, 𝑦2),………, (𝑥𝑛, 𝑦𝑛) is called as Interpolation and the curve
obtained is called as Interpolating curve. Interpolating polynomial
passing through the given set of points is Unique. Let 𝑥0, 𝑥1, 𝑥2,….. , 𝑥𝑛
be given set of observations y=f(x) and be the given function, then the
method to find f(𝑥𝑚)∀ 𝑥0 ≤ 𝑥𝑚 ≤ 𝑥𝑛 is called as an INTERPOLATION.
(𝑥0, 𝑦0)
(𝑥1, 𝑦1)
(𝑥𝑛, 𝑦𝑛)
(𝑥𝑛−1, 𝑦𝑛−1)
(𝑥3, 𝑦3)
(𝑥2, 𝑦2)
f(x)
Interpolation

19. Finite Difference Concept
The Interpolation depends upon Finite Difference Concept. If
be given set of observations and let 𝒚𝟎= f (𝒙𝟎),…., 𝒚𝒏= f
(𝒙𝒏) be their corresponding values for the curve y = f(x) , then
𝒚𝟏 − 𝒚𝟎,….., 𝒚𝒏 − 𝒚𝒏−𝟏 is called as Finite Difference.
Interpolation

20. Equally Spaced Arguments
When the arguments are equally spaced i.e. 𝒙𝒊 - 𝒙𝒊−𝟏 = h ∀ i then
we can use one of the following differences:
Interpolation
Forward
Differences
Backward
differences
Central
differences

21. Equally Spaced Arguments
To calculate Equally Spaced Arguments , we use following
methods:
 Newton’s interpolation method for equal interval
• Newton’s Forward Interpolation Formula
• Newton’s Backward Interpolation Formula
 Gauss’s interpolation method for equal interval
• Gauss’s Forward Interpolation Formula
• Gauss’s Backward Interpolation Formula
Interpolation

22. Equally Spaced Arguments
 To calculate Unequally Spaced Arguments , we use following
method:
Interpolation
Lagrange’s interpolation
method

23. Newton’s Interpolation

24. Introduction
 Newton’s Interpolation Method:
• Newton’s Forward Interpolation Formula
• Newton’s Backward Interpolation Formula
Newton’s Interpolation

25. Newton’s forward Interpolation Method
 Statement:
If 𝑥0 , 𝑥1 , 𝑥2 ,….., 𝑥𝑛 are given set of observations with common
difference h and let 𝑦0, 𝑦1, 𝑦2,….., 𝑦𝑛 are their corresponding values,
where y = f(x) be the given function then
f (𝑥0 + ph) = 𝑦0 + p∆ 𝑦0 +
𝑝(𝑝−1)
2!
∆2
𝑦0+…..+
𝑝 𝑝−1 ……..(𝑝−𝑛−1 )
𝑛!
∆ 𝑛
𝑦0
where,
p =
(𝒙−𝒙𝟎)
𝒉
Newton’s Interpolation

26. Newton’s backward Interpolation Method
 Statement:
If 𝑥0 , 𝑥1 , 𝑥2 ,….., 𝑥𝑛 are given set of observations with common
difference h and let 𝑦0, 𝑦1, 𝑦2,….., 𝑦𝑛 are their corresponding values,
where y = f(x) be the given function then
f (𝑥𝑛 + ph) = 𝑦𝑛 + p∇ 𝑦𝑛 +
𝑝(𝑝+1)
2!
∇ 2
𝑦𝑛+…..+
𝑝 𝑝+1 ……..(𝑝+ 𝑛−1 )
𝑛!
∇ 𝑛
𝑦𝑛
where,
p =
(𝒙−𝒙𝟎)
𝒉
Newton’s Interpolation

27. Gauss’s Interpolation

28. Introduction
 Gauss’s Interpolation Method:
• Gauss’s Forward Central Difference Formula
• Gauss’s Backward Central Difference Formula
Gauss’s Interpolation

29.  Statement:
If … 𝑥−2 , 𝑥−1, 𝑥0 , 𝑥1 ,𝑥2 ,….. are given set of observations with common
difference h and let … 𝑦−2 , 𝑦−1, 𝑦0 , 𝑦1 ,𝑦2 ,….. are their corresponding
values, where y = f(x) be the given function then
𝑦𝑝 = 𝑦0 + p∆ 𝑦0 +
𝑝(𝑝−1)
2!
∆2𝑦−1+
𝑝(𝑝−1) 𝑝+1
3!
∆ 3𝑦−1 +……
where,
p =
(𝒙−𝒙𝟎)
𝒉
Gauss’s Interpolation
Gauss Forward Central Difference Operator

30.  Statement:
If … , 𝑥−2 , 𝑥−1, 𝑥0 , 𝑥1 ,𝑥2 ,….. are given set of observations with common
difference h and let … 𝑦−2 , 𝑦−1, 𝑦0 , 𝑦1 ,𝑦2 ,….. are their corresponding values,
where y = f(x) be the given function then
𝑦𝑝 = 𝑦0 + p∆ 𝑦−1 +
𝑝(𝑝+1)
2!
∆2
𝑦−1+
𝑝(𝑝−1) 𝑝+1
3!
∆ 3
𝑦−2 +……
where,
p =
(𝒙−𝒙𝟎)
𝒉
Gauss’s Interpolation
Gauss Backward Central Difference Operator

31. Stirling’s Formula

32. Introduction
 Statement:
This formula gives the average of the values obtained by Gauss forward
and backward interpolation formulae. For using this formula we should
have – ½ < p< ½.
32
We can get very good estimates if - ¼ < p < ¼.
The formula is:
where,
u =
𝒙−𝒙𝟎
𝒉
Stirling’s Formula
y = 𝒚𝟎 + u(
∆𝒚𝟎 + ∆𝒚−𝟏
𝟐
) +
𝒖𝟐
𝟐!
∆𝟐 𝒚−𝟏 +
𝒖(𝒖𝟐−𝟏)
𝟑!
[
∆𝟑𝒚−𝟏+∆ 𝟑𝒚−𝟐
𝟐
]+……

33. Lagrange’s Interpolation

34. Lagrange’s Formula
 Statement:
If 𝑥0 , 𝑥1 , 𝑥2 ,….., 𝑥𝑛 are given set of observations which are not be
equally spaced and let 𝑦0, 𝑦1, 𝑦2,….., 𝑦𝑛 are their corresponding values,
where y = f(x) be the given function then
f(x) =
(𝒙−𝒙𝟏)……..(𝒙−𝒙𝒏)
(𝒙𝟎−𝒙𝟏)……..(𝒙𝟎−𝒙𝒏)
𝒚𝟎 + ……………. +
(𝒙−𝒙𝟎)……..(𝒙−𝒙𝒏−𝟏)
(𝒙𝒏−𝒙𝟎)……..(𝒙𝒏−𝒙𝒏−𝟏)
𝒚𝒏
Lagrange’s Interpolation

35. Formulas
Difference Operators
 Forward Difference operator:
f(x)=f(x+h)-f(x)
 Backward Difference operator :
 f(x)=f(x)-f(x-h)
 Central Difference operator :
 f(x) = f(x +
𝒉
𝟐
)-f(x -
𝒉
𝟐
)
 Averaging Operator :
μ f(x) =
𝟏
𝟐
[f(x +
𝒉
𝟐
) - f(x -
𝒉
𝟐
)]
 Shift operator :
E f(x)=f(x+h)

36. Formulas
Interpolation
 Newton’s forward Interpolation Method:
f (𝒙𝟎 + ph) = 𝒚𝟎 + p∆ 𝒚𝟎 +
𝒑(𝒑−𝟏)
𝟐!
∆𝟐
𝒚𝟎+…..+
𝒑 𝒑−𝟏 ……..(𝒑−𝒏−𝟏 )
𝒏!
∆ 𝒏
𝒚𝟎
 Newton’s backward Interpolation Method:
f (𝒙𝒏 + ph) = 𝒚𝒏 + p∇ 𝒚𝒏 +
𝒑(𝒑+𝟏)
𝟐!
∇ 𝟐
𝒚𝒏+…..+
𝒑 𝒑+𝟏 ……..(𝒑+ 𝒏−𝟏 )
𝒏!
∇ 𝒏
𝒚𝒏
 Gauss Forward Central Difference Operator:
𝒚𝒑 = 𝒚𝟎 + p∆ 𝒚𝟎 +
𝒑(𝒑−𝟏)
𝟐!
∆𝟐
𝒚−𝟏+
𝒑(𝒑−𝟏) 𝒑+𝟏
𝟑!
∆ 𝟑
𝒚−𝟏 +……
 Gauss Backward Central Difference Operator:
𝒚𝒑 = 𝒚𝟎 + p∆ 𝒚−𝟏 +
𝒑(𝒑+𝟏)
𝟐!
∆𝟐
𝒚−𝟏+
𝒑(𝒑−𝟏) 𝒑+𝟏
𝟑!
∆ 𝟑𝒚−𝟐 +……

37. Formulas
Interpolation
 Stirling’s Formula:
y = 𝒚𝟎 + u(
∆𝒚𝟎 + ∆𝒚−𝟏
𝟐
) +
𝒖𝟐
𝟐!
∆𝟐 𝒚−𝟏 +
𝒖(𝒖𝟐−𝟏)
𝟑!
[
∆𝟑𝒚−𝟏+∆ 𝟑𝒚−𝟐
𝟐
]+……
 Lagrange’s Formula
f(x) =
(𝒙−𝒙𝟏)……..(𝒙−𝒙𝒏)
(𝒙𝟎−𝒙𝟏)……..(𝒙𝟎−𝒙𝒏)
𝒚𝟎 + ……………. +
(𝒙−𝒙𝟎)……..(𝒙−𝒙𝒏−𝟏)
(𝒙𝒏−𝒙𝟎)……..(𝒙𝒏−𝒙𝒏−𝟏)
𝒚𝒏