Interpolation and Finite Difference Operators, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Chapter 5

Interpolation

Chapter 5

Interpolation

IntroductionIntroduction

Finite differences play anFinite differences play animportant role in numericalimportant role in numericaltechniques, wheretechniques, wheretabulated values of thetabulated values of thefunctions are available.functions are available.For instance, consider aFor instance, consider afunctionfunction

y^

f^

x

That is, for given a table ofvalues,^ (^

,^

),^

k^

k

x^

y^

k^

n

^

the process of estimatingthe value of

y , for any

intermediate value of

x , is

called interpolation.

The method of computingthe value of

y , for a given

value of

x , lying outside

the table of values of

x^

is

known as extrapolation.

To be explicit, we write

0

1

0

1

2

1

1

1

n^

n^

n

y^

y^

y

y^

y^

y

y^

y^

y

^



^

^



^

^



^

^



^

^



Similarly, the differences ofthe first differences arecalled second differences,defined by

2

2

0

1

0

1

2

1

,

y^

y^

y^

y^

y^

y

^

 

 

^

 

 

Thus, in general

2

1

i^

i^

i

y^

y^

y 

^

 

 

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BackwardBackwardDifferencesDifferences

1

,^ (

1),

,

i^

i^

i

y^

y^

y^

i^

n^

n



^

^

^

^

^



1

1

0

2

2

1 1

n^

n^

n

y^

y^

y

y^

y^

y

y^

y^

y^ 

^

^

^

^

^

^

^

^

OR

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Thus, in general, the secondbackward differences are^2

,^1

, (^

1),..., 2

i^

i^

i

y^

y^

y^

i^

n^ n 

^

 

 

^



while the

k-th

backward

differences are given as

1

1 1

,^

, (^

1),...,

k^

k^

k

i^

i^

i

y^

y^

y^

i^ n

n^

k

^

 

^

 

 ^

^



1 2^

1

0

3 2^

2

1

,^

,

y^

y^

y^

y^

y^

y

^

^

^



In general

(1 2)

(1 2)

i^

i^

i

y^

y^

y

^

^



^



Higher order differences aredefined as follows:

2

(1 2)

(1 2)

i^

i^

i

y^

y^

y

^



 ^



^

 1

1

(1 2)

(1 2)

n^

n^

n

i^

i^

i

y^

y^

y

^





^



^



^



Shift operator, E Let

y^ =

f^ ( x

) be a function of

x ,

and let

x^ takes the consecutive values

x ,^

x + h

,^ x^

+ 2

h , etc. We

then define an operator havingthe property

( )

(^

)

E f

x

f^

x^

h

^



Thus, when

E^

operates on

f^ (

x ),

the result is the next value ofthe function. Here,

E^

is called

the shift operator. If we applythe operator

E^

twice on

f^ (

x ),

we get

2

( )

[^

( )]

[^

(^

)]^

(^

2 )

E^

f^ x

E E f

x

E f

x

h

f^ x

h



^

^

^

