Numerical differentiaon  Formulas of numerical differentiation
Formulas of numerical differentiation
1. On a basis of Newton’s first interpolating formula
For finding of the first and second derivatives of function, given in equidistant points (i = 0, 1, 2, …, n) of segment [a, b] by values, it is approximately replaced with Newton’s interpolating polynomial constructed for system of nodes [1]:
(5)
Removing brackets and considering, that
we’ll receive:
. (6)
Similarly, considering
we’ll receive:
. (7)
In the same way if necessary, it is possible to calculate any order derivative of function. We’ll notice, that at calculation of derivatives in fixed point х as it is necessary to take the nearest tabular value of argument.
It is possible to deduce also formulas of numerical differentiation based on Newton’s second interpolating formula [1].
2. On a basis of Newton’s second interpolating formula
Let – system of equidistant points with step and corresponding values of given function. Puttingand replacing the function by Stirling’s interpolating polynomial, we’ll receive:
(8)
where for brevity of record the following designations are entered:
and so on.
From (8) in view of that , follows:
(9)
. (10)
