Range and Domain Partitioning in Piecewise Polynomial Approximation

Abstract

Abstract: Error control in piecewise polynomial interpolation of a smooth univariate function f requires that the interval of approximation be subdivided into many subintervals, on each of which an interpolating polynomial is determined. The number of such subintervals is often over- estimated through the use of a high-order derivative of f . We report on a partitioning algorithm, in which we attempt to reduce the number of subintervals required, by imposing conditions on f and its relevant higher derivative. One of these conditions facilitates a distinction between the need for absolute or relative error control. Two examples demonstrate the effectiveness of this partitioning algorithm.
Key Words: Piecewise Polynomial; Range Partitioning; Domain Partitioning; Error Control