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![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](https://image.slidesharecdn.com/prernaactual-220716123049-8a20a5c9/75/Prerna-actual-pptx-8-2048.jpg)
![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
]+……](https://image.slidesharecdn.com/prernaactual-220716123049-8a20a5c9/75/Prerna-actual-pptx-32-2048.jpg)
![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)](https://image.slidesharecdn.com/prernaactual-220716123049-8a20a5c9/75/Prerna-actual-pptx-35-2048.jpg)
![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](https://image.slidesharecdn.com/prernaactual-220716123049-8a20a5c9/75/Prerna-actual-pptx-37-2048.jpg)
More Related Content
Prerna actual.pptx
- 1.
- 2. Introduction Types of Operators: Forward Difference Operator Backward Difference Operator Central Difference Operator Shift Operator Averaging Operator Difference Operators
- 3. Types of operators DifferenceOperators Forward Difference Operator
- 4. Forward Difference Operator Definition: ForwardDifference 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: BackwardDifference 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: CentralDifference 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 AveragingOperator, 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 DifferentRelations: 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 ThirdDifferences ∆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 DifferenceOperators 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.
- 17. Introduction Topics tobe 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 offinding 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 TheInterpolation 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 Whenthe 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 Tocalculate 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.
- 24. Introduction Newton’s InterpolationMethod: • Newton’s Forward Interpolation Formula • Newton’s Backward Interpolation Formula Newton’s Interpolation
- 25. Newton’s forward InterpolationMethod 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 InterpolationMethod 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.
- 28. Introduction Gauss’s InterpolationMethod: • 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.
- 32. Introduction Statement: This formulagives 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.
- 34. Lagrange’s Formula Statement: If𝑥0 , 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 ForwardDifference 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 forwardInterpolation 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