Question

Newton forward difference formula derivation

Answer 1

Newton's Interpolation Formulae

As stated earlier, interpolation is the process of approximating a given function, whose values are known at  tabular points, by a suitable polynomial,  of degree  which takes the values  at  for  Note that if the given data has errors, it will also be reflected in the polynomial so obtained.

In the following, we shall use forward and backward differences to obtain polynomial function approximating  when the tabular points  's are equally spaced. Let



where the polynomial  is given in the following form: 

   (11.4.1)


for some constants  to be determined using the fact that  for 

So, for  substitute  in (11.4.1) to get This gives us  Next,



So,  For   or equivalently



Thus,  Now, using mathematical induction, we get



Thus, 

    


As this uses the forward differences, it is called NEWTON'S FORWARD DIFFERENCE FORMULA for interpolation, or simply, forward interpolation formula.

EXERCISE 11.4.1   Show that

   and

and in general,



For the sake of numerical calculations, we give below a convenient form of the forward interpolation formula.

Let  then



With this transformation the above forward interpolation formula is simplified to the following form: 

        (11.4.2)


If  =1, we have a linear interpolation given by

(11.4.3)


For  we get a quadratic interpolating polynomial:

(11.4.4)


and so on.

It may be pointed out here that if  is a polynomial function of degree  then coincides with  on the given interval. Otherwise, this gives only an approximation to the true values of 

If we are given additional point  also, then the error, denoted by  is estimated by



Similarly, if we assume,  is of the form



then using the fact that  we have


       


Thus, using backward differences and the transformation  we obtain the Newton's backward interpolation formula as follows:

(11.4.5)


EXERCISE 11.4.2   Derive the Newton's backward interpolation formula (11.4.5) for 

Remark 11.4.3   If the interpolating point lies closer to the beginning of the interval then one uses the Newton's forward formula and if it lies towards the end of the interval then Newton's backward formula is used.

Remark 11.4.4   For a given set of n tabular points, in general, all the n points need not be used for interpolating polynomial. In fact N is so chosen that  forward/backward difference almost remains constant. Thus N is less than or equal to n.

EXAMPLE 11.4.5  Obtain the Newton's forward interpolating polynomial,  for the following tabular data and interpolate the value of the function at 

x00.0010.0020.0030.0040.005y1.1211.1231.12551.1271.1281.1285
Solution: For this data, we have the Forward difference difference table

 01.1210.0020.0005-0.00150.002-.0025 .0011.1230.0025-0.00100.0005-0.0005  .0021.12550.0015-0.00050.0   .0031.1270.001-0.0005    .0041.1280.0005     .0051.1285      

Thus, for  where  and  we get

    


Thus, 

        


Using the following table for approximate its value at  Also, find an error estimate (Note  ).

0.70720.740.760.78 0.842290.877070.913090.950450.98926 

Solution: As the point  lies towards the initial tabular values, we shall use Newton's Forward formula. The forward difference table is:

  0.700.842290.034780.001240.00010.00001  0.720.877070.036020.001340.00011   0.740.913090.037360.00145    0.760.950450.03881     0.780.98926      

In the above table, we note that  is almost constant, so we shall attempt  degree polynomial interpolation.

Note that  gives  Thus, using forward interpolating polynomial of degree  we get




      


An error estimate for the approximate value is



Note that exact value of  (upto  decimal place) is  and the approximate value, obtained using the Newton's interpolating polynomial is very close to this value. This is also reflected by the error estimate given above.Apply  degree interpolation polynomial for the set of values given in Example 11.2.15, to estimate the value of  by taking



Also, find approximate value of  
Solution: Note that  is closer to the values lying in the beginning of tabular values, while  is towards the end of tabular values. Therefore, we shall use forward difference formula for  and the backward difference formula for  Recall that the interpolating polynomial of degree  is given by



Therefore,for  and  we have  This gives, 

   


for  and  we have  This gives, 

   


Note: as  is closer to  we may expect estimate calculated using  to be a better approximation.for  we use the backward interpolating polynomial, which gives,



Therefore, taking  and  we have  This gives,