Considerations on computing real logarithms of matrices, hamiltonian logarithms, and skew-symmetric logarithms

Abstract

The issue of computing a real logarithm of a real matrix is addressed. After a brief review of some known methods, more attention is paid to three: (1) Padé approximation techniques, (2) Newton's method, and (3) a series expansion method. Newton's method has not been previously treated in the literature; we address commutativity issues, and simplify the algorithmic formulation. We also address general structure-preserving issues for two applications in which we are interested: finding the real Hamiltonian logarithm of a symplectic matrix, and finding the skew-symmetric logarithm of an orthogonal matrix. The diagonal Padé approximants and the proposed series expansion technique are proven to be structure-preserving. Some algorithmic issues are discussed.