SKEDSOFT
nterpolation With Evenly Spaced Points : It is the simplified method of Newton’s Divided Difference Interpolation.
Let the data (xi, f(xi)) be given with uniform spacing, that is, the nodal points are given by xi = x0 ih, i = 0, 1, 2, ..., n. In this case, Lagrange and divided difference interpolation polynomials can also be used for interpolation. However, we can derive simpler interpolation formulas for the uniform mesh case. We define finite difference operators and finite differences to derive these formulas.
We define the following difference operators:
Shift operator E : When the operator E is applied on f(xi), we obtain
.................... 1.1
That is,
Therefore, the operator E when applied on f(x) shifts it to the value at the next nodal point. We have
In general, we have,
.................... 1.2
where k is any real number. For example, we define
Forward difference operator Δ When the operator Δ is applied on f(xi), we obtain
.......................1.3
That is,
These differences are called the first forward differences.
The second forward difference is defined by,
The third forward difference is defined by,
Now, from (1.2) and (1.3), we get
Comparing, we obtain the operator relation
........................1.4
