Stability of multiquadric quasi-interpolation to approximate high order derivatives

Abstract

Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation, especially for solving the differential equations numerically. The classical method is the divided difference method. However, it has been shown strongly unstable in practice. Actually, it can only be used to simulate the lower order derivatives in applications. To simulate the high order derivatives, this paper suggests a new method using multiquadric quasi-interpolation. The stability of the multiquadric quasi-interpolation method is compared with the classical divided difference method. Moreover, some numerical examples are presented to confirm the theoretical results. Both theoretical results and numerical examples show that the multiquadric quasi-interpolation method is much stabler than the divided difference method. This property shows that multiquadric quasi-interpolation method is an efficient tool to construct an approximation of high order derivatives based on scattered sampling data even with noise.