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Contents >> Applied Mathematics >> Numerical Methods >> Numerical Differentiaon >> Formulas of numerical differentiaon

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]:


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:


where for brevity of record the following designations are entered:

and so on.

From (8) in view of that  , follows:


. (10)

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