Newton's divided difference interpolations

NEWTON'S DIVIDED DIFFERENCE INTERPOLATIONS

Let, y = f(x) take values f (x₀), f(x₁) ... f (x_n) corresponding to the arguments x₀, x₁, ... X_n

By definition,

1. Using Newton's divided difference formula, find u(3) given u(1) = -26, ad u(2) = 12, u(4) = 256, u(6) = 844.

[A.U. April/May 2004]

Solution:

We form the divided difference table, since the intervals are unequal

2. Find f (x) as a polynomial in x for the following data by Newton's divided difference formula

3. Finf f(8) by Newton’s divided difference formula for the data :

4. Find the function f(x) from the following table using Newton’s divided difference formula :

Solution :

We from the divided difference table, since the intervals are unequal

5. Given values

Evalute f(9) using Newton’s divided difference formula

Solution :

The divided difference table is as follows :

6. Given the data :

 Find the cubic function of x

Solution :

The divided difference table is as follows :

7. Using Newton’s divided difference formula, find f (x) and f (6) from the following data :

Solution :

The divided difference table is

8. Using Newton’s divided difference formula, find f (5)  from the following data :

Solution :

The divided difference table is

9. Given the following data find y (6), y (5) and the maximum value of y.

Solution : since the arguments are not equally spaced, we will use Newton’s divided difference formula

The divided Difference Table is

y(x) is maximum if y’ (x) = 0 3x² + 4x + 3 = 0. But the roots are imaginary. Therefore, there is no extremum value in the range. In fact it is an increasing curve.

10. Find the fourth degree curve y = f(x) passing through points (2,3), (4,43), (5,138), (7,778) and (8, 1515) using Newton’s divided difference formula.

Solution :