Information-based complexity
On stable numerical differentiation
Mathematics of Computation
One-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computational Physics
The use of higher order finite difference schemes is not dangerous
Journal of Complexity
A new approach to numerical differentiation and integration
Mathematical and Computer Modelling: An International Journal
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In this paper we discuss the problem of approximation of the first derivative of a function at the endpoint of its definition interval. This problem is motivated by diabetes therapy management, where it is important to provide estimations of the future blood glucose trend from current and past measurements. A natural way to approach the problem is to use one-sided finite difference schemes for numerical differentiation, but, following this way, one should be aware that the values of the function to be differentiated are noisy and available only at given fixed points. Then (as we argue in the paper) the number of used point values is the only parameter to be employed for regularization of the above mentioned ill-posed problem of numerical differentiation. In this paper we present and theoretically justify an adaptive procedure for choosing such a parameter. We also demonstrate some illustrative tests, as well as the results of numerical experiments with simulated clinical data.