Selection of parameters of generalized cubic splines with convexity preserving interpolation

It is shown that computation of generalized interpolating cubic splines is reduced to solving a tridiagonal system of linear equations with column diagonal dominance with respect to knot values of the second derivative of a spline. The non-negativity conditions of the solution for such systems are found. The general scheme for choosing tension parameters of the generalized splines for convexity-preserving interpolation is offered. The resulting spline minimally differs from the classical cubic one and coincides with it if sufficient convexity conditions for the last one are satisfied. The algorithms specified are considered for different generalized cubic splines such as rational, exponential, variable power, hyperbolic splines and splines with additional knots.