Interpolation Methods 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.

For interpolation of atabulated function, theconcept of finite differencesis important. The knowledgeabout various finitedifference operators andtheir symbolic relations arevery much needed toestablish variousinterpolation formulae.

Finite

DifferenceOperators

Finite

DifferenceOperators

Forward Differences

Forward Differences

(^

,^

),^

k^

k

x^

y^

k^

n

( ), y^

f^

x 

For a given table of valueswith equally spaced abscissasof a function we define the forward differenceoperator

as follows 

These differences arecalled

first differences of the function y

and are

denoted by the symbolHere,

is called the first

difference operator

y ^ i

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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By defining a difference tableas a convenient device fordisplaying variousdifferences, the abovedefined differences can bewritten down systematicallyby constructing a differencetable for values

(^

,^

),^

0,1,..., 6

k^

k x^

y^

k^ 

Forward Difference Table