Numerical differentiation

In numerical mathematics is denoted by numerical differentiation of the approximate calculation of the derivative of the given function values, mostly by means of a difference quotient. This is necessary if the derivative function is not given or the function itself only indirectly, for example through readings, is available. In contrast, the automatic differentiation of the code defining the function considered, by an expanded inference engine.

Is the distance (h ) of the function values ​​is small, as in any more detail, the approximation calculation would initially better. However, occurs in the calculation by means of floating-point extinction, so must not be below the selected hour of the machine accuracy -related barriers.

Alternatively, you can also use approximations of differentiable functions such as cubic splines. If you are not interested in the entire function curve, but only in certain places, so there are special formulas.

In practical application, the function values ​​are often error-prone. Therefore Sobel operators are used, for example, for edge detection, which simultaneously perform a smoothing. Another possibility is the use of smoothed splines (also Ausgleichssplines ).

Difference quotient

One obvious approach is the use of forward difference quotient:

However, the approximation when compared to the extinction is relatively poor. A better approximation is obtained by using the central difference quotient:

By means of local polynomial interpolation can be further improve this approximation. For the notation see Landau symbols.