Accuracy analysis and applications of the Sterling interpolation method for nonlinear function error propagation

Abstract

The differential is replaced by the Sterling interpolation method with a finite difference, without calculating the first and second derivatives of the nonlinear function. The mean and variance or covariance matrix of the nonlinear function can then be obtained with the similar precision as that of the second-order Taylor expansion by the Sterling interpolation method. Existing studies neither give accurate proof on the means and covariance matrices of the multiple nonlinear functions solved by the Sterling interpolation method that can achieve second-order precision nor give a theoretical proof of the choice of the step factor h. In this paper, the mean of any nonlinear function calculated by the Sterling interpolation method that can be achieved second-order precision is proposed by formula deduction. The optimal value of step factor h is 3, which is researched from the perspective of formula deduction when using the Sterling interpolation method to calculate the variance or covariance matrix of nonlinear functions. As the mean of dependent variable of nonlinear function can be calculated by the Sterling interpolation through error propagation, the Sterling interpolation method is used to calculate the bias of random quantity. The Sterling interpolation method is applied in a forward intersection, a positive computation of Gaussian projection coordinates and the bias of displacements of the rectangular dislocation model. The case studies in this paper indicate the correctness of the theoretical proof and selection of the step factor, and the applicability and advantages of the Sterling interpolation method for error propagation and calculation of bias are verified.