One-Step Semi-Explicit Methods Based on the Cayley Transform for Solving Isospectral Flows

This note deals with the numerical solution of the matrix differential system Y′ = [B(t, Y), Y], Y(0) = Y0, t ≥ 0, (1) where Y0 is a real constant symmetric matrix, B maps symmetric into skew-symmetric matrices, and [B(t, Y), Y] is the Lie bracket commutator of B(t, Y) and Y, i.e. [B(t, Y), Y] = B(t, Y)Y - YB(t, Y). The unique solution of (1) is isospectral, that is the matrix Y(t) preserves the eigenvalues of Y0 and is symmetric for all t (see [1, 5]). Isospectral methods exploit the Flaschka formulation of (1) in which Y(t) is written as Y(t) = U(t)Y0UT(t), for t ≥ 0, where U(t) is the orthogonal solution of the differential system U′ = B(t, UY0UT)U, U(0) = I, t ≥ 0, (2) (see [5]). Here a numerical procedure based on the Cayley transform is proposed and compared with known isospectral methods.