Piecewise polynomial approximation of the sixth order with automatic knots detection

Coefficients of a local segment model for piecewise polynomial approximation of the sixth order are evaluated using values of the function and of its first derivative at three knots of the support. Formulae for coefficients of the function expansion in degrees of $x-x_0$ on a three-point grid are obtained within the framework of the recently proposed basic element method. An algorithm for automatic knot detection is developed. Numerical calculations applying quite complicated tests have shown high efficiency of the model with respect to the calculation stability, accuracy and smoothness of approximation.