Consider a function f(x) tabulated for equally spaced points x0, x1, x2, . . ., xn with step length h. In many problems one may be interested to know the behaviour of f(x) in the neighbourhood of xr (x0 + rh). If we take the transformation X = (x - (x0 + rh)) / h, the data points for X and f(X) can be written as
now the central difference table can be generated using the definition of central differences:
Gauss and Stirling formulae :
Consider the central difference table interms of forward difference operator D and with Sheppard's Zigzag rule
Now by divided difference formula along the solid line interms of forward difference operator
is called the Gauss forward difference formula.
Now if we repeat the same along dotted line weget
is called the Gauss backward difference formula.
Now changing these two formulae to d notation produces respectively
Now by adding these two expression and dividing by two gives
where the averaging operator m is defined as
This formula is called the Stirling's interpolation formula.
Using Stirling's formula compute f(12.2) from the data
= 31788 + 714.9 - 6.14 - 1.648 + 0.208
Þf(x) = 10-5fx = 0.32495
Combining the Gauss forward formula with Gauss Backward formula based on a zigzag line just one unit below the earlier one gives the Bessel formula. This is equivalent to
Then the Bessel formula is
set x = z + 1/2
for z = 0 we have
Now by choosing proper choise of origin x, one can take the central difference formula in the range
Compute 344.51/3 for the equation f(x) = x1/3
Þf(x) = 7.010189