A Fundamental study on second order complex-step Derivative Approximation by Quadruple precision floating point arithmetics

Abstract

Methods for improving the accuracy of second order derivative calculations using the complex first-order derivative approximation (CSDA method) is proposed. In this paper, the following two approaches are investigated: (1) perturbation direction on the complex number plane, (2) use of double and quadruple precision floating point arithmetics. The above approaches were applied to several numerical examples to discuss the accuracy and computational costs. As the results, the perturbation of CSDA on θ = 45° had the best accuracy and stability throughout all the examples. Moreover, it was also found that when combined with perturbation on θ = 45° and quadruple precision floating point arithmetics, the second-order derivatives could be computed with more than accuracy of the double precision arithmetic in the examples.