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 :
Removing brackets and considering, that
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 .
2. On a basis of Newton’s second interpolating formula
Let – system of equidistant points with step and corresponding values of given function. Putting and replacing the function by Stirling’s interpolating polynomial, we’ll receive:
where for brevity of record the following designations are entered:
and so on.
From (8) in view of that , follows:
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