Interpolation is the process of estimating the value of function for any intermediate value of the variable with the help of its given set of values. Let us assume that the function y=f(x) is known for certain values of x say a for x 0,x 1,x 2,………x n. As f(x 0 ),f(x 1 ),……..f(x n ). The process of finding the value of f(x) corresponding to x=x i. Where x 0
1.Finite Difference Operators 2.Newton’s Forward Difference Interpolation Formula 3.Newton’s Backward Difference Interpolation Formula 4.Lagrange’s Interpolation Formula
rth forward difference kth backward difference
Shift operator, E
The inverse operator E -1 is defined as Similarly,
Differential Operator, D
Newton’s Forward Difference Interpolation Formula
This is known as Newton’s forward difference formula for interpolation, which gives the value of f (x 0 + ph) in terms of f (x 0 ) and its leading differences.
This formula is also known as Newton-Gregory forward difference interpolation formula. Here p=(x-x 0 )/h. An alternate expression is
The required cubic polynomial.
Let y = f (x) be a function which takes on values f (x n ), f (x n -h), f (x n -2h), …, f (x 0 ) corresponding to equispaced values x n, x n -h, x n -2h, …, x 0. Suppose, we wish to evaluate the function f (x) at (x n + ph),
Binomial expansion yields,
This formula is known as Newton’s backward interpolation formula. This formula is also known as Newton’s-Gregory backward difference interpolation formula.
Example:- For the following table of values, estimate f (7.5).
In this problem,
Newton’s interpolation formula gives Therefore,
Let y = f (x) be a function which takes the values, y 0, y 1,…y n corresponding to x 0, x 1, …x n. Since there are (n + 1) values of y corresponding to (n + 1) values of x, we can represent the function f (x) by a polynomial of degree n. DERIVATION:-
or in the form
Here, the coefficients a k are so chosen as to satisfy this equation by the (n + 1) pairs (x i, y i ). Thus we get Therefore,
The Lagrange’s formula for interpolation
We can easily observe that, and Thus introducing Kronecker delta notation