Numerical integration of a class of ordinary differential equations on the general linear group of matrices
This paper is concerned with the numerical solution of initial value problems for matrix differential equations on \( { GL}(n)\) with the form \[ Y'(t) = Y^{-T} (t) F( Y(t) ), \quad t \geq t_0,\;Y(t_0)= Y_0, \] where \(f\) is assumed to be a continuous and globally Lipschitz matrix function. By using the theory of ordinary differential equations it can be seen that the solution \( Y(t)\) may have a escape for some finite \( \tau \) and then \( Y(t)\) approaches to a singular matrix when \( t \to \tau. \) The authors propose two alternative approaches to deal with this kind of problems when finite escapes are encountered. In the first one they use a low order explicit Runge-Kutta method on some modified scheme to avoid the unstable behaviour in the neighbourhood of the singularity. In the second one, they use the singular value decomposition of \( Y(t) = U(t) \Sigma (t) V(t)\) to integrate the components of \( U, \Sigma \) and \( V\). The paper ends with the results of a numerical experiment to test the behaviour of both approaches for some low order Runge-Kutta methods.
scientific article; zbMATH DE number 2026778