On the computation of derivatives within LD factorization of parametrized matrices

The paper presents a new method for calculating the values of derivatives in the LD factorization of parametrized matrices, based on the direct procedure for the modified weighted Gram–Schmidt orthogonalization.
The need for calculating the values of derivatives in matrix orthogonal transformations arises in the theory of perturbations and control, in differential geometry, in solving problems such as the Lyapunov exponential calculation, the problems of automatic differentiation, the calculation of the numerical solution of the matrix differential Riccati equation, the calculation of high-order derivatives in the optimal input design. In the theory of parameter identification of mathematical models of discrete linear stochastic systems, such problems are solved by developing numerically effective algorithms for finding the solution of the matrix difference Riccati sensitivity equation.
In this paper, we have posed and solved a new problem of calculating the values of derivatives. Lemma 1 represents the main theoretical result. The practical result is the computational algorithm 2. The software implementation of the algorithm allows us to calculate the values of derivatives of the parametrized matrices that are the result of a direct procedure of the LD factorization quickly and with high accuracy. It is not necessary to calculate the values of derivatives of the matrix of weighted orthogonal transformation. The algorithm has a simple structure and does not contain complex operations of symbolic or numerical differentiation. Only one inversion of the triangular matrix and simple matrix operations of addition and multiplication are required.
Two numerical examples are considered that show the operability and numerical efficiency of the proposed algorithm 2.
The results obtained in this paper will be used to construct new classes of adaptive LD filters in the area of parameter identification of mathematical models of discrete linear stochastic systems.