Unit-4

Numerical analysis

4.1 Introduction and concept: Finite differences – forward difference operator

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:  Then  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.

Central differences-

The central difference operator is is defined by the relations as-

The difference table is-

Example: Evaluate .

Sol.

Or

Example: Evaluate-

Sol.

We have-

4.2 Newton’s forward difference formula with simple examples, backward difference operator & Newton’s backward interpolation formula with simple examples.

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 forward  interpolation  method.

Let , then

The table of forward finite difference is given below:

By Newton 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 

References-