Download - Complex Variable & Numerical Method
Presentationon
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)
G ANDH INAG AR INSTITUTE O F TECH NO LO G Y
INTERPOLATION AND EXTRAPOLATION
The process of finding the values inside the interval ๐ฅ0<๐ฅ< ๐ฅ๐ is known as interpolation.
The process of finding the values outside the interval ๐ฅ0<๐ฅ< ๐ฅ๐ is known as extrapolation.
Interpolation
Forward interpolation
Backward interpolation
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.
FINITE DIFFERENCES
FINITE DIFFRENCES
FORWARD DIFFERENCES
CENTRAL DIFFERENCES
BACKWARD DIFFERENCES
Finite differences are of three types :-
RULES OF INTERPOLATIONInterpolation formulas can be used only when the values of the argument ๐ฅ are equidistant.
The point ๐ฅ0 should be selected very close to the point at which interpolation is required.
Usually in the forward interpolation the very first value of ๐ฅ is taken equal to ๐ฅ0.
Backward interpolation is suitable for interpolation near the end of tabulated values in the backward interpolation.
In backward interpolation the last value of ๐ฅ is taken equal to ๐ฅ๐.
FIRST FORWARD DIFFERENCES
The ๐ฆ1- ๐ฆ0 , ๐ฆ2 - ๐ฆ1 , ๐ฆ๐ - ๐ฆ๐โ1.differences are called the first forward differences of the function.
y = f (x) and we denote these difference by
โ๐ฆ0 , โ๐ฆ1 , โ๐ฆ2โฆโฆโฆ.., โ๐ฆ๐respectively, where ฮ is called the descending or forward difference operator.
In general, the first forward differences is defined by
ฮ๐ฆ๐ฅ= ๐ฆ๐ฅ+1โ ๐ฆ๐ฅ.
where ฮ is called first forward difference operator.
SECOND FORWARD DIFFRENCE OPERATOR
The differences of first forward differences are called second forward differences.
โ๐๐ฆ0=โ๐ฆ1- โ๐ฆ0.
โ๐๐ฆ1 =โ๐ฆ2 - โ๐ฆ1.
โ๐๐๐โ๐= โ๐ฆ๐ - โ๐ฆ๐โ1.
โ๐๐ฆ0 , โ๐๐ฆ1,โฆโฆโฆ..,โ๐๐๐โ๐are called second forward differences.
where โ๐is called second forward difference order.
TABLE
Argument
x
Entry
y = f(x)
First
Differences
โ
Second
Differences
โ๐
Third
Differences
โ๐
Fourth
Differences
โ๐
๐ฅ0
๐ฅ1
๐ฅ2
๐ฅ3
๐ฅ4
๐ฆ0
๐ฆ1
๐ฆ2
๐ฆ3
๐ฆ4
โ๐ฆ0
โ๐ฆ1
โ๐ฆ2
โ๐ฆ3
โ๐๐ฆ0
โ๐๐ฆ1
โ๐๐ฆ2
โ๐๐ฆ0
โ๐๐ฆ1
โ๐๐ฆ0
FIRST BACKWARD DIFFRENCES
The ๐ฆ1- ๐ฆ0 , ๐ฆ2 - ๐ฆ1 , ๐ฆ๐ - ๐ฆ๐โ1.differences are called the first forward differences of the function.
y = f (x) and we denote these difference by
๐๐ฆ1 , ๐๐ฆ2 , ๐๐ฆ3โฆโฆโฆ..,๐๐ฆ๐respectively, where ๐is called the descending or forward difference operator.
In general, the first forward differences is defined by
๐๐ฆ๐= ๐ฆ๐ โ ๐ฆ๐โ1.
where is called first backward difference operator.
SECOND BACKWARD DIFFRENCE OPERATOR
The differences of first forward differences are called second backward differences.
๐๐๐ฆ1=โ๐ฆ1- โ๐ฆ0.
๐๐๐ฆ2 =โ๐ฆ2 - โ๐ฆ1.
๐๐๐๐= โ๐ฆ๐ - โ๐ฆ๐โ1.
๐๐๐ฆ1 , โ๐๐ฆ2,โฆโฆโฆ..,โ๐๐๐are called second forward differences.
where ๐is called second backward difference operator.
TABLEArgument
x
Entry
y = f(x)
First
Differences
๐
Second
Differences
๐ป๐
Third
Differences
๐ป๐
Fourth
Differences
๐ป๐
๐ฅ0
๐ฅ1
๐ฅ2
๐ฅ3
๐ฅ4
๐ฆ0
๐ฆ1
๐ฆ2
๐ฆ3
๐ฆ4
๐๐ฆ0
๐๐ฆ1
๐๐ฆ2
๐๐ฆ3
๐๐๐ฆ0
๐๐๐ฆ1
๐๐๐ฆ2
๐๐๐ฆ0
๐๐๐ฆ1
๐๐๐ฆ0
THE DIFFERENT TYPES OF OPERATORS.
CENTRAL DIFFERNCES (ฮด)
If we denote the differences ๐ฟ๐ฆ1/2 , ฮด๐ฆ3/2 ,โฆ..., ฮด๐ฆ๐โ1/2 respectively,
then we have
๐ฟ๐ฆ1/2=๐ฆ1- ๐ฆ0 , ฮด๐ฆ3/2=๐ฆ2 - ๐ฆ1 , โฆโฆ..,
ฮด๐ฆ๐โ1/2=๐ฆ๐ - ๐ฆ๐โ1.
Where ฮด is called first central difference operator.
Where ๐ฟ๐ฆ1/2 , ฮด๐ฆ3/2 ,โฆโฆโฆ.., ฮด๐ฆ๐โ1/2 are called first central
differences.
GENERAL ๐๐๐ป TERM FOR CENTRAL DIFFRENCES
In the general , the ๐๐กโ central differences can be written as:-
๐น๐๐๐โ(
๐
๐)
=๐น๐โ๐๐๐ - ๐น๐โ๐๐๐โ๐ .
where n = 1,2,3โฆโฆโฆn.
following table shows how the central difference can be written.
TABLE
Argument
x
Entry
y = f(x)
First
Differences
๐
Second
Differences
๐ ๐
Third
Differences
๐ ๐
Fourth
Differences
๐ ๐
๐ฅ0
๐ฅ1
๐ฅ2
๐ฅ3
๐ฅ4
๐ฆ0
๐ฆ1
๐ฆ2
๐ฆ3
๐ฆ4
๐ฟ๐ฆ1/2
ฮด๐ฆ3/2
ฮด๐ฆ5/2
ฮด๐ฆ7/2
๐ ๐y1
๐ ๐๐ฆ2
๐ ๐๐ฆ3
๐ ๐๐ฆ3/2
๐ ๐๐ฆ5/2
๐ ๐๐ฆ2
TYPES OF OPERATORS
Operators
Shifting operator Unit operatorInverse
operator
Differential
operator
Forward difference operators
Backward difference operator
FORWARD AND BACKWARD DIFFERNCES OPERATORS EQUATIONS
โ๐(๐ฅ) = ๐(๐ฅ + โ)- ๐(๐ฅ).This equation is known as forward difference operators equation.
โ๐(๐ฅ) = ๐(๐ฅ ) - ๐(๐ฅ โ โ).
This equation is known as backward difference operators equation.
SHIFTING OPERATOR (โ)
E๐(๐ฅ) = ๐(๐ฅ + h).
E2๐(๐ฅ) = ๐(๐ฅ+ 2h).
E3 ๐(๐ฅ) = ๐(๐ฅ+ 3h).
โฎ โฎ
E๐๐(๐ฅ)= ๐(๐ฅ + nh).
E is also known as displacement or translation operator.
INVERSE OPERATOR
๐ธโ1๐(๐ฅ+ h) = ๐(๐ฅ โ h).
๐ธโ2๐(๐ฅ+ h) = ๐(๐ฅ โ 2h).
๐ธโ3๐(๐ฅ+ h) = ๐(๐ฅ โ 3h).
โฎ โฎ
๐ธโ๐๐(๐ฅ+ h) = ๐(๐ฅ โ nh).
where ๐โ1 is known as inverse operator.
DIFFERNTIAL OPERATOR
๐ท๐(๐ฅ) =๐
๐๐ฅ๐(๐ฅ).
๐ท2๐ ๐ฅ =๐2
๐๐ฅ2๐(๐ฅ).
โฎ โฎ
๐ท๐๐ ๐ฅ =๐๐
๐๐ฅ๐๐ ๐ฅ .
where ๐ท is known as differential operator.
UNIT OPERATOR
The unit operator 1 is defined as 1.f(x)= f(x).
RELATION BETWEEN FORWARD AND SHIFTING OPERATOR
โ=E-1.
By definition ,
โ๐(๐ฅ)=๐ ๐ฅ + h โ ๐(๐ฅ).
โ๐(๐ฅ)=E ๐ ๐ฅ โ 1. ๐(๐ฅ).
โ๐(๐ฅ)=(E-1)๐(๐ฅ).
โ= ๐ธ โ 1.
RELATION BETWEEN THE BACKWARD AND INVERSE OPERATOR
โ=1-๐ธโ1.
By definition ,
โ๐(๐ฅ)=๐(๐ฅ)-๐(๐ฅ โ h).
โ๐(๐ฅ)=1.๐(๐ฅ)-๐ธโ1๐(๐ฅ).
โ๐ ๐ฅ =(1-๐ธโ1)๐(๐ฅ).
โ=(1-๐ธโ1).
RELATION BETWEEN THE CENTRE AND INVERSE OPERATOR
ฮด=๐ธโ1/2โ.
By definition ,
ฮด๐(๐ฅ)=๐(๐ฅ +h
2)-๐ ๐ฅ โ
h
2.
ฮด๐(๐ฅ)=๐ธ1/2๐(๐ฅ)-๐ธโ1
2๐ ๐ฅ .
ฮด๐ ๐ฅ = (๐ธ1/2-๐ธโ1/2)๐ ๐ฅ .
ฮด= ๐ธโ1/2(E-1).
ฮด= ๐ธโ1/2โ.
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