  Spline interpolation has been used in several applications due to its
favorable properties regarding smoothness and accuracy of the interpolant.
However, when there exists a discontinuity or a steep gradient in the data,
some artifacts can appear due to the Gibbs phenomenon. Also, preservation of
data monotonicity is a requirement in some applications, and that property is
not automatically verified by the interpolator. In this paper, we study
sufficient conditions to obtain monotone cubic splines based on Hermite cubic
interpolators and propose different ways to construct them using non-linear
formulas. The order of approximation, in each case, is calculated and several
numerical experiments are performed to contrast the theoretical results.
