Accurate evaluation of the \(k\)-th derivative of a polynomial and its application (Q1936186)

A compensated algorithm for the evaluation of the \(k\)-th derivative of a polynomial in power basis is presented. The proposed algorithm makes the direct evaluation without obtaining the \(k\)-th derivative expression of the polynomial itself possible, with a very accurate result to all but the most ill-conditioned evaluation. Forward error analysis and running error analysis are performed by an approach based on the data dependency graph. Numerical experiments illustrate the accuracy and efficiency of the presented algorithm.