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 𝑥0<𝑥< 𝑥𝑛 is known as interpolation.

The process of finding the values outside the interval 𝑥0<𝑥< 𝑥𝑛 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 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 𝑥𝑛.

FIRST FORWARD DIFFERENCES

The 𝑦1- 𝑦0 , 𝑦2 - 𝑦1 , 𝑦𝑛 - 𝑦𝑛−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= 𝑦𝑥+1– 𝑦𝑥.

where Δ is called first forward difference operator.

SECOND FORWARD DIFFRENCE OPERATOR

The differences of first forward differences are called second forward differences.

∆𝟐𝑦0=∆𝑦1- ∆𝑦0.

∆𝟐𝑦1 =∆𝑦2 - ∆𝑦1.

∆𝟐𝒚𝒏−𝟏= ∆𝑦𝑛 - ∆𝑦𝑛−1.

∆𝟐𝑦0 , ∆𝟐𝑦1,………..,∆𝟐𝒚𝒏−𝟏are called second forward differences.

where ∆𝟐is called second forward difference order.

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

∆𝟐𝑦0

∆𝟐𝑦1

∆𝟐𝑦2

∆𝟑𝑦0

∆𝟑𝑦1

∆𝟒𝑦0

FIRST BACKWARD DIFFRENCES

The 𝑦1- 𝑦0 , 𝑦2 - 𝑦1 , 𝑦𝑛 - 𝑦𝑛−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 – 𝑦𝑛−1.

where is called first backward difference operator.

SECOND BACKWARD DIFFRENCE OPERATOR

The differences of first forward differences are called second backward differences.

𝛁𝟐𝑦1=∆𝑦1- ∆𝑦0.

𝛁𝟐𝑦2 =∆𝑦2 - ∆𝑦1.

𝛁𝟐𝒚𝒏= ∆𝑦𝑛 - ∆𝑦𝑛−1.

𝛁𝟐𝑦1 , ∆𝟐𝑦2,………..,∆𝟐𝒚𝒏are called second forward differences.

where 𝛁is called second backward difference operator.

TABLEArgument

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

𝛁𝟐𝑦0

𝛁𝟐𝑦1

𝛁𝟐𝑦2

𝛁𝟑𝑦0

𝛁𝟑𝑦1

𝛁𝟒𝑦0

THE DIFFERENT TYPES OF OPERATORS.

CENTRAL DIFFERNCES (δ)

If we denote the differences 𝛿𝑦1/2 , δ𝑦3/2 ,…..., δ𝑦𝑛−1/2 respectively,

then we have

𝛿𝑦1/2=𝑦1- 𝑦0 , δ𝑦3/2=𝑦2 - 𝑦1 , ……..,

δ𝑦𝑛−1/2=𝑦𝑛 - 𝑦𝑛−1.

Where δ is called first central difference operator.

Where 𝛿𝑦1/2 , δ𝑦3/2 ,……….., δ𝑦𝑛−1/2 are called first central

differences.

GENERAL 𝑁𝑇𝐻 TERM FOR CENTRAL DIFFRENCES

In the general , the 𝑛𝑡ℎ central differences can be written as:-

𝜹𝒏𝒚𝒊−(

𝟏

𝟐)

=𝜹𝒏−𝟏𝒚𝒊 - 𝜹𝒏−𝟏𝒚𝒊−𝟏 .

where n = 1,2,3………n.

following table shows how the central difference can be written.

TABLE

Argument

x

Entry

y = f(x)

First

Differences

𝛅

Second

Differences

𝛅𝟐

Third

Differences

𝛅𝟑

Fourth

Differences

𝛅𝟒

𝑥0

𝑥1

𝑥2

𝑥3

𝑥4

𝑦0

𝑦1

𝑦2

𝑦3

𝑦4

𝛿𝑦1/2

δ𝑦3/2

δ𝑦5/2

δ𝑦7/2

𝛅𝟐y1

𝛅𝟐𝑦2

𝛅𝟐𝑦3

𝛅𝟑𝑦3/2

𝛅𝟑𝑦5/2

𝛅𝟒𝑦2

TYPES OF OPERATORS

Operators

Shifting operator Unit operatorInverse

operator

Differential

operator

Forward difference operators

Backward difference operator

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.

SHIFTING OPERATOR (∈)

E𝑓(𝑥) = 𝑓(𝑥 + h).

E2𝑓(𝑥) = 𝑓(𝑥+ 2h).

E3 𝑓(𝑥) = 𝑓(𝑥+ 3h).

⋮ ⋮

E𝑛𝑓(𝑥)= 𝑓(𝑥 + nh).

E is also known as displacement or translation operator.

INVERSE OPERATOR

𝐸−1𝑓(𝑥+ h) = 𝑓(𝑥 – h).

𝐸−2𝑓(𝑥+ h) = 𝑓(𝑥 – 2h).

𝐸−3𝑓(𝑥+ h) = 𝑓(𝑥 – 3h).

⋮ ⋮

𝐸−𝑛𝑓(𝑥+ h) = 𝑓(𝑥 – nh).

where 𝜖−1 is known as inverse operator.

DIFFERNTIAL OPERATOR

𝐷𝑓(𝑥) =𝑑

𝑑𝑥𝑓(𝑥).

𝐷2𝑓 𝑥 =𝑑2

𝑑𝑥2𝑓(𝑥).

⋮ ⋮

𝐷𝑛𝑓 𝑥 =𝑑𝑛

𝑑𝑥𝑛𝑓 𝑥 .

where 𝐷 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

∆=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.

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−1𝑓(𝑥).

∇𝑓 𝑥 =(1-𝐸−1)𝑓(𝑥).

∇=(1-𝐸−1).

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)=𝐸1/2𝑓(𝑥)-𝐸−1

2𝑓 𝑥 .

δ𝑓 𝑥 = (𝐸1/2-𝐸−1/2)𝑓 𝑥 .

δ= 𝐸−1/2(E-1).

δ= 𝐸−1/2∆.

THANK

YOU