MatrixODE tool for solution of matrix-valued ODE with application to online time-varying matrix inversion

In modern engineering researches, dynamics of systems are often studied by well-design computer simulations, and these dynamics are usually depicted in the form of vector-or matrix-valued ordinary differential equation (ODE) systems. A common approach for solution of matrix-valued ODE in MATLAB is to use the Kronecker product and vectorization technique. However, the vectorization procedure may be troublesome, and it can be difficult if not impossible to vectorize a complicated matrix-valued ODE (especially when it is nonlinear). Differing from the traditional approach, this paper proposes an elegant, compact and effective tool (named MatrixODE) for solving a matrix-valued ODE with arbitrary right hand side. As a complement and extension of the built-in solvers in MATLAB, the proposed MatrixODE tool allows solving a matrix-valued ODE using matrix language directly, i.e., users do not need to perform matrix-vector transformations. Application of the proposed MatrixODE tool to online time-varying matrix inversion further substantiates its effectiveness for matrix-valued ODE solving.