Error analysis for piecewise quadratic curve fitting algorithms (Q1819886)

\textit{D. F. McAllister} and \textit{J. A. Roulier} [ACM Trans. Math. Software 7, 331-347 (1981; Zbl 0464.65003)] introduced a piecewise quadratic curve fitting algorithm (MR-algorithm) which locally preserves geometric properties of the data such as monotonicity and convexity. The piecewise quadratics employed in the MR-algorithm belong to the class \(C^ 1\) and therefore a convergence order of \(O(h^ 3)\) is generally expected for a three times continuously differentiable function f. But, it has been observed by McAllister and Roulier that the convergence order is only \(O(h^ 2)\) near the zeros of f'. In view of this, the authors of the present paper suggest two new algorithms which modify the interpolant near the zeros of f' in the sense that the algorithms generate \(C^ 1\) piecewise quadratics which produce the convergence order of \(O(h^ 3)\), uniformly for all points of the interval as well as they preserve the geometric properties of the function.