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:
So, for substitute in (11.4.1) to get This gives us Next,
So, For or equivalently
Thus, Now, using mathematical induction, we get
and in general,
For the sake of numerical calculations, we give below a convenient form of the forward interpolation formula.
With this transformation the above forward interpolation formula is simplified to the following form:
If =1, we have a linear interpolation given by
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:
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
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.
Also, find approximate value of
Therefore, taking and we have This gives,
Compute the value of the function at and
Find the approximate speed of the train at the mid point between the two stations.
Obtain a fifth degree interpolating polynomial for Compute and also find an error estimate for it.
|Time||8 am||12 noon||4 pm||8pm|
Obtain Newton's backward interpolating polynomial of degree to compute the temperature in Kanpur on that day at 5.00 pm.