  In this paper, we continue to study the derivative-free extended Kalman
filtering (DF-EKF) framework for state estimation of continuous-discrete
nonlinear stochastic systems. Having considered the Euler-Maruyama and
It\^{o}-Taylor discretization schemes for solving stochastic differential
equations, we derive the related filters' moment equations based on the
derivative-free EKF principal. In contrast to the recently derived MATLAB-based
continuous-discrete DF-EKF techniques, the novel DF-EKF methods preserve an
information about the underlying stochastic process and provide the estimation
procedure for a fixed number of iterates at the propagation steps.
Additionally, the DF-EKF approach is particularly effective for working with
stochastic systems with highly nonlinear and/or nondifferentiable drift and
observation functions, but the price to be paid is its degraded numerical
stability (to roundoff) compared to the standard EKF framework. To eliminate
the mentioned pitfall of the derivative-free EKF methodology, we develop the
conventional algorithms together with their stable square-root implementation
methods. In contrast to the published DF-EKF results, the new square-root
techniques are derived within both the Cholesky and singular value
decompositions. A performance of the novel filters is demonstrated on a number
of numerical tests including well- and ill-conditioned scenarios.
