Unit-2

Interpolation

2.1 Introduction Finite Differences

Interpolation

Definition: Interpolation is a technique of estimating the value of a  function for any  intermediate value of the  independent  variable while  the process  of  computing the  value of the function outside the given range is called   extrapolation.

Let be a function of x.

The table given below gives corresponding values of y for different values of x.

 The process of finding the values of y corresponding to any value of x which lies between   is called interpolation.

If the given function is a polynomial, it is polynomial interpolation and given function is known as interpolating polynomial.

Conditions for Interpolation

1)     The function must be   a polynomial of independent variable.

2)     The function   should be either increasing or decreasing function.

3)     The value of   the function should be increase or decrease uniformly.

Finite Difference

Let be a function of x. The table given below gives corresponding values of y for different values of x.

a) Forward Difference:  The  are called differences of y, denoted by 

The symbol  is called the forward difference operator.  Consider the forward difference table below:

Where

 And third forward difference so on.

b) Backward Difference:

The difference are called first backward difference and  is denoted by  Consider the backward  difference  table  below:

Where

And third backward differences so on.

Key takeaways-

a)     Forward Difference:  The  are called differences of y, denoted by 

The symbol  is called the forward  difference operator

3.     Backward Difference:

The difference are called first  backward difference and  is denoted by 

2.2 Difference tables Polynomial Interpolation: Newton’s forward and backward formula

Newton Forward Difference formula:

This method is useful for interpolation near the beginning of a set of tabular  values.

Where

Example1: Using Newton’s forward difference formula, find the sum

Putting

It   follows that

Since is a  fourth  degree polynomial in  n.

Further,

By Newton Forward Difference Method

Example2: Given  find   , by using Newton’s forward interpolation method.

Let , then

The table of forward finite difference is given below:

By Newton’s forward difference method

Here initial value = 45, difference of interval h = 5 and  the value to be calculated at x=52.

By Formula

Example3: Find the missing term in the following:

Let

First, we construct the forward difference table:

Now,

Newton Backward Difference Method:

This method is useful for interpolation near the ending of a set of tabular values.

Where 

Example1:  Find from the following table:

 Consider the backward difference method

  Here

By Newton backward difference formula

Example2: The following table give the amount of a chemical dissolved in water:

  Compute the amount dissolve at

Consider the following backward difference table:

Here

By Newton  Backward  difference formula

Example3: The following are the marks obtained by 492 candidates in a certain examination

Find out the number of candidates:

a)     Who secured more than 48 but not more than 50 marks?

b)    Who secured less than 48 but not less than 45 marks?

Consider the forward difference table given below:

Here 

 By Newton Forward Difference formula

f

a)     No. of candidate secured more than 48 but not more than 50 marks

b)    No. of candidate secured less than 48 but not less than 45 marks

Key takeaways-

 Where

2.     Newton Backward Difference Method:

Where 

2.3 Central Difference Formulae: Gauss forward and backward formula

Gauss forward difference formula-

The Gauss forward difference formula is defined as-

Gauss backward difference formula-

The Gauss backward difference formula is defined as-

Example: By using Gauss forward difference formula obtain f (32) given that-

f (25) = 0.2707   f (35) = 0.3386

f (30) = 0.3027   f (40) = 0.3794

Sol.

Here

Let us take the origin at 30, a = 30 then we get-

u = 0.4

The forward difference table-

By using Gauss forward difference formula, we get-

Example: Find by using Gauss forward formula if-

Sol.

Let a = 8, h = 4,

We have

a + hu = 9

and we get-

u = 0.25

The forward difference table-

By using Gauss forward difference formula, we get-

Example: Using Gauss’s backward interpolation formula, find the population for the year 1936 given that-

Sol.

Let origin is at 1941 and h = 10

Then-

Which gives-

The backward difference table-

By using Gauss backward formula, we get-

Hence the pop. of the year 1936 is 32625

Key takeaways-

2.     Gauss backward difference formula-

2.4 Stirling’s, Bessel’s, Everett’s formula

Stirling’s formula-

The Stirling’s formula is the average of Gauss forward and backward interpolation formulae.

It is defined as-

Bessel’s formula-

Example: By using Stirling’s formula to compute from the table ( given below-

Sol.

Taking the origin at

We get the following central table-

At x = 12.2, p = 0.2

Stirling’s formula-

When p = 0.2, we get-

Example: By using Bessel’s formula to find the value of f (27.5) from the table given below-

Sol.

Taking the origin at

We have p = x – 27

The central table will be as follows-

At x = 27.5, p =0.5

Bessel’s formula is-

When p = 0.5, we get-

So that-

f (27.5) = 3.585

Example: Using Everett’s formula, evaluate f (30) if

f (20) = 2854,                              f (28) = 3162

f (36) = 7088,                                f (44) = 7984.

Sol.

Let the origin is 28, a = 28, h = 8

Then

And

The difference table is-

By using Everette’s formula-

Therefore f (30) = 4064

Key takeaways-

2.     Bessel’s formula-

2.5 Interpolation with unequal intervals: Lagrange’s Interpolation, Newton Divided difference formula, Hermite’s Interpolation

Divided Difference:

In the case of interpolation, when the value of the argument is not equi-spaced (unequal intervals) we use the class of differences called divided differences.

Definition:   The   difference which are defined by taking into consideration   the change in the value of the argument is known as divided differences.

Let be a function defined as

Where are unequal i.e., it is case of unequal interval.

The first order divided differences are:

And so on.

The second order divided difference is:

And so on.

Similarly, the nth order divided difference is:

With the help of these we construct the divided difference table:

Newton’s Divided difference Formula:

Let be a function defined as

Where are unequal i.e., it is case of unequal interval.

.

Example1: By means of Newton’s divided difference formula, find the values of from the following table:

We construct the divided difference table is given by:

By Newton’s Divided difference formula

.

Now, putting in above we get

.

Example2: The following values of the function f(x) for values of x are given:

Find the value of and also the value of x for which f(x) is maximum or minimum.

We construct the divide difference table:

By Newton’s divided difference formula

.

Putting  in above we get

For maximum and minimum of , we have

Or 

Example3: Find a polynomial satisfied by

, by the use of Newton’s interpolation formula with divided difference.

Here

We will construct the divided difference table:

By Newton’s divided difference formula

.

This is the required polynomial.

Lagrange’s   Interpolation of polynomial:

Let , be defined function we get

Where the interval is not necessarily equal. We assume f(x) is a polynomial od degree n.  Then Lagrange’s interpolation formula is given by

Example1: Deduce Lagrange’s formula for interpolation.  The observed values of a function are respectively 168,120,72 and 63 at the four position3,7,9 and 10 of the independent variables. What is the best estimate you can for the value of the function at the position6 of the independent variable?

We construct the table for the given data:

We need to calculate for x = 6, we need f(6)=?

Here

We get

  By Lagrange’s interpolation formula, we have

Hence the estimated value for x=6 is 147.

Example2: By means of Lagrange’s formula, prove that

We construct the table:

Here x = 3, f(x)=?

  By Lagrange’s formula for   interpolation

Hence proved.

Example3: Find the polynomial of fifth degree from the following data

Here

We get 

  By Lagrange’s interpolation formula

Hermite’s interpolation formula-

The Hermite’s interpolation formula is defined as-

Example: Find the cubic polynomial by using Hermite’s interpolation formula, given-

Sol.

We know that Hermite’s interpolation formula-

..........(1)

Now

And

Hence

From equation (1)-

Key takeaways-

.

3.      Hermite’s interpolation formula

References