Automatic knot finding for piecewise-cubic approximation

The paper proposes within the frame of four point transforms a method for piecewise-cubic approximation that detects the knots of the segments in auto-tracking mode. A 3-point cubic parametric spline (TPS) is used as a model of a local approximant. The free parameter $\theta $ (a coefficient at $x^{3}$) is searching using step-by-step averaging. An analytical expression for $\theta $ is received via a length of the segment and values of a function and derivatives that shows the dependence of the $C^{1}$-smoothness on the accuracy of the $\theta$-estimate. The stability of the method w.r.t. input errors is shown as well. The key parameters are: the parameters of the basis functions, the variance of the input errors, and a sampling step. The efficiency of the method is shown by numerical calculations on test examples.