Algorithm 1008: Multicomplex Number Class for Matlab, with a Focus on the Accurate Calculation of Small Imaginary Terms for Multicomplex Step Sensitivity Calculations

摘要

A Matlab class for multicomplex numbers was developed with particular attention paid to the robust and accurate handling of small imaginary components. This is primarily to allow the class to be used to obtain n-order derivative information using the multicomplex step method for, among other applications, gradient-based optimization and optimum control problems. The algebra of multicomplex numbers is described, as is its accurate computational implementation, considering small term approximations and the identification of principal values. The implementation of the method in Matlab is studied, and a class definition is constructed. This new class definition enables Matlab to handle n-order multicomplex numbers and perform arithmetic functions. It was found that with this method, the step size could be arbitrarily decreased toward machine precision. Use of the method to obtain up to the seventh derivative of functions is presented, as is timing data to demonstrate the efficiency of the class implementation.