C2 monotone spline interpolation based on one-parameter groups of diffeomorphism

This paper discusses the use of one-parameter groups consisting of the diffeomorphisms of the unit interval [0, 1] for constructing the 𝐶2 spline interpolants preserving the monotonicity of the given data sets. The choice of local interpolants in the form of superpositions of the diffeomorphisms belonging to definite one-parameter groups guarantees the monotonicity of the obtained interpolant for strictly monotone data. As in the case of cubic splines, for achieving the extra degree of smoothness (of 𝐶2 class) for the interpolant it must satisfy to conditions of the continuity for its second derivative at the inner knots of the grid used. The explicit form of the corresponding equations for the proposed interpolants is obtained, the method of its numerical solution is proposed. For one sort of these interpolants the equations are nonlinear, but one can easily solve them using the Newton's method; another sort of these interpolants does not require solving of any equations: the values of the first derivative at the inner knots are to be computed using the Harmonic Mean Method (HMM). The efficiency of the proposed approach is verified in a series of computational experiments