The failure of splines to preserve monotonicity has prompted fundamental research in this area since 1960s. One of the first methods to remedy this problem were splines in tension by Schweikert \cite{Sch}, where a tension parameter controlled the shape of exponential splines \cite{Spath1969}. Later on several monotonicity preserving polynomial spline algorithms were proposed \cite{Schumaker1983,PasRoul1977,AndElf1987,Andersson1991_JAT,McAllister1981_ACM,PasRoul1977}. These algorithms typically rely on introducing additional spline knots between the abscissae of the data. Algorithmic developments are active to this day, see for example \cite{Kvasov2000_book,Abbas2011}.
-When in addition to the pairs $(x_i, y_i)$ the slopes of the function are available, i.e., the data comes in triples $(x_i, y_i, p_i)$, the interpolation problem is called Hermite, and the Hermite splines are used. However, even when the sequence $y_i$ is increasing and the slopes $p_i$ are non-negative, cubic Hermite splines may still fail to be monotone, as illustrated in Figure \ref{ch11:fig2}. Thus monotone Hermite splines are needed \cite{Gregory1982}. \index{Hermite spline}
+When in addition to the pairs $(x_i, y_i)$ the slopes of the function are available, i.e., the data comes in triples $(x_i, y_i, p_i)$, the interpolation problem is called Hermite, and the Hermite splines are used. However, even when the sequence $y_i$ is increasing and the slopes $p_i$ are non-negative, cubic Hermite splines may still fail to be monotone, as illustrated in Figure \ref{ch11:fig2}. Thus monotone Hermite splines are needed \cite{Gregory1982}. \index{Hermite splines}
Another issue with monotone approximation is noisy data. In this case, inaccuracies in the data make the input sequence $y_i$ itself non-monotone, and hence monotone spline interpolation algorithms will fail. Monotone spline smoothing algorithms are available, e.g. \cite{Andersson1991_JAT,Elfving1989_NM}. Such algorithms are based on solving a quadratic (or another convex) programming problem numerically, and have not been yet adapted to parallel processing.
